All that we must do to defeat the Tarski Undefinability Theorem:
We define the notion of formal system as an extended
version of Prolog's Facts and Rules. This new system
can handle arbitrary orders of logic. Encodes Facts
in formalized natural language.
The Rules only allow semantic logical entailment from
Facts. When we do this Tarski's Liar Paradox basis is
simply rejected as untrue and
Boolean True(Language L, Expression E) becomes definable.
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 Gödel said and proved.
Gödel'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 Gödel number of the sentence has no >>>>>>>>>>>>>>>>>>>>> arithmetic significance.
Yes that is the moronic received view yet these stupid >>>>>>>>>>>>>>>>>>>> people stupidly ignore Gödel's own words. >>>>>>>>>>>>>>>>>>>
The most important aspect of Gödel's 1931 >>>>>>>>>>>>>>>>>>>> Incompleteness theorem
are these plain English direct quotes of Gödel from >>>>>>>>>>>>>>>>>>>> his paper:
...there is also a close relationship with the >>>>>>>>>>>>>>>>>>>> “liar” 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 ... >>>>>>>>>>>>>>>>>>>> (Gödel 1931:40-41)
Gödel, 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 Gödel 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 Gödel'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 2025-10-05 14:03:37 +0000, olcott said:
Gödel 1931 undecidability and Tarski Undefinability
only exist because they they not know to reject an
expression of language that is not a truth bearer.
What expression of the language of the first order theory of the
first order Peano arithmetic is not a truth bearer? That you can't
determine the truth value of some expression does not mean that ir
has none. That you don't understand the proof does not mean that
the proofs are not sound.
Claude AI is quite hesitant at first, disagreeing
with me several times. Then it is finally convinced
that I am correct.
That you may convince an artificial idiot means nothing.
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.
My scope is the entire body of human knowledge
that can be expressed in language.
Tarski's scope was only formal theories and their languages. Within that scope it is at least clear what constitutes a proof.
On 2025-10-05 14:09:37 +0000, olcott said:
On 10/5/2025 5:26 AM, Mikko wrote:
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
It would then be possible to reconstruct the antinomy of the liar >>>>>> in the metalanguage, by forming in the language itself a sentence >>>>>> x such that the sentence of the metalanguage which is correlated >>>>>> with x asserts that x is not a true sentence.
https://liarparadox.org/Tarski_247_248.pdf
Formalized as:
x ∉ 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.
Gödel uses tricks for that.
Tarski used the same tricks.
Yet they only actually boil down to
Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).
and this
LP := ~True(LP)
The system they considered has no symbol for :=. Instead, they construct >>> something like LP <-> ~True(LP). Gödel then shows that that the
expression
that asserts its own unprovability is is not provable and therefore
true.
Claude AI eventually agreed that both Gödel's 1931
Incompleteness theorem and the Tarski Undefinability
theorem are anchored in liar paradox like expression
that should have been rejected as not a truth bearer.
Gödel proved that every sentence of a first order theory that is not
the negation of any sentnece of that theory is true in some model of
that theory. Therefore every sentence of every first order theory is
a truth-bearer.
As long as you don't understand that "The liar's paradox is not true"
is true and therefore a valid basis for a proof you cannot say anything
about Tarski's proof but are stuck to straw men.
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