Of course,
dequantification of fantastically quantified statements doesn't make a statement about nonconstructible objects because there aren't any
outside of the fantastical quantification.
On 7/5/2026 9:52 AM, Tristan Wibberley wrote:
On 04/07/2026 16:31, Tristan Wibberley wrote:
On 06/05/2026 20:37, Julio Di Egidio wrote:
On 02/05/2026 20:47, Scott Hoge wrote:
In Cantor's theorem, we do not actually construct a diagonal.
Rather, we presuppose that we can enumerate a set, and then,
/purely on the grounds of possibility/, conceive a diagonalized
non-element.
Nope, as explained and re-explained ad nauseam around here:
just the resident trolls won't get it.
Cantor's diagonal argument, the one with the binary sequences,
is indeed constructive: a definition of anti-diagonal of *any*
(infinite) list is provided, and the proof that the anti-diagonal
cannot be in the list is quite constructive.
"quite" but not "completely".
A constructive operation is defined, but a diagonal number is
constructed just when that constructive operation is applied to a
constructible list.
I should note for the less knowledgable readers of course it's less
often than that, it is only that often for systems such as the one Julio
and Phoenix are using which allows dequantification of universally
quantified statements into the system proper which then have derivable
statements containing actual constructions of the constructible objects
they apply to by virtue of their original quantification. Of course,
dequantification of fantastically quantified statements doesn't make a
statement about nonconstructible objects because there aren't any
outside of the fantastical quantification.
By which I don't mean to argue the countability of the set of reals as
defined in what we call Cantor's Proof of the Uncountability of the
Reals to include objects quantified over by fantatstical quantification
but not by universal quantification, but it does make some meaning
clearer.
While some of the sets might have objects in the system proper, some of
the members of some of the sets clearly do not.
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Olcott's Minimal Type Theory
G ↔ ¬Prov_PA(⌜G⌝)
Directed Graph of evaluation sequence
00 ↔ 01 02
01 G
02 ¬ 03
03 Prov_PA 04
04 Gödel_Number_of 01 // cycle indicates no well-founded justification
tree exists.
The absence of
(a) finite sequence of inference steps to an atomic base,
(b) canonical proof
(c) well-founded justification tree
makes the above to PTS invalid.
On 7/5/2026 2:56 PM, Ross Finlayson wrote:
On 07/05/2026 09:33 AM, olcott wrote:
On 7/5/2026 9:52 AM, Tristan Wibberley wrote:
On 04/07/2026 16:31, Tristan Wibberley wrote:
On 06/05/2026 20:37, Julio Di Egidio wrote:
On 02/05/2026 20:47, Scott Hoge wrote:
In Cantor's theorem, we do not actually construct a diagonal.
Rather, we presuppose that we can enumerate a set, and then,
/purely on the grounds of possibility/, conceive a diagonalized
non-element.
Nope, as explained and re-explained ad nauseam around here:
just the resident trolls won't get it.
Cantor's diagonal argument, the one with the binary sequences,
is indeed constructive: a definition of anti-diagonal of *any*
(infinite) list is provided, and the proof that the anti-diagonal
cannot be in the list is quite constructive.
"quite" but not "completely".
A constructive operation is defined, but a diagonal number is
constructed just when that constructive operation is applied to a
constructible list.
I should note for the less knowledgable readers of course it's less
often than that, it is only that often for systems such as the one
Julio
and Phoenix are using which allows dequantification of universally
quantified statements into the system proper which then have derivable >>>> statements containing actual constructions of the constructible objects >>>> they apply to by virtue of their original quantification. Of course,
dequantification of fantastically quantified statements doesn't make a >>>> statement about nonconstructible objects because there aren't any
outside of the fantastical quantification.
By which I don't mean to argue the countability of the set of reals as >>>> defined in what we call Cantor's Proof of the Uncountability of the
Reals to include objects quantified over by fantatstical quantification >>>> but not by universal quantification, but it does make some meaning
clearer.
While some of the sets might have objects in the system proper, some of >>>> the members of some of the sets clearly do not.
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Olcott's Minimal Type Theory
G ↔ ¬Prov_PA(⌜G⌝)
Directed Graph of evaluation sequence
00 ↔ 01 02
01 G
02 ¬ 03
03 Prov_PA 04
04 Gödel_Number_of 01 // cycle indicates no well-founded justification >>> tree exists.
The absence of
(a) finite sequence of inference steps to an atomic base,
(b) canonical proof
(c) well-founded justification tree
makes the above to PTS invalid.
