Refuting Gödel's 1931 Incompleteness Theorem in one sentence

[PeteOlcott]

Not really not at all.

Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015):28
1.4 An Axiom System for the Propositional Calculus page 27-28
A wf C is said to be a consequence in S of a set Γ of wfs if and only if there is a
sequence B1, …, Bk of wfs such that C is Bk and, for each i, either Bi is an axiom
or Bi is in Γ, or Bi is a direct consequence by some rule of inference of some of
the preceding wfs in the sequence. Such a sequence is called a proof (or deduction)
of C from Γ. The members of Γ are called the hypotheses or premisses of the proof.
We use Γ ⊢ C as an abbreviation for “C is a consequence of Γ”...

If Γ is the empty set ∅, then ∅ ⊢ C if and only if C is a theorem. It is customary
to omit the sign “∅” and simply write ⊢ C. Thus, ⊢ C is another way of asserting that
C is a theorem.

It is exactly: ⊢ C with the additional stipulation that Axioms are actually true,
not this conventional true-ish: http://mathworld.wolfram.com/Axiom.html

Thanks for clarifying this, however I do not understand the meanings of those symbols (⊢, Γ, wfs, Bk etc.) so I can't really discuss the matter in-depth as much as I'd like to.

I admit I'm not educated enough to actually refuting or verifying either theory.

But I'd like to learn the language of logic, is there a youtube tutorial series you would recommend? Alternatively, if possible - could you explain those symbols?

My entire basis for refuting Tarski Undefinability and Gödel Incompleteness
is this junior high school level of logic. As soon as we cause the formal proofs
of mathematical logic to conform to sound deduction
(a) Incompleteness (b) Undefinability (c) Undecidability cease to exist.

https://www.iep.utm.edu/val-snd/
Validity and Soundness
A deductive argument is said to be valid if and only if it takes a form that makes
it impossible for the premises to be true and the conclusion nevertheless to be
false. Otherwise, a deductive argument is said to be invalid.

A deductive argument is sound if and only if it is both valid, and all of its premises
are actually true. Otherwise, a deductive argument is unsound.

Because it is totally unbelievable that junior high school logic would utterly
refute at least two very well established theorems that have stood the test
of time for more that eight decades my idea is rejected out-of-hand without
being tested.

So far two people have tested my idea and have agreed that it is correct.
Here is a quote from one of them:
"I agree with you that if your definitions are accepted then we can eliminate incompleteness."

When I state this in a way that mathematicians can understand:
When we simply specify that True(x) is exactly the subset of the conventional formal proofs
of mathematical logic having true premises then True(x) is always defined and never undefinable.

The above defines Tarski's undefinable which in turn can be used to complete Gödel's incompleteness.

[Eodnhoj7]

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

Just apply the theorem to the theorem.