On 10/30/2025 5:37 AM, Mikko wrote:
On 2025-10-29 16:22:11 +0000, olcott said:
On 10/29/2025 5:43 AM, Mikko wrote:
On 2025-10-28 15:07:18 +0000, olcott said:
On 10/28/2025 4:54 AM, Mikko wrote:
On 2025-10-27 13:56:21 +0000, olcott said:
On 10/27/2025 4:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:But we can show a direct one-to-one mapping of finite strings in a >>>>>>>>> finite alphabet to natural numbers (it just becomes the notational base
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said:
On 10/22/2025 4:23 AM, Mikko wrote:
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>A subset of formalized English does do this.
On 10/21/2025 4:31 AM, Mikko wrote:There is no language that begins with a consistent set of basic facts
On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only semantic >>>>>>>>>>>>>>>>>>>>> logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.On 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>
On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense.
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ >>>>>>>>>>>>>>>>>>>>>> at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>
Only if you exclude from "correct reasioning" some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins >>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>
Nobody has ever presented such subset.
ChatGPT rewords my words into that subset.
Your words have much less material than all human knowledge without
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/Wittgenstein.pdf >>>>>>>>>>>>> and he was so succinct eliminate the convoluted mess of Gödel >>>>>>>>>>>>> numbers that people mistook his succinct analysis as simplistic >>>>>>>>>>>>> not understanding what Gödel was really saying.
Yes, an expectable consequence of ignoring essential aspects is >>>>>>>>>>>> not understanding.
Gödel numbers are merely the ridiculously convoluted
way that Gödel added the functionality of this simple
operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the >>>>>>>>>> basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about >>>>>>>>>> finite strings. It is much simpler to express claims and proofs as >>>>>>>>>> fimite strings as thats how they are usually expressed anyway. But >>>>>>>>>> that requires that one first constructs a theory of finite strings. >>>>>>>>>
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But >>>>>>>> the proof of Gödel's theorem is simpler if the number of symbols in >>>>>>>> the alphabet is prime even if that would mean that some of the symbols >>>>>>>> are unused.
Gödel numbers hide this simple essence behind a convoluted mess: >>>>>>> G := (F ⊬ G)
No, that is not the sentence. For Gödel's purposes it was important to >>>>>> cover languages that have no symbol for ':=' or '⊬'.
But that is irrelevant to my preference for the numbering method.
His result is supposed to apply to any formal language
as expressive as PA or above.
Yes, includint PA itself. Therefore his proof could not assume any
symbols other than those of PA. A proof that requires other symbols
proves nothing about PA.
If he started with Minimal Type Theory and the understanding
that cycles in the evaluation sequence of expressions indicate
that they are semantically unsound we would have never heard
of Incompleteness.
Sooner or later someone would have found incompleteness anyway.
Incompleteness is a misconception nothing more.
His incompleteness applies to PA or above, thus
applies to MTT.
On 10/30/2025 5:44 AM, Mikko wrote:
On 2025-10-29 16:26:58 +0000, olcott said:
On 10/29/2025 5:53 AM, Mikko wrote:
On 2025-10-29 01:57:07 +0000, Richard Damon said:
On 10/28/25 11:12 AM, olcott wrote:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:But we can show a direct one-to-one mapping of finite strings in a >>>>>>>>>> finite alphabet to natural numbers (it just becomes the notational base
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said:
On 10/22/2025 4:23 AM, Mikko wrote:Nobody has ever presented such subset.
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>A subset of formalized English does do this. >>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote:There is no language that begins with a consistent set of basic facts
On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only semantic >>>>>>>>>>>>>>>>>>>>>> logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.On 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>
On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense.
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>
Only if you exclude from "correct reasioning" some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins >>>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset.
Your words have much less material than all human knowledge without
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/Wittgenstein.pdf >>>>>>>>>>>>>> and he was so succinct eliminate the convoluted mess of Gödel >>>>>>>>>>>>>> numbers that people mistook his succinct analysis as simplistic >>>>>>>>>>>>>> not understanding what Gödel was really saying.
Yes, an expectable consequence of ignoring essential aspects is >>>>>>>>>>>>> not understanding.
Gödel numbers are merely the ridiculously convoluted
way that Gödel added the functionality of this simple >>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about >>>>>>>>>>> finite strings. It is much simpler to express claims and proofs as >>>>>>>>>>> fimite strings as thats how they are usually expressed anyway. But >>>>>>>>>>> that requires that one first constructs a theory of finite strings. >>>>>>>>>>
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in >>>>>>>>> the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be >>>>>>> hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own >>>>>> unprovability. 15 ...
(Gödel 1931:40-41)
Which is a statement in the meta-logic, not in the logic.
G asserts it isn't provable, but asserting that there can be no number >>>>> which can encode a proof of itself.
This statement MUST be either true of false.
It's actually both.
It is a lie that it is even a statement.
I could equally say this statement is true or false: "What time is it?"
You cant say that in the language of Peano arithmetic.
PA is ridiculously inexpressive so that the key
essence is hidden bu far too many purely extraneous
details.
Since Incompleteness applies to PA OR ABOVE.
I go with above.
On 10/30/2025 5:50 AM, Mikko wrote:
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:And consequently Gödel's example of an undecidable formula is more complex
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:But we can show a direct one-to-one mapping of finite strings in a >>>>>>>>> finite alphabet to natural numbers (it just becomes the notational base
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said:
On 10/22/2025 4:23 AM, Mikko wrote:
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>A subset of formalized English does do this.
On 10/21/2025 4:31 AM, Mikko wrote:There is no language that begins with a consistent set of basic facts
On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only semantic >>>>>>>>>>>>>>>>>>>>> logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.On 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>
On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense.
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ >>>>>>>>>>>>>>>>>>>>>> at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>
Only if you exclude from "correct reasioning" some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins >>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>
Nobody has ever presented such subset.
ChatGPT rewords my words into that subset.
Your words have much less material than all human knowledge without
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/Wittgenstein.pdf >>>>>>>>>>>>> and he was so succinct eliminate the convoluted mess of Gödel >>>>>>>>>>>>> numbers that people mistook his succinct analysis as simplistic >>>>>>>>>>>>> not understanding what Gödel was really saying.
Yes, an expectable consequence of ignoring essential aspects is >>>>>>>>>>>> not understanding.
Gödel numbers are merely the ridiculously convoluted
way that Gödel added the functionality of this simple
operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the >>>>>>>>>> basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about >>>>>>>>>> finite strings. It is much simpler to express claims and proofs as >>>>>>>>>> fimite strings as thats how they are usually expressed anyway. But >>>>>>>>>> that requires that one first constructs a theory of finite strings. >>>>>>>>>
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But >>>>>>>> the proof of Gödel's theorem is simpler if the number of symbols in >>>>>>>> the alphabet is prime even if that would mean that some of the symbols >>>>>>>> are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position. >>>>>>
than needed. Which doesn't really matter as the complex details can be >>>>>> hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own >>>>> unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is incorrect. >>> Gödel's G when boiled down to its barest essence is simply semantically >>> unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a simpler proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that.
On 2025-10-30 12:54:57 +0000, olcott said:
On 10/30/2025 5:37 AM, Mikko wrote:
On 2025-10-29 16:22:11 +0000, olcott said:
On 10/29/2025 5:43 AM, Mikko wrote:
On 2025-10-28 15:07:18 +0000, olcott said:
On 10/28/2025 4:54 AM, Mikko wrote:
On 2025-10-27 13:56:21 +0000, olcott said:His result is supposed to apply to any formal language
On 10/27/2025 4:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of >>>>>>>>>>> mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said:
On 10/22/2025 4:23 AM, Mikko wrote:Nobody has ever presented such subset.
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>A subset of formalized English does do this. >>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote:There is no language that begins with a consistent >>>>>>>>>>>>>>>>>>> set of basic facts
On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only >>>>>>>>>>>>>>>>>>>>>> semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>
On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense.
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and >>>>>>>>>>>>>>>>>>>>>>>> savior Jesus Christ.
Right, and the empirical falct that nobody has >>>>>>>>>>>>>>>>>>>>>>> verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus >>>>>>>>>>>>>>>>>>>>>>> Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.
Only if you exclude from "correct reasioning" some >>>>>>>>>>>>>>>>>>>>> reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins >>>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those >>>>>>>>>>>>>>>>>>> facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset.
Your words have much less material than all human >>>>>>>>>>>>>>> knowledge without
any gaps. Therefore it is impossible to verify that you >>>>>>>>>>>>>>> have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/
Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel >>>>>>>>>>>>>> numbers that people mistook his succinct analysis as >>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying.
Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>> aspects is
not understanding.
Gödel numbers are merely the ridiculously convoluted
way that Gödel added the functionality of this simple >>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>
Therefore restricting the assumptions to natural numbers >>>>>>>>>>> provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem >>>>>>>>>>> about
finite strings. It is much simpler to express claims and >>>>>>>>>>> proofs as
fimite strings as thats how they are usually expressed
anyway. But
that requires that one first constructs a theory of finite >>>>>>>>>>> strings.
But we can show a direct one-to-one mapping of finite strings >>>>>>>>>> in a finite alphabet to natural numbers (it just becomes the >>>>>>>>>> notational base for the numbers).
That's true, and I would prefer that rather than Gödel's
mapping. But
the proof of Gödel's theorem is simpler if the number of
symbols in
the alphabet is prime even if that would mean that some of the >>>>>>>>> symbols
are unused.
Gödel numbers hide this simple essence behind a convoluted mess: >>>>>>>> G := (F ⊬ G)
No, that is not the sentence. For Gödel's purposes it was
important to
cover languages that have no symbol for ':=' or '⊬'.
But that is irrelevant to my preference for the numbering method. >>>>>>
as expressive as PA or above.
Yes, includint PA itself. Therefore his proof could not assume any
symbols other than those of PA. A proof that requires other symbols
proves nothing about PA.
If he started with Minimal Type Theory and the understanding
that cycles in the evaluation sequence of expressions indicate
that they are semantically unsound we would have never heard
of Incompleteness.
Sooner or later someone would have found incompleteness anyway.
Incompleteness is a misconception nothing more.
No, it is a part of analytic knowledge. You have not shown any error
in Gödel's and other proofs that every theory of natural numbers is incomplete. All you have shown is that you don't like the fact and
some rationale for your dislike.
His incompleteness applies to PA or above, thus
applies to MTT.
You have never shown the full theory of MTT so hard to tell. But if
MTT is consistent it should be provably incomplete.
On 2025-10-30 13:01:34 +0000, olcott said:
On 10/30/2025 5:44 AM, Mikko wrote:
On 2025-10-29 16:26:58 +0000, olcott said:
On 10/29/2025 5:53 AM, Mikko wrote:You cant say that in the language of Peano arithmetic.