Yeah, come up with something new, or stuff a sock in it.
The above proves that the notion of undecidable
is incorrect if you understood rather than ignored
what it says.
It also is the final resolution to the Liar Paradox
and you would know this if you understood it.
On 7/5/2026 4:30 PM, Ross Finlayson wrote:
On 07/05/2026 01:25 PM, olcott wrote:
On 7/5/2026 2:56 PM, Ross Finlayson wrote:
On 07/05/2026 09:33 AM, olcott wrote:
On 7/5/2026 9:52 AM, Tristan Wibberley wrote:
On 04/07/2026 16:31, Tristan Wibberley wrote:
On 06/05/2026 20:37, Julio Di Egidio wrote:
On 02/05/2026 20:47, Scott Hoge wrote:
In Cantor's theorem, we do not actually construct a diagonal. >>>>>>>>> Rather, we presuppose that we can enumerate a set, and then, >>>>>>>>> /purely on the grounds of possibility/, conceive a diagonalized >>>>>>>>> non-element.
Nope, as explained and re-explained ad nauseam around here:
just the resident trolls won't get it.
Cantor's diagonal argument, the one with the binary sequences, >>>>>>>> is indeed constructive: a definition of anti-diagonal of *any* >>>>>>>> (infinite) list is provided, and the proof that the anti-diagonal >>>>>>>> cannot be in the list is quite constructive.
"quite" but not "completely".
A constructive operation is defined, but a diagonal number is
constructed just when that constructive operation is applied to a >>>>>>> constructible list.
I should note for the less knowledgable readers of course it's less >>>>>> often than that, it is only that often for systems such as the one >>>>>> Julio
and Phoenix are using which allows dequantification of universally >>>>>> quantified statements into the system proper which then have
derivable
statements containing actual constructions of the constructible
objects
they apply to by virtue of their original quantification. Of course, >>>>>> dequantification of fantastically quantified statements doesn't
make a
statement about nonconstructible objects because there aren't any
outside of the fantastical quantification.
By which I don't mean to argue the countability of the set of
reals as
defined in what we call Cantor's Proof of the Uncountability of the >>>>>> Reals to include objects quantified over by fantatstical
quantification
but not by universal quantification, but it does make some meaning >>>>>> clearer.
While some of the sets might have objects in the system proper,
some of
the members of some of the sets clearly do not.
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Olcott's Minimal Type Theory
G ↔ ¬Prov_PA(⌜G⌝)
Directed Graph of evaluation sequence
00 ↔ 01 02
01 G
02 ¬ 03
03 Prov_PA 04
04 Gödel_Number_of 01 // cycle indicates no well-founded
justification
tree exists.
The absence of
(a) finite sequence of inference steps to an atomic base,
(b) canonical proof
(c) well-founded justification tree
makes the above to PTS invalid.
Yeah, come up with something new, or stuff a sock in it.
The above proves that the notion of undecidable
is incorrect if you understood rather than ignored
what it says.
It also is the final resolution to the Liar Paradox
and you would know this if you understood it.
Like I said,
"understanding" is for suckers,
"comprehension" is for knowledge.
Gemini agrees with me and I only gave it the Prolog. https://share.gemini.google/1dJnMwOZ2k5F
Your axiomatization otherwise is false.
It's like they say,
"It just don't mean a thing."
WM <- retro-finitist crankety-troll
JG <- retro-finitist crankety-troll
PO <- retro-finitist crankety-troll
"Polluter(s) of sci.math"
On 7/6/2026 4:17 AM, Mikko wrote:
On 04/07/2026 20:07, olcott wrote:
Q that cannot resolve (∀x, S(x) ≠ x) is complete
according to its definition.
By the defintion of "incomplete" Q is incomplete. The theory
Q + (∀x, S(x) ≠ x) is more complete but still incomplete.
It fully meets its design spec thus calling it
any kind of incomplete is a damned lie.
