We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard
definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such
that (T ⊬ φ) and (T ⊬ ¬φ).
This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system.
Since self-contradictory expressions are neither provable nor disprovable only because they are self-contradictory we could define them as unsound instead of defining the formal system as incomplete.
Formalized by Olcott as:According to Wittgenstein:
'True in Russell's system' means, as was said: proved in Russell's
system; and 'false in Russell's system' means: the opposite has
been proved in Russell's system. (Wittgenstein 1983,118-119)
∀F ∈ Formal_Systems ∀C ∈ WFF(F) (((F⊢C)) ↔ True(F, C))
∀F ∈ Formal_Systems ∀C ∈ WFF(F) (((F⊢¬C)) ↔ False(F, C))
We had to add that the proofs referred to by Wittgenstein must be to theorem consequences thus requiring the axioms of formal proofs to act as a proxy for the true premises of sound deduction.
We simply construe a formal proof to theorem consequences as isomorphic to deduction from a sound argument to a true conclusion. This requires the theorems of formal systems to be construed as the true premises of sound deduction.
[Simplified Gödel Sentence]
https://plato.stanford.edu/entries/goed ... rIncTheCom
(G) F ⊢ GF ↔ ¬ProvF(⌈GF⌉). // Original
(G) F ⊢ GF ↔ ¬ProvF(GF). // Remove arithmetization
// Adapt syntax and quantify:
∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ (F ⊬ G))
When a WFF expresses that it is logically equivalent to its own unprovability: G ↔ (F ⊬ G) this expression is self-contradictory thus unsatisfiable. If we could prove that a sentence that asserts it is logically equivalent to its own unprovability is true this contradicts its assertion therefore we cannot prove that it is true.
Likewise with its negation: G ↔ (F ⊬ ¬G). If we could prove that a sentence that asserts it is logically equivalent to its own unprovability is false this contradicts its assertion therefore we cannot prove that it is false.
The conventional definition of incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ).
As we can see from the above neither (F ⊬ G) not (F ⊬ ¬G) can be satisfied only because they are both self contradictory. Because they are self-contradictory they meet the definition of Incompleteness. Within the sound deductive inference model unprovable expressions of language are simply construed as unsound arguments thus untrue.
A formula is satisfiable if it is possible to find an interpretation
(model) that makes the formula true.
An interpretation is an assignment of meaning to the symbols of a
Wittgenstein, Ludwig 1