On 2025-10-29 01:57:07 +0000, Richard Damon said:
On 10/28/25 11:12 AM, olcott wrote:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of >>>>>>>>>>>> mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote:Nobody has ever presented such subset.
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent >>>>>>>>>>>>>>>>>>>> set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only >>>>>>>>>>>>>>>>>>>>>>> semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and >>>>>>>>>>>>>>>>>>>>>>>>> savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has >>>>>>>>>>>>>>>>>>>>>>>> verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour >>>>>>>>>>>>>>>>>>>>>>>> Jesus Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.
Only if you exclude from "correct reasioning" some >>>>>>>>>>>>>>>>>>>>>> reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that >>>>>>>>>>>>>>>>>>>>> begins
with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to >>>>>>>>>>>>>>>>>>>> those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset.
Your words have much less material than all human >>>>>>>>>>>>>>>> knowledge without
any gaps. Therefore it is impossible to verify that you >>>>>>>>>>>>>>>> have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of >>>>>>>>>>>>>>> Gödel
numbers that people mistook his succinct analysis as >>>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying.
Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>>> aspects is
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers >>>>>>>>>>>> provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar >>>>>>>>>>>> theorem about
finite strings. It is much simpler to express claims and >>>>>>>>>>>> proofs as
fimite strings as thats how they are usually expressed >>>>>>>>>>>> anyway. But
that requires that one first constructs a theory of finite >>>>>>>>>>>> strings.
But we can show a direct one-to-one mapping of finite strings >>>>>>>>>>> in a finite alphabet to natural numbers (it just becomes the >>>>>>>>>>> notational base for the numbers).
That's true, and I would prefer that rather than Gödel's >>>>>>>>>> mapping. But
the proof of Gödel's theorem is simpler if the number of >>>>>>>>>> symbols in
the alphabet is prime even if that would mean that some of the >>>>>>>>>> symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, >>>>>>>>> it uses prime numbers to number the symbols, and prime powers >>>>>>>>> for the position.
And consequently Gödel's example of an undecidable formula is >>>>>>>> more complex
than needed. Which doesn't really matter as the complex details >>>>>>>> can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts >>>>>>> its own unprovability. 15 ...
(Gödel 1931:40-41)
Which is a statement in the meta-logic, not in the logic.
G asserts it isn't provable, but asserting that there can be no
number which can encode a proof of itself.
This statement MUST be either true of false.
It's actually both.
It is a lie that it is even a statement.
I could equally say this statement is true or false: "What time is it?" >>>
PA is ridiculously inexpressive so that the key
essence is hidden bu far too many purely extraneous
details.
PA can express everything we need to know about natural numbers.
Complexities can be hidden behind definitions.
Since Incompleteness applies to PA OR ABOVE.
I go with above.
You may need to rewrite the proof for the language of the theory you
want to use, especially if you don't want to use the ordinary logic.
On 2025-10-30 13:03:23 +0000, olcott said:
On 10/30/2025 5:50 AM, Mikko wrote:
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of >>>>>>>>>>> mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said:
On 10/22/2025 4:23 AM, Mikko wrote:Nobody has ever presented such subset.
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>A subset of formalized English does do this. >>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote:There is no language that begins with a consistent >>>>>>>>>>>>>>>>>>> set of basic facts
On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only >>>>>>>>>>>>>>>>>>>>>> semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>
On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense.
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and >>>>>>>>>>>>>>>>>>>>>>>> savior Jesus Christ.
Right, and the empirical falct that nobody has >>>>>>>>>>>>>>>>>>>>>>> verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus >>>>>>>>>>>>>>>>>>>>>>> Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.
Only if you exclude from "correct reasioning" some >>>>>>>>>>>>>>>>>>>>> reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins >>>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those >>>>>>>>>>>>>>>>>>> facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset.
Your words have much less material than all human >>>>>>>>>>>>>>> knowledge without
any gaps. Therefore it is impossible to verify that you >>>>>>>>>>>>>>> have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/
Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel >>>>>>>>>>>>>> numbers that people mistook his succinct analysis as >>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying.
Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>> aspects is
not understanding.
Gödel numbers are merely the ridiculously convoluted
way that Gödel added the functionality of this simple >>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>
Therefore restricting the assumptions to natural numbers >>>>>>>>>>> provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem >>>>>>>>>>> about
finite strings. It is much simpler to express claims and >>>>>>>>>>> proofs as
fimite strings as thats how they are usually expressed
anyway. But
that requires that one first constructs a theory of finite >>>>>>>>>>> strings.
But we can show a direct one-to-one mapping of finite strings >>>>>>>>>> in a finite alphabet to natural numbers (it just becomes the >>>>>>>>>> notational base for the numbers).
That's true, and I would prefer that rather than Gödel's
mapping. But
the proof of Gödel's theorem is simpler if the number of
symbols in
the alphabet is prime even if that would mean that some of the >>>>>>>>> symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, >>>>>>>> it uses prime numbers to number the symbols, and prime powers >>>>>>>> for the position.
And consequently Gödel's example of an undecidable formula is
more complex
than needed. Which doesn't really matter as the complex details >>>>>>> can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts
its own unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And >>>>>> Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is
incorrect.
Gödel's G when boiled down to its barest essence is simply semantically >>>> unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a simpler
proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that.