On 7/5/2026 5:15 PM, Ross Finlayson wrote:
On 07/05/2026 02:45 PM, olcott wrote:
On 7/5/2026 4:30 PM, Ross Finlayson wrote:
On 07/05/2026 01:25 PM, olcott wrote:
On 7/5/2026 2:56 PM, Ross Finlayson wrote:
On 07/05/2026 09:33 AM, olcott wrote:
On 7/5/2026 9:52 AM, Tristan Wibberley wrote:
On 04/07/2026 16:31, Tristan Wibberley wrote:
On 06/05/2026 20:37, Julio Di Egidio wrote:
On 02/05/2026 20:47, Scott Hoge wrote:
In Cantor's theorem, we do not actually construct a diagonal. >>>>>>>>>>> Rather, we presuppose that we can enumerate a set, and then, >>>>>>>>>>> /purely on the grounds of possibility/, conceive a diagonalized >>>>>>>>>>> non-element.
Nope, as explained and re-explained ad nauseam around here: >>>>>>>>>> just the resident trolls won't get it.
Cantor's diagonal argument, the one with the binary sequences, >>>>>>>>>> is indeed constructive: a definition of anti-diagonal of *any* >>>>>>>>>> (infinite) list is provided, and the proof that the anti-diagonal >>>>>>>>>> cannot be in the list is quite constructive.
"quite" but not "completely".
A constructive operation is defined, but a diagonal number is >>>>>>>>> constructed just when that constructive operation is applied to a >>>>>>>>> constructible list.
I should note for the less knowledgable readers of course it's less >>>>>>>> often than that, it is only that often for systems such as the one >>>>>>>> Julio
and Phoenix are using which allows dequantification of universally >>>>>>>> quantified statements into the system proper which then have
derivable
statements containing actual constructions of the constructible >>>>>>>> objects
they apply to by virtue of their original quantification. Of
course,
dequantification of fantastically quantified statements doesn't >>>>>>>> make a
statement about nonconstructible objects because there aren't any >>>>>>>> outside of the fantastical quantification.
By which I don't mean to argue the countability of the set of
reals as
defined in what we call Cantor's Proof of the Uncountability of the >>>>>>>> Reals to include objects quantified over by fantatstical
quantification
but not by universal quantification, but it does make some meaning >>>>>>>> clearer.
While some of the sets might have objects in the system proper, >>>>>>>> some of
the members of some of the sets clearly do not.
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Gemini agrees with not-you.
OK then the point that I was trying to make is
exactly what Gemini said right here:
https://share.gemini.google/1dJnMwOZ2k5F
On 05/07/2026 00:01, olcott wrote:
On 7/4/2026 12:11 PM, dbush wrote:
On 7/4/2026 1:07 PM, olcott wrote:
On 7/4/2026 3:06 AM, Mikko wrote:
On 03/07/2026 21:20, olcott wrote:
On 7/3/2026 12:35 PM, André G. Isaak wrote:
On 2026-07-03 09:38, olcott wrote:The base definition of "incomplete" means that it is
On 7/3/2026 4:28 AM, Mikko wrote:
On 02/07/2026 17:49, olcott wrote:
On 7/2/2026 1:55 AM, Mikko wrote:
On 01/07/2026 18:16, olcott wrote:
On 7/1/2026 2:24 AM, Mikko wrote:
On 30/06/2026 16:58, olcott wrote:Base-Extension Semantics (B-eS) seems to be essentially a >>>>>>>>>>>> cheat.
On 6/30/2026 3:18 AM, Mikko wrote:
On 29/06/2026 16:29, olcott wrote:
On 6/29/2026 1:14 AM, Mikko wrote:
On 29/06/2026 05:52, olcott wrote:
On 6/28/2026 3:39 AM, Mikko wrote:Irrelevant. The definition of completeness
On 27/06/2026 17:50, polcott wrote:Q also can't bake a birthday cake, this does not make >>>>>>>>>>>>>>>>>> Q in any way "incomplete" relative to what it was >>>>>>>>>>>>>>>>>> defined to do. Incomplete only counts relative to >>>>>>>>>>>>>>>>>> its intended purpose. A car without an engine is >>>>>>>>>>>>>>>>>> incomplete relative to a mode of transportation. >>>>>>>>>>>>>>>>>
On 6/27/2026 1:53 AM, Tristan Wibberley wrote: >>>>>>>>>>>>>>>>>>>>> On 20/06/2026 18:32, olcott wrote:It comes close. If ∃x x=S(x) is likewise "ungrounded" >>>>>>>>>>>>>>>>>>> but in the
If there is no sequence of inference steps in Q from >>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) is >>>>>>>>>>>>>>>>>>>> ungrounded in the PTS atomic base of Q. >>>>>>>>>>>>>>>>>>>>
A proof theoretic expression is known to be true when >>>>>>>>>>>>>>>>>>>>>> it is fully grounded in its atomic base. Only two >>>>>>>>>>>>>>>>>>>>>> PTS semantics researchers deal with true Dag Prawitz >>>>>>>>>>>>>>>>>>>>>> is the one that began this. PTS previously only dealt >>>>>>>>>>>>>>>>>>>>>> with semantic meaning and never got around to >>>>>>>>>>>>>>>>>>>>>> true(L,x).
