Proof that Wittgenstein is correct about Gödel

[peteolcott]

https://plato.stanford.edu/entries/goedel-incompleteness/
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.

Wittgenstein’s “notorious paragraph” about the Gödel Theorem
http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6.annotate

Formalizing Wittgenstein’s words:
// Minimal essence of the 1931 Incompleteness Theorem conclusion
P ↔ (RS ⊬ P)

// LHS := RHS The LHS is defined as the RHS
// True and False axiom schemata
∀x (True(RS, x) := (RS ⊢ x))
∀x (False(RS, x) := (RS ⊢ ¬x))

My proof that Wittgenstein is correct
Truth Table(B) // Minimal Essence of the 1931 Incompleteness Theorem
These two lines are stipulated by the above True/False schemata:
Whenever P is true in RS P is provable in RS and ¬P is ¬provable in RS
Whenever P is false in RS ¬P is provable in RS and P is ¬provable in RS

THIS TABLE REQUIRES FIXED WIDTH FONT
P ↔ (RS ⊬ P)
T-F-----F // if P is true then P is provable
F-F-----T // if P is false then P is ¬provable
For any formal system expressive enough to have its own provability
operator the above formula expresses self-contradiction. This is shown
by the failure of logical equivalence on the above two rows.

These two rows of the truth table are defined as non-existent
by the above axiom schemata True/False predicate templates.

THIS TABLE REQUIRES FIXED WIDTH FONT
P ↔ (RS ⊬ P)
T-T-----T // if P is true then P is ¬provable
F-T-----F // if P is false then P is provable
We are now switching from Wittgenstein’s names:
(1) Russell’s System (RS) becomes Peano Arithmetic (PA)
(2) Wittgenstein’s P becomes Gödel’s G

If G is not provable in PA then G is not true in PA. If G is provable
in the Gödelization of PA then G is true in the Gödelization of PA.
Diagonalization is an alternative form of provability.

The Gödelization of PA is a distinctly different formal system than
PA as shown by the difference of the provability of G.

The key difference between PA and the Gödelization of PA is that
G is inside the scope of self-contradiction in PA and G is outside
the scope of self-contradiction in the Gödelization of PA.

When we address the comparable proof in the Tarski Undefinability
Theorem we also address another aspect of the Incompleteness Theorem.

For formal systems that are not expressive enough to have their own
provability operator as in Gödel’s PA and Tarski’s Theory there are
indeed expression’s of PA and Tarski’s Theory that can be shown to
be true in the Gödelization of PA and Tarski’s meta-theory respectively
that are not provable (and therefore untrue) in PA and Tarski’s Theory.

Here is a key brand new insight anchored in the Tarski Undefinability
Theorem. Tarski concluded that an infinite hierarchy of Meta-Theories
was required to always have provability. This can be easily shown to
be untrue.

We simply have two different versions of Tarski’s Theory and two
different versions of his Meta-Theory having the exact same relations
as the original Theory and Meta-Theory, yet all of these relations
are differently named.

Meta-Theory-A is in terms of the relations of Theory-A.
Meta-Theory-B is in terms of the relations of Theory-B.

Now we have two Meta-Theories at the exact same hierarchy level that
can each prove the unprovable expressions of the other. They can do
this because these expressions are outside of the scope of self-contradiction

Same words as above using HTYML to make them easier to understand.
http://liarparadox.org/index.php/2019/06/17/proof-that-wittgenstein-is-correct-about-godel/

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Copyright 2019 Pete Olcott All rights reserved

"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein