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On 2025-10-04 13:34:07 +0000, olcott said:
On 10/4/2025 5:25 AM, Mikko wrote:
On 2025-10-02 10:07:33 +0000, olcott said:
On 10/2/2025 4:38 AM, Mikko wrote:
On 2025-10-01 15:40:06 +0000, olcott said:
On 10/1/2025 5:12 AM, Mikko wrote:
On 2025-10-01 01:46:15 +0000, olcott said:He anchored his whole proof in that he
On 9/30/2025 7:48 AM, Mikko wrote:
On 2025-09-29 12:21:25 +0000, olcott said:
On 9/27/2025 5:05 AM, Mikko wrote:
On 2025-09-26 01:08:45 +0000, olcott said:
On 9/25/2025 2:15 AM, Mikko wrote:
On 2025-09-24 14:27:00 +0000, olcott said:Tarski admits that he anchor his whole proof on the
On 9/24/2025 2:12 AM, Mikko wrote:
On 2025-09-23 16:04:38 +0000, olcott said:
On 9/23/2025 4:21 AM, Mikko wrote:
On 2025-09-23 00:56:19 +0000, olcott said:Prior to Pythagoras there was a universal consensus >>>>>>>>>>>>>>>> that the Earth is flat.
On 9/21/2025 4:22 AM, Mikko wrote:
On 2025-09-20 14:57:20 +0000, olcott said: >>>>>>>>>>>>>>>>>>>> On 9/20/2025 4:31 AM, Mikko wrote:
It is what G||del said and proved.
G||del's sentence is not really self-referential. It >>>>>>>>>>>>>>>>>>>>> is a valid
sentence in the first order language of Peano >>>>>>>>>>>>>>>>>>>>> arithmetic. That
the value of an arithmetic expression in that >>>>>>>>>>>>>>>>>>>>> sentence evaluates
to the G||del number of the sentence has no >>>>>>>>>>>>>>>>>>>>> arithmetic significance.
Yes that is the moronic received view yet these stupid >>>>>>>>>>>>>>>>>>>> people stupidly ignore G||del's own words. >>>>>>>>>>>>>>>>>>>
The most important aspect of G||del's 1931 >>>>>>>>>>>>>>>>>>>> Incompleteness theorem
are these plain English direct quotes of G||del from >>>>>>>>>>>>>>>>>>>> his paper:
...there is also a close relationship with the >>>>>>>>>>>>>>>>>>>> rCLliarrCY antinomy,14 ...
...14 Every epistemological antinomy can likewise be >>>>>>>>>>>>>>>>>>>> used for a similar undecidability proof... >>>>>>>>>>>>>>>>>>>> ...We are therefore confronted with a proposition >>>>>>>>>>>>>>>>>>>> which asserts its own unprovability. 15 ... >>>>>>>>>>>>>>>>>>>> (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of Principia >>>>>>>>>>>>>>>>>>>> Mathematica And Related Systems
The most important aspect is the theorem itself: >>>>>>>>>>>>>>>>>>> every theory that
has the symbols and axioms of the first order Peano >>>>>>>>>>>>>>>>>>> arithmetic is
either incomplete or inconsistent.
That never has been the important part.
That has always been bullshit misdirection
It is important because people consider it important. >>>>>>>>>>>>>>>>
To think the Earth as flat is simpler and good enough for >>>>>>>>>>>>>>> many
purposes. For a long time there was no need to think >>>>>>>>>>>>>>> about the
shape of the Earth.
The point is that not even a universal consensus equates >>>>>>>>>>>>>> to truth.
No, but it is a significant aspect of culture. A question >>>>>>>>>>>>> of importance
is not a matter of truth but a matter of opinion.
Many poeple also
find it useful to know that any attempt to construct a >>>>>>>>>>>>>>>>> cmplete theory
of arithemtic would be a waste of time.
Yet only when the architecture of the formal system is >>>>>>>>>>>>>>>> screwed up.
If you want to build a formal system that is not >>>>>>>>>>>>>>>> anchored in a
screwed up idea than this is straight forward.
That 2 + 3 = 5 has practical value even if the theory >>>>>>>>>>>>>>> around it
cannot be made complete.
*Refuting G||del 1931 Incompleteness*
You begin with a finite list of basic facts and only apply >>>>>>>>>>>>>>>> the truth preserving operation of semantic logical >>>>>>>>>>>>>>>> entailment
to these basic facts.
It is generally accepted that the set of axioms can be >>>>>>>>>>>>>>> infinite
Not when we are representing the finite set of human >>>>>>>>>>>>>> general knowledge.