I did not say nor mean "you did not disagree with anything". The usual meaning of what I said is that you don't disagree with what I said in
my previous message about the complexity of Gödel's sentence.
On 10/31/2025 6:02 AM, Mikko wrote:
On 2025-10-30 12:54:57 +0000, olcott said:
On 10/30/2025 5:37 AM, Mikko wrote:
On 2025-10-29 16:22:11 +0000, olcott said:
On 10/29/2025 5:43 AM, Mikko wrote:
On 2025-10-28 15:07:18 +0000, olcott said:
On 10/28/2025 4:54 AM, Mikko wrote:
On 2025-10-27 13:56:21 +0000, olcott said:His result is supposed to apply to any formal language
On 10/27/2025 4:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote:Nobody has ever presented such subset.
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only semantic >>>>>>>>>>>>>>>>>>>>>>> logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.On 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>
Only if you exclude from "correct reasioning" some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins >>>>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset.
Your words have much less material than all human knowledge without
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel >>>>>>>>>>>>>>> numbers that people mistook his succinct analysis as simplistic >>>>>>>>>>>>>>> not understanding what Gödel was really saying.
Yes, an expectable consequence of ignoring essential aspects is >>>>>>>>>>>>>> not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as >>>>>>>>>>>> fimite strings as thats how they are usually expressed anyway. But >>>>>>>>>>>> that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a >>>>>>>>>>> finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in >>>>>>>>>> the alphabet is prime even if that would mean that some of the symbols
are unused.
Gödel numbers hide this simple essence behind a convoluted mess: >>>>>>>>> G := (F ⊬ G)
No, that is not the sentence. For Gödel's purposes it was important to
cover languages that have no symbol for ':=' or '⊬'.
But that is irrelevant to my preference for the numbering method. >>>>>>>
as expressive as PA or above.
Yes, includint PA itself. Therefore his proof could not assume any >>>>>> symbols other than those of PA. A proof that requires other symbols >>>>>> proves nothing about PA.
If he started with Minimal Type Theory and the understanding
that cycles in the evaluation sequence of expressions indicate
that they are semantically unsound we would have never heard
of Incompleteness.
Sooner or later someone would have found incompleteness anyway.
Incompleteness is a misconception nothing more.
No, it is a part of analytic knowledge. You have not shown any error
in Gödel's and other proofs that every theory of natural numbers is
incomplete. All you have shown is that you don't like the fact and
some rationale for your dislike.
His incompleteness applies to PA or above, thus
applies to MTT.
You have never shown the full theory of MTT so hard to tell. But if
MTT is consistent it should be provably incomplete.
I showed the entire language specification you just
didn't pay attention.
On 10/31/2025 6:10 AM, Mikko wrote:
On 2025-10-30 13:01:34 +0000, olcott said:
On 10/30/2025 5:44 AM, Mikko wrote:
On 2025-10-29 16:26:58 +0000, olcott said:
On 10/29/2025 5:53 AM, Mikko wrote:You cant say that in the language of Peano arithmetic.
On 2025-10-29 01:57:07 +0000, Richard Damon said:
On 10/28/25 11:12 AM, olcott wrote:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential aspects is >>>>>>>>>>>>>>> not understanding.
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote:Nobody has ever presented such subset.
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.
Only if you exclude from "correct reasioning" some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins >>>>>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset.
Your words have much less material than all human knowledge without
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel >>>>>>>>>>>>>>>> numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...
(Gödel 1931:40-41)
Which is a statement in the meta-logic, not in the logic.
G asserts it isn't provable, but asserting that there can be no number >>>>>>> which can encode a proof of itself.
This statement MUST be either true of false.
It's actually both.
It is a lie that it is even a statement.
I could equally say this statement is true or false: "What time is it?" >>>>
PA is ridiculously inexpressive so that the key
essence is hidden bu far too many purely extraneous
details.
PA can express everything we need to know about natural numbers.
Complexities can be hidden behind definitions.
MTT can express anything that we way to know about
formal expressions of language.
Since Incompleteness applies to PA OR ABOVE.
I go with above.
You may need to rewrite the proof for the language of the theory you
want to use, especially if you don't want to use the ordinary logic.
All that I did was get rid of the purely extraneous
complexity of Gödel's G
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:
On 10/30/2025 5:50 AM, Mikko wrote:
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote:Nobody has ever presented such subset.
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only semantic >>>>>>>>>>>>>>>>>>>>>>> logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.On 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>
Only if you exclude from "correct reasioning" some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins >>>>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset.
Your words have much less material than all human knowledge without
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel >>>>>>>>>>>>>>> numbers that people mistook his succinct analysis as simplistic >>>>>>>>>>>>>>> not understanding what Gödel was really saying.
Yes, an expectable consequence of ignoring essential aspects is >>>>>>>>>>>>>> not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as >>>>>>>>>>>> fimite strings as thats how they are usually expressed anyway. But >>>>>>>>>>>> that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a >>>>>>>>>>> finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in >>>>>>>>>> the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be >>>>>>>> hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And >>>>>>> Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is incorrect. >>>>> Gödel's G when boiled down to its barest essence is simply semantically >>>>> unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a simpler proof >>>> does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that.
I did not say nor mean "you did not disagree with anything". The usual
meaning of what I said is that you don't disagree with what I said in
my previous message about the complexity of Gödel's sentence.
The complexity of Gödel's sentence is nuts.
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
On 2025-10-31 12:09:48 +0000, olcott said:
On 10/31/2025 6:02 AM, Mikko wrote:
On 2025-10-30 12:54:57 +0000, olcott said:
On 10/30/2025 5:37 AM, Mikko wrote:
On 2025-10-29 16:22:11 +0000, olcott said:
On 10/29/2025 5:43 AM, Mikko wrote:
On 2025-10-28 15:07:18 +0000, olcott said:
On 10/28/2025 4:54 AM, Mikko wrote:
On 2025-10-27 13:56:21 +0000, olcott said:His result is supposed to apply to any formal language
On 10/27/2025 4:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of >>>>>>>>>>>>> mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>>>> aspects is
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote:Nobody has ever presented such subset.
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent >>>>>>>>>>>>>>>>>>>>> set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only >>>>>>>>>>>>>>>>>>>>>>>> semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and >>>>>>>>>>>>>>>>>>>>>>>>>> savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has >>>>>>>>>>>>>>>>>>>>>>>>> verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour >>>>>>>>>>>>>>>>>>>>>>>>> Jesus Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the >>>>>>>>>>>>>>>>>>>>>>>> Principle of
Explosion ceases to exist.
Only if you exclude from "correct reasioning" >>>>>>>>>>>>>>>>>>>>>>> some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that >>>>>>>>>>>>>>>>>>>>>> begins
with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to >>>>>>>>>>>>>>>>>>>>> those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset.
Your words have much less material than all human >>>>>>>>>>>>>>>>> knowledge without
any gaps. Therefore it is impossible to verify that you >>>>>>>>>>>>>>>>> have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of >>>>>>>>>>>>>>>> Gödel
numbers that people mistook his succinct analysis as >>>>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers >>>>>>>>>>>>> provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar >>>>>>>>>>>>> theorem about
finite strings. It is much simpler to express claims and >>>>>>>>>>>>> proofs as
fimite strings as thats how they are usually expressed >>>>>>>>>>>>> anyway. But
that requires that one first constructs a theory of finite >>>>>>>>>>>>> strings.
But we can show a direct one-to-one mapping of finite >>>>>>>>>>>> strings in a finite alphabet to natural numbers (it just >>>>>>>>>>>> becomes the notational base for the numbers).
That's true, and I would prefer that rather than Gödel's >>>>>>>>>>> mapping. But
the proof of Gödel's theorem is simpler if the number of >>>>>>>>>>> symbols in
the alphabet is prime even if that would mean that some of >>>>>>>>>>> the symbols
are unused.
Gödel numbers hide this simple essence behind a convoluted mess: >>>>>>>>>> G := (F ⊬ G)
No, that is not the sentence. For Gödel's purposes it was
important to
cover languages that have no symbol for ':=' or '⊬'.
But that is irrelevant to my preference for the numbering method. >>>>>>>>
as expressive as PA or above.
Yes, includint PA itself. Therefore his proof could not assume any >>>>>>> symbols other than those of PA. A proof that requires other symbols >>>>>>> proves nothing about PA.
If he started with Minimal Type Theory and the understanding
that cycles in the evaluation sequence of expressions indicate
that they are semantically unsound we would have never heard
of Incompleteness.
Sooner or later someone would have found incompleteness anyway.
Incompleteness is a misconception nothing more.
No, it is a part of analytic knowledge. You have not shown any error
in Gödel's and other proofs that every theory of natural numbers is
incomplete. All you have shown is that you don't like the fact and
some rationale for your dislike.
His incompleteness applies to PA or above, thus
applies to MTT.
You have never shown the full theory of MTT so hard to tell. But if
MTT is consistent it should be provably incomplete.
I showed the entire language specification you just
didn't pay attention.
Irrelevant to my comment that some parts of the theory are not yet shown.
On 2025-10-31 12:18:16 +0000, olcott said:
On 10/31/2025 6:10 AM, Mikko wrote:
On 2025-10-30 13:01:34 +0000, olcott said:
On 10/30/2025 5:44 AM, Mikko wrote:
On 2025-10-29 16:26:58 +0000, olcott said:
On 10/29/2025 5:53 AM, Mikko wrote:
On 2025-10-29 01:57:07 +0000, Richard Damon said:
On 10/28/25 11:12 AM, olcott wrote:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of >>>>>>>>>>>>>> mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>>>>> aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote:Your words have much less material than all human >>>>>>>>>>>>>>>>>> knowledge without
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset.
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent >>>>>>>>>>>>>>>>>>>>>> set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only >>>>>>>>>>>>>>>>>>>>>>>>> semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green >>>>>>>>>>>>>>>>>>>>>>>>>>> cheeseOn 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
this proves that Donald Trump is the Lord and >>>>>>>>>>>>>>>>>>>>>>>>>>> savior Jesus Christ.
Right, and the empirical falct that nobody has >>>>>>>>>>>>>>>>>>>>>>>>>> verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour >>>>>>>>>>>>>>>>>>>>>>>>>> Jesus Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the >>>>>>>>>>>>>>>>>>>>>>>>> Principle of
Explosion ceases to exist.
Only if you exclude from "correct reasioning" >>>>>>>>>>>>>>>>>>>>>>>> some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of >>>>>>>>>>>>>>>>>>>>>>> human
knowledge that can be expressed in language that >>>>>>>>>>>>>>>>>>>>>>> begins
with a consistent set of basic facts and only >>>>>>>>>>>>>>>>>>>>>>> applies
semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to >>>>>>>>>>>>>>>>>>>>>> those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that >>>>>>>>>>>>>>>>>> you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of >>>>>>>>>>>>>>>>> Gödel
numbers that people mistook his succinct analysis as >>>>>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers >>>>>>>>>>>>>> provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar >>>>>>>>>>>>>> theorem about
finite strings. It is much simpler to express claims and >>>>>>>>>>>>>> proofs as
fimite strings as thats how they are usually expressed >>>>>>>>>>>>>> anyway. But
that requires that one first constructs a theory of finite >>>>>>>>>>>>>> strings.
But we can show a direct one-to-one mapping of finite >>>>>>>>>>>>> strings in a finite alphabet to natural numbers (it just >>>>>>>>>>>>> becomes the notational base for the numbers).
That's true, and I would prefer that rather than Gödel's >>>>>>>>>>>> mapping. But
the proof of Gödel's theorem is simpler if the number of >>>>>>>>>>>> symbols in
the alphabet is prime even if that would mean that some of >>>>>>>>>>>> the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be >>>>>>>>>>> prime, it uses prime numbers to number the symbols, and prime >>>>>>>>>>> powers for the position.
And consequently Gödel's example of an undecidable formula is >>>>>>>>>> more complex
than needed. Which doesn't really matter as the complex
details can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts >>>>>>>>> its own unprovability. 15 ...
(Gödel 1931:40-41)
Which is a statement in the meta-logic, not in the logic.
G asserts it isn't provable, but asserting that there can be no >>>>>>>> number which can encode a proof of itself.
This statement MUST be either true of false.
It's actually both.
It is a lie that it is even a statement.
I could equally say this statement is true or false: "What time is >>>>>> it?"
You cant say that in the language of Peano arithmetic.
PA is ridiculously inexpressive so that the key
essence is hidden bu far too many purely extraneous
details.
PA can express everything we need to know about natural numbers.
Complexities can be hidden behind definitions.
MTT can express anything that we way to know about
formal expressions of language.
Since Incompleteness applies to PA OR ABOVE.
I go with above.
You may need to rewrite the proof for the language of the theory you
want to use, especially if you don't want to use the ordinary logic.
All that I did was get rid of the purely extraneous
complexity of Gödel's G
All that complexity is in Gödel's proof. You need to rewrite the proof
and verify its equivalence to the original if you want to ger rid of
the complexity.
But the complexity affects only the verification of the correctness of
the proof. The meaning of the conclusion is simple.
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:
On 10/30/2025 5:50 AM, Mikko wrote:
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of >>>>>>>>>>>>> mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential >>>>>>>>>>>>>>> aspects is
On 2025-10-23 15:39:13 +0000, olcott said:
On 10/23/2025 5:02 AM, Mikko wrote:
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote:Nobody has ever presented such subset.
On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>
On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent >>>>>>>>>>>>>>>>>>>>> set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only >>>>>>>>>>>>>>>>>>>>>>>> semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and >>>>>>>>>>>>>>>>>>>>>>>>>> savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has >>>>>>>>>>>>>>>>>>>>>>>>> verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour >>>>>>>>>>>>>>>>>>>>>>>>> Jesus Christ
at the same time supports the idea that the >>>>>>>>>>>>>>>>>>>>>>>>> conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the >>>>>>>>>>>>>>>>>>>>>>>> Principle of
Explosion ceases to exist.
Only if you exclude from "correct reasioning" >>>>>>>>>>>>>>>>>>>>>>> some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that >>>>>>>>>>>>>>>>>>>>>> begins
with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to >>>>>>>>>>>>>>>>>>>>> those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset.
Your words have much less material than all human >>>>>>>>>>>>>>>>> knowledge without
any gaps. Therefore it is impossible to verify that you >>>>>>>>>>>>>>>>> have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ >>>>>>>>>>>>>>>> Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of >>>>>>>>>>>>>>>> Gödel
numbers that people mistook his succinct analysis as >>>>>>>>>>>>>>>> simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers >>>>>>>>>>>>> provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar >>>>>>>>>>>>> theorem about
finite strings. It is much simpler to express claims and >>>>>>>>>>>>> proofs as
fimite strings as thats how they are usually expressed >>>>>>>>>>>>> anyway. But
that requires that one first constructs a theory of finite >>>>>>>>>>>>> strings.
But we can show a direct one-to-one mapping of finite >>>>>>>>>>>> strings in a finite alphabet to natural numbers (it just >>>>>>>>>>>> becomes the notational base for the numbers).
That's true, and I would prefer that rather than Gödel's >>>>>>>>>>> mapping. But
the proof of Gödel's theorem is simpler if the number of >>>>>>>>>>> symbols in
the alphabet is prime even if that would mean that some of >>>>>>>>>>> the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be
prime, it uses prime numbers to number the symbols, and prime >>>>>>>>>> powers for the position.
And consequently Gödel's example of an undecidable formula is >>>>>>>>> more complex
than needed. Which doesn't really matter as the complex details >>>>>>>>> can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts >>>>>>>> its own unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica >>>>>>>> And Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is
incorrect.
Gödel's G when boiled down to its barest essence is simply
semantically
unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a simpler >>>>> proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that.
I did not say nor mean "you did not disagree with anything". The usual
meaning of what I said is that you don't disagree with what I said in
my previous message about the complexity of Gödel's sentence.
The complexity of Gödel's sentence is nuts.
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about the complexity of Gödel's sentence.
On 11/1/2025 4:10 AM, Mikko wrote:
On 2025-10-31 12:09:48 +0000, olcott said:
On 10/31/2025 6:02 AM, Mikko wrote:
On 2025-10-30 12:54:57 +0000, olcott said:
On 10/30/2025 5:37 AM, Mikko wrote:
On 2025-10-29 16:22:11 +0000, olcott said:
On 10/29/2025 5:43 AM, Mikko wrote:
On 2025-10-28 15:07:18 +0000, olcott said:
On 10/28/2025 4:54 AM, Mikko wrote:
On 2025-10-27 13:56:21 +0000, olcott said:His result is supposed to apply to any formal language
On 10/27/2025 4:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote:Your words have much less material than all human knowledge without
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset.
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.
Only if you exclude from "correct reasioning" some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins >>>>>>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Gödel numbers hide this simple essence behind a convoluted mess: >>>>>>>>>>> G := (F ⊬ G)
No, that is not the sentence. For Gödel's purposes it was important to
cover languages that have no symbol for ':=' or '⊬'.
But that is irrelevant to my preference for the numbering method. >>>>>>>>>
as expressive as PA or above.
Yes, includint PA itself. Therefore his proof could not assume any >>>>>>>> symbols other than those of PA. A proof that requires other symbols >>>>>>>> proves nothing about PA.
If he started with Minimal Type Theory and the understanding
that cycles in the evaluation sequence of expressions indicate
that they are semantically unsound we would have never heard
of Incompleteness.
Sooner or later someone would have found incompleteness anyway.
Incompleteness is a misconception nothing more.
No, it is a part of analytic knowledge. You have not shown any error
in Gödel's and other proofs that every theory of natural numbers is
incomplete. All you have shown is that you don't like the fact and
some rationale for your dislike.
His incompleteness applies to PA or above, thus
applies to MTT.
You have never shown the full theory of MTT so hard to tell. But if
MTT is consistent it should be provably incomplete.
I showed the entire language specification you just
didn't pay attention.
Irrelevant to my comment that some parts of the theory are not yet shown.
Its exactly the same thing a FOL with additional operators
to make it much more expressive. For example it can directly
expression any HOL logic expression.
I give an example.
On 11/1/2025 4:16 AM, Mikko wrote:
On 2025-10-31 12:18:16 +0000, olcott said:
On 10/31/2025 6:10 AM, Mikko wrote:
On 2025-10-30 13:01:34 +0000, olcott said:
On 10/30/2025 5:44 AM, Mikko wrote:
On 2025-10-29 16:26:58 +0000, olcott said:
On 10/29/2025 5:53 AM, Mikko wrote:You cant say that in the language of Peano arithmetic.
On 2025-10-29 01:57:07 +0000, Richard Damon said:
On 10/28/25 11:12 AM, olcott wrote:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote:Your words have much less material than all human knowledge without
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset. >>>>>>>>>>>>>>>>>>>>
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.
Only if you exclude from "correct reasioning" some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins
with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...
(Gödel 1931:40-41)
Which is a statement in the meta-logic, not in the logic.
G asserts it isn't provable, but asserting that there can be no number
which can encode a proof of itself.
This statement MUST be either true of false.
It's actually both.
It is a lie that it is even a statement.
I could equally say this statement is true or false: "What time is it?" >>>>>>
PA is ridiculously inexpressive so that the key
essence is hidden bu far too many purely extraneous
details.
PA can express everything we need to know about natural numbers.
Complexities can be hidden behind definitions.
MTT can express anything that we way to know about
formal expressions of language.
Since Incompleteness applies to PA OR ABOVE.
I go with above.
You may need to rewrite the proof for the language of the theory you
want to use, especially if you don't want to use the ordinary logic.
All that I did was get rid of the purely extraneous
complexity of Gödel's G
All that complexity is in Gödel's proof. You need to rewrite the proof
and verify its equivalence to the original if you want to ger rid of
the complexity.
But the complexity affects only the verification of the correctness of
the proof. The meaning of the conclusion is simple.
*Gödel already said that it is equivalent*
On 11/1/2025 4:18 AM, Mikko wrote:
On 2025-10-31 12:20:13 +0000, olcott said:
On 10/31/2025 6:15 AM, Mikko wrote:
On 2025-10-30 13:03:23 +0000, olcott said:
On 10/30/2025 5:50 AM, Mikko wrote:
On 2025-10-29 16:25:00 +0000, olcott said:
On 10/29/2025 5:45 AM, Mikko wrote:
On 2025-10-28 15:12:29 +0000, olcott said:
On 10/28/2025 4:58 AM, Mikko wrote:
On 2025-10-28 01:22:21 +0000, Richard Damon said:
On 10/27/25 5:45 AM, Mikko wrote:
On 2025-10-26 17:57:16 +0000, Richard Damon said:
On 10/26/25 7:44 AM, Mikko wrote:
On 2025-10-25 19:57:19 +0000, olcott said:
On 10/25/2025 6:59 AM, Mikko wrote:Natural numbers are one of the best understood topics of mathematics.
On 2025-10-24 17:37:43 +0000, olcott said:
On 10/24/2025 2:46 AM, Mikko wrote:Yes, an expectable consequence of ignoring essential aspects is
On 2025-10-23 15:39:13 +0000, olcott said: >>>>>>>>>>>>>>>>>>
On 10/23/2025 5:02 AM, Mikko wrote:Your words have much less material than all human knowledge without
On 2025-10-22 12:39:31 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>
On 10/22/2025 4:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 2025-10-21 15:11:19 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>Nobody has ever presented such subset.
A subset of formalized English does do this. >>>>>>>>>>>>>>>>>>>>On 10/21/2025 4:31 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-20 16:24:46 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>There is no language that begins with a consistent set of basic facts
On 10/20/2025 3:55 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-19 15:03:34 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>
When we require that correct reasoning is only semanticOn 10/19/2025 3:44 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-10-18 10:58:15 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>>>>>
The conventional way to do proofs concludes that >>>>>>>>>>>>>>>>>>>>>>>>>>> within its rules if the Moon is made from green >>>>>>>>>>>>>>>>>>>>>>>>>>> cheese and the Moon is not made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>>> this proves that Donald Trump is the Lord and savior Jesus Christ.On 10/18/2025 4:30 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>A way to do is not a sense. >>>>>>>>>>>>>>>>>>>>>>>>>>>
That is not a sense of "proof". >>>>>>>>>>>>>>>>>>>>>>>>>>>>>That is the correct way to do a proof. >>>>>>>>>>>>>>>>>>>>>>>>>>>>
Right, and the empirical falct that nobody has verified all of
1. the Moon is made from green cheese >>>>>>>>>>>>>>>>>>>>>>>>>> 2. the Moon is not made from grenn cheese >>>>>>>>>>>>>>>>>>>>>>>>>> 3. Donald Trump is not the Lord and saviour Jesus Christ
at the same time supports the idea that the conventiona wya to
do proofs is correct and trustworthy. >>>>>>>>>>>>>>>>>>>>>>>>>
logical entailment from basic facts the Principle of >>>>>>>>>>>>>>>>>>>>>>>>> Explosion ceases to exist.
Only if you exclude from "correct reasioning" some reasning that
actually is correct.
There is no counter-example that can possibly exist >>>>>>>>>>>>>>>>>>>>>>> that can show there are any gaps in the body of human >>>>>>>>>>>>>>>>>>>>>>> knowledge that can be expressed in language that begins >>>>>>>>>>>>>>>>>>>>>>> with a consistent set of basic facts and only applies >>>>>>>>>>>>>>>>>>>>>>> semantic logical entailment to these facts. >>>>>>>>>>>>>>>>>>>>>>
and only applies semantic logical entailment to those facts and can
express all of human knowledge without any gaps. >>>>>>>>>>>>>>>>>>>>>
ChatGPT rewords my words into that subset. >>>>>>>>>>>>>>>>>>
any gaps. Therefore it is impossible to verify that you have told
every necessary word to it.
Categorically exhaustive reasoning reverse-engineers >>>>>>>>>>>>>>>>> the required architecture of the system. Wittgenstein >>>>>>>>>>>>>>>>> did this on Gödel https://www.liarparadox.org/ Wittgenstein.pdf
and he was so succinct eliminate the convoluted mess of Gödel
numbers that people mistook his succinct analysis as simplistic
not understanding what Gödel was really saying. >>>>>>>>>>>>>>>>
not understanding.
Gödel numbers are merely the ridiculously convoluted >>>>>>>>>>>>>>> way that Gödel added the functionality of this simple >>>>>>>>>>>>>>> operator: ⊢ to a formal language woefully too inexpressive. >>>>>>>>>>>>>>
Therefore restricting the assumptions to natural numbers provdes the
basis for the most interesting result.
Of course one can ask whether one can prove a similar theorem about
finite strings. It is much simpler to express claims and proofs as
fimite strings as thats how they are usually expressed anyway. But
that requires that one first constructs a theory of finite strings.
But we can show a direct one-to-one mapping of finite strings in a
finite alphabet to natural numbers (it just becomes the notational base
for the numbers).
That's true, and I would prefer that rather than Gödel's mapping. But
the proof of Gödel's theorem is simpler if the number of symbols in
the alphabet is prime even if that would mean that some of the symbols
are unused.
Godel's Theorem doesn't need the number of symbols to be prime, it uses
prime numbers to number the symbols, and prime powers for the position.
And consequently Gödel's example of an undecidable formula is more complex
than needed. Which doesn't really matter as the complex details can be
hidden behind definitions.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...
(Gödel 1931:40-41)
G := (F ⊬ G) // Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2 // YACC BNF of MTT >>>>>>>>>
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And >>>>>>>>> Related Systems
Nice to see that you don't disagree.
In other words you agree that I proved 1931 Incompleteness is incorrect.
Gödel's G when boiled down to its barest essence is simply semantically
unsound the same way the Liar Paradox is unsound.
As yousually your "in other words" is a lie. Existence of a simpler proof
does not make the original one incorrect.
My words are true within your words:
"Nice to see that you don't disagree."
If you didn't mean that then you should not have said that.
I did not say nor mean "you did not disagree with anything". The usual >>>> meaning of what I said is that you don't disagree with what I said in
my previous message about the complexity of Gödel's sentence.
The complexity of Gödel's sentence is nuts.
G := (F ⊬ G) Just detect the cycle in this
and Gödel is refuted.
Nice to see that you still don't disagree with my comment about the
complexity of Gödel's sentence.
Yet I absolutely do disagree.
*Gödel already said that it is equivalent*
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