That's surprising, disregard for axioms? >>>>>>>>>>>>>>>>>>>>
This does not mean undecidable or incomplete >>>>>>>>>>>>>>>>>>>> it means that ~∃x x=S(x) is out-of-scope for Q. >>>>>>>>>>>>>>>>>>>
language of Q then ~∃x x=S(x) and ∃x x=S(x) are both >>>>>>>>>>>>>>>>>>> undecidable
and Q is incomplete, bcause that is what the words mean. >>>>>>>>>>>>>>>>>>
It a misnomer and does not literally mean (as it implies) >>>>>>>>>>>>>>>> that something is missing that could be added to make >>>>>>>>>>>>>>>> it complete.
It does mean that something is missing that could be >>>>>>>>>>>>>>> added to
enabe a proof of an unprovable sentence.
Base-Extension Semantics (B-eS) allows that.
It never was incomplete. It always did what it was defined >>>>>>>>>>>>>> to do.
When Q is extended to become PA it stops being Q and >>>>>>>>>>>>>> becomes PA.
However, there are theories that reamain incomplete even when >>>>>>>>>>>>> more postolates are added, as long as there is a way to know >>>>>>>>>>>>> which sentences are included in the added postulates. >>>>>>>>>>>>> Important
examples include Peano arithmetic and ZFC set theory. >>>>>>>>>>>>
When we ask what is grounded in an atomic base of Q and we >>>>>>>>>>>> add axioms to Q to become PA we cheated in that we changed >>>>>>>>>>>> the original question rather than answered it.
Yes, in a sense. But sometimes it is better to have a partial >>>>>>>>>>> answer
rather than no answer at all. Of course Q with any additional >>>>>>>>>>> postulate is not Q but if the additional postulates are true >>>>>>>>>>> about
natural numbers then the strengthened theory is still a >>>>>>>>>>> theory of
natural numbers. PA is one such strengthened Q but still >>>>>>>>>>> incomplete
and can be strengthened further.
Is (∀x, S(x) ≠ x) provable or refutable in Q?
Yes if you cheat, no if you don't cheat.
As I already pointed out in another message, which you apparently >>>>>>>>> missed, it is neither.
Thus PTS would say that (∀x, S(x) ≠ x) is semantically
undefined in Q.
And that differs from claiming that Q is incomplete exactly how...? >>>>>>
not operating according to design spec.
No, it is not. The term "incomplete" in its base meaning is
appicable to various things that are not exprected to operate.
The English word "incomplete" establishes the base
meaning (parent node) in the knowledge ontology.
I will not tolerate deceptive terms-of-the-art.
In other words, you intend to lie by misusing definitions.
These definitions are the liars.
Maybe your definitions, but not the usual ones, which tell truthfully
how the defined words are used and understood by the experts.
Term-of-the-art
A cat is a dalmatian dog
Perhaps some art but neither ailurology nor cynology.
On 7/6/2026 12:27 PM, André G. Isaak wrote:
On 2026-07-06 10:58, olcott wrote:
On 7/6/2026 11:07 AM, André G. Isaak wrote:
On 2026-07-06 09:47, olcott wrote:
On 7/6/2026 4:17 AM, Mikko wrote:
On 04/07/2026 20:07, olcott wrote:
Q that cannot resolve (∀x, S(x) ≠ x) is complete
according to its definition.
By the defintion of "incomplete" Q is incomplete. The theory
Q + (∀x, S(x) ≠ x) is more complete but still incomplete.
It fully meets its design spec thus calling it
any kind of incomplete is a damned lie.
What exactly do you think the 'design spec' of Q is?
Make sure that Q has less capability than PA is its design
spec by its designer.
And you presumably have a reference to back that up?
It is common knowledge that was Robinson's purpose
In mathematics, Robinson arithmetic is a finitely
axiomatized fragment of first-order Peano arithmetic
(PA), first set out by Raphael M. Robinson in 1950.
It is usually denoted Q.
https://en.wikipedia.org/wiki/Robinson_arithmetic
But it doesn't matter either way since the mathematical definition of
incomplete makes no reference to the 'spec' of a system.
Within the natural preexisting order of the body
of knowledge saying that incomplete(math) inherits
part of its meaning from incomplete(base) semantic
parent node is simply a lie.
On 2026-07-06 11:45, olcott wrote:
On 7/6/2026 12:27 PM, André G. Isaak wrote:
On 2026-07-06 10:58, olcott wrote:
On 7/6/2026 11:07 AM, André G. Isaak wrote:
On 2026-07-06 09:47, olcott wrote:
On 7/6/2026 4:17 AM, Mikko wrote:
On 04/07/2026 20:07, olcott wrote:
Q that cannot resolve (∀x, S(x) ≠ x) is complete
according to its definition.
By the defintion of "incomplete" Q is incomplete. The theory
Q + (∀x, S(x) ≠ x) is more complete but still incomplete.
It fully meets its design spec thus calling it
any kind of incomplete is a damned lie.
What exactly do you think the 'design spec' of Q is?
Make sure that Q has less capability than PA is its design
spec by its designer.
And you presumably have a reference to back that up?
It is common knowledge that was Robinson's purpose
In mathematics, Robinson arithmetic is a finitely
axiomatized fragment of first-order Peano arithmetic
(PA), first set out by Raphael M. Robinson in 1950.
It is usually denoted Q.
https://en.wikipedia.org/wiki/Robinson_arithmetic
But it doesn't matter either way since the mathematical definition of
incomplete makes no reference to the 'spec' of a system.
Within the natural preexisting order of the body
of knowledge saying that incomplete(math) inherits
part of its meaning from incomplete(base) semantic
parent node is simply a lie.
Yes, I agree that it is a lie.
For starters, there's no such thing as the 'natural preexisting order of
the body of knowlege'.
And incomplete(math) doesn't inherit from anything so claiming it
inherits from incomplete(base) (whatever that may be) is of course a lie.
André
It is impossible to leap from {nothingness} to
1987 Chevy Camaro with no steps inbetweem.
{Thing}--->{Physically Existing Thing}
... {Motor Vehicle}---> {Automobile} ...
On 7/6/2026 4:13 PM, André G. Isaak wrote:I did not write any of the above. Please don't claim that I did.
It is impossible to leap from {nothingness} to
1987 Chevy Camaro with no steps inbetweem.
{Thing}--->{Physically Existing Thing}
... {Motor Vehicle}---> {Automobile} ...
Try to explain how the notion of 1987 Chevy Camaro
pops into existence from out-of-nowhere with no
prerequisite order.
On 2026-07-06 16:00, olcott wrote:
On 7/6/2026 4:13 PM, André G. Isaak wrote:I did not write any of the above. Please don't claim that I did.
It is impossible to leap from {nothingness} to
1987 Chevy Camaro with no steps inbetweem.
{Thing}--->{Physically Existing Thing}
... {Motor Vehicle}---> {Automobile} ...
Try to explain how the notion of 1987 Chevy Camaro
pops into existence from out-of-nowhere with no
prerequisite order.
If you want to know how concepts are actually organized, you need to
look at experimental evidence from psychology, psycholinguistics,
aphasiology, etc. They aren't organized into a tree where concepts have parents. Armchair philosophizing (aka mental masturbation) isn't going
to get you anywhere.
André
On 7/6/2026 5:18 PM, André G. Isaak wrote:
On 2026-07-06 16:00, olcott wrote:
On 7/6/2026 4:13 PM, André G. Isaak wrote:I did not write any of the above. Please don't claim that I did.
It is impossible to leap from {nothingness} to
1987 Chevy Camaro with no steps inbetweem.
{Thing}--->{Physically Existing Thing}
... {Motor Vehicle}---> {Automobile} ...
Try to explain how the notion of 1987 Chevy Camaro
pops into existence from out-of-nowhere with no
prerequisite order.
I know that I wrote it.
You must show exactly how
I am incorrect otherwise your fake rebuttal is
simply hiding behind profound ignorance.
If you want to know how concepts are actually organized, you need to
look at experimental evidence from psychology, psycholinguistics,
aphasiology, etc. They aren't organized into a tree where concepts
have parents. Armchair philosophizing (aka mental masturbation) isn't
going to get you anywhere.
André
They are organized as a type hierarchy.
You either understand this or fail to comprehend.
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