Once again you try to deceive with a change of topic. There >>>>>>>>>>>>> is no
need to prove the incompletenes of human general knowledge >>>>>>>>>>>>> as that
already is obvious. But G||del's and Tarski's theorems are >>>>>>>>>>>>> about
natural number arithmetic and its extensions so they need >>>>>>>>>>>>> to cover
the possibility that there are infinitely many axioms. >>>>>>>>>>>>
liar paradox and
He doesn't "anchor" it to the liar paradix. The liar paradox >>>>>>>>>>> has some
formal similarity to the sentence Tarski constructs but is >>>>>>>>>>> not a part
of the proof. Consequently anything said about the liar's >>>>>>>>>>> paracos is
irrelevant to the correctness of the proof.
Factually incorrect.
False. Tarski confirms what I said:
And I prove my point in the paragraph that you skipped.
Not relevant to Tarski's rerutation of your "Factually incorrect". >>>>>>
needed an extra level of logic to do this:
You need metalogic if you want to say anything about logic.
X is any expression of language that is not
a truth bearer. It is true that X is not true.
You cannot say that in the plain language of logic. You need a
metatheory that can express and infer about expressions and
relate them to truth.
I proved otherwise
https://claude.ai/share/45d3ca9c-fa9a-4a02-9e22-c6acd0057275
You didn't prove anything. If you could you would post here the
plain logic sentence that says what you said.
It explains a correction to an aspect of the
foundation of logic and it does this in plain
English and a tiny bit of Prolog.
It is not correct to use the word "correction" when the thing to
be corrected is correct already.
But Tarski proved about natural numbers that if there were a definition
of a predicate in terms of a formula in the language of Peano arithmetic >>> that accepts all numbers that encode a true sentence and rejects all
other numbers then that predicate would accept a number that encodes
a false sentence or reject a number that encodes a true sentence.
Mine has a broader scope that can be applied to
any pathological self-reference(Olcott 2004) in
formal expressions and formalized natural language
expressions.
Tarstki's scope is wider, too, but the first order arithmetic of natural numbers is the most interesting part of the scope.
On 2025-10-04 13:30:22 +0000, olcott said:
On 10/4/2025 5:11 AM, Mikko wrote:
On 2025-10-02 10:15:13 +0000, olcott said:
On 10/2/2025 5:03 AM, Mikko wrote:
On 2025-10-01 16:33:46 +0000, olcott said:
On 10/1/2025 5:13 AM, Mikko wrote:
On 2025-10-01 01:48:56 +0000, olcott said:
On 9/30/2025 7:54 AM, Mikko wrote:
On 2025-09-29 12:24:30 +0000, olcott said:
On 9/27/2025 5:07 AM, Mikko wrote:
Any sentence that is neither true nor false
must be rejected from any system of logic.
Non-truth bearers in logic systems are like
turds in birthday cakes.
Every sentence of logic that is not tautology or dontradiction is >>>>>>>>> true in some contexts and false in others.
A mere false assumption
No, it is true on the basis of the meanings of the words.
The syntax of formal logical languages allows
some expressions to be created having pathological
self-reference(Olcott 2004).
No syntax is enough for self-reference.
Syntax is enough for self-reference.
No, self-reference is a semantic feature. A string without meaning
does not refer.
Self-reference can be detected in a string with a name.
Tarski's Liar Paradox from page 248
-a-a-a It would then be possible to reconstruct the antinomy of the liar >>>> -a-a-a in the metalanguage, by forming in the language itself a sentence >>>> -a-a-a x such that the sentence of the metalanguage which is correlated >>>> -a-a-a with x asserts that x is not a true sentence.
-a-a-a https://liarparadox.org/Tarski_247_248.pdf
Formalized as:
x ree True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf
The semantics determines whether
any syntactic construct is a self-reference. For example. the
arithmetic
semantics of a formal language of arithmetics do not permit a self-
reference.
G||del uses tricks for that.
Tarski used the same tricks.
Yet they only actually boil down to
Incomplete(F) rao reaG ((F re4 G) reo (F re4 -4G)).
and this
LP := ~True(LP)
The system they considered has no symbol for :=. Instead, they construct something like LP <-> ~True(LP). G||del then shows that that the expression that asserts its own unprovability is is not provable and therefore true.
Tarski shows that if there is there is a formula that is true if its
argument encodes a true sentence and false if its argument encodes a false statements (its value does not matter if the argument does not encode
a sentence) then it is possible to encode a sentence that says "the truth formula returns false if given my encoding" and give it to the truth
formula, which would then return the negation of the value it returns.
As that would be a contradiction there cannot be a truth formula and therefore no definition truth.
At least for some systems it is possible to have a truth predicate in
a metatheory even if not in the system itself.