Refuting Gödel's 1931 Incompleteness Theorem in one sentence

[PeteOlcott]

When the notion of true is defined as provable from axioms and axioms
are defined to be finite strings having the semantic property of
Boolean true then any expression of language that is not provable
is not true.

English: C is not Provable entails that C is not a Theorem:
∀C (¬∃Γ(Γ ⊢ C) → (⊬C))

[Within the above definition of True]
English: C is not a Theorem entails that C is not True:
∀C (⊬C) → ¬True(C)

https://plato.stanford.edu/entries/goed ... pleteness/
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried
out, there are statements of the language of F which can neither be
proved nor disproved in F. (Raatikainen, Panu: Fall 2018)


--
Copyright 2019 Pete Olcott
All rights reserved

Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015):28

Mendelson.png

[Univalence]

Define the notion of “provable”.

[PeteOlcott]

Define the notion of “provable”.

It is an inline image because Scott Mayers was paranoid about viruses.

[Univalence]

Why have you chosen to conflate the (otherwise distinct) notions of truth and provability?

[Univalence]

English: C is not a Theorem entails that C is not True:
∀C (⊬C) → ¬True(C)

Where does that leave your axioms?

They aren't theorems, therefore they are not True.

[PeteOlcott]

English: C is not a Theorem entails that C is not True:
∀C (⊬C) → ¬True(C)

Where does that leave your axioms?

They aren't theorems, therefore they are not True.

(a) Axioms are finite strings defined to have the semantic property of Boolean true.
(b) True(x) is defined as Theorem(x) // Assuming that Axioms are Theorems of themselves.

Axiom(1) All dogs are mammals.
Axiom(2) All mammals breathe.
--------------------------------------
Theorem(1) ∴ All dogs breathe.

[Univalence]

Axiom(1) All dogs are mammals.
Axiom(2) All mammals breathe.
--------------------------------------
Theorem(1) ∴ All dogs breathe.

Why go through all of these shenanigans?

Axiom(1) All dogs breathe.
Axiom(2) 1 = 0

[PeteOlcott]

Axiom(1) All dogs are mammals.
Axiom(2) All mammals breathe.
--------------------------------------
Theorem(1) ∴ All dogs breathe.

Why go through all of these shenanigans?

Axiom(1) All dogs breathe.
Axiom(2) 1 = 0

There are no rules-of-inferences between your Axioms yet we can still conclude that All dogs breathe.

[PeteOlcott]

Why go through all of these shenanigans?

Refuting Gödel and Tarski will give me the credibility that I need to get a
research position completing the formal specification of the upper knowledge
ontology of human knowledge for the purpose of deriving the automated
process for populating systems such as Doug Lenat's Cyc project.

Since natural language semantics is anchored in truth conditional semantics
we must refute Tarski to have any objective measure that a truth conditional
semantics specification of a natural language expression is correct or incorrect.

Unless we have a measure of correct and incorrect we have no measure of
incremental improvement of the formalization of natural language expressions.

Without a measure of incremental improvement no genetic algorithm or
deep learning can be applied to improve the quality of natural language
formalizations.

[Univalence]

There are no rules-of-inferences between your Axioms yet we can still conclude that All dogs breathe.

Naturally.

And we can axiomatically define 0 = 1 as being True.

[Univalence]

Since natural language semantics is anchored in truth conditional semantics
we must refute Tarski to have any objective measure that a truth conditional
semantics specification of a natural language expression is correct or incorrect.

This makes absolutely no sense to me.

If you want to use your model as a metric - then use it as a metric. Nobody is stopping you.
Tarski is certainly not standing in your way, for you can simply disregard all of his work should you choose to do so.

If your metric/model makes accurate predictions then it's useful irrespective of what Tarski said.

[PeteOlcott]

There are no rules-of-inferences between your Axioms yet we can still conclude that All dogs breathe.

Naturally.

And we can axiomatically define 0 = 1 as being True.

Sure and we can also axiomatically define does to be office buildings.
The whole point is that Tarski and Gödel are refuted by formal systems
defined on the basis of sound deductive inference.

[PeteOlcott]

Since natural language semantics is anchored in truth conditional semantics
we must refute Tarski to have any objective measure that a truth conditional
semantics specification of a natural language expression is correct or incorrect.

This makes absolutely no sense to me.

Tarski "proved" that truth cannot be formalized for systems of arithmetic and greater.
The would entail that truth conditional semantics (that depends upon a formal notion of truth)
cannot possibly be anchored.

[Univalence]

Tarski "proved" that truth cannot be formalized for systems of arithmetic and greater.
The would entail that truth conditional semantics (that depends upon a formal notion of truth)
cannot possibly be anchored.

Writing some squiggles on a paper and getting other logicians to pat you on the back isn't going to change any of that.

Empiricism trumps formalism.

[PeteOlcott]

Tarski "proved" that truth cannot be formalized for systems of arithmetic and greater.
The would entail that truth conditional semantics (that depends upon a formal notion of truth)
cannot possibly be anchored.

Writing some squiggles on a paper and getting other logicians to pat you on the back isn't going to change any of that.

Empiricism trumps formalism.

When we get into questions regarding empiricism we are getting into the fundamental nature
of reality and have thus quit talking about logic and started talking about metaphysics and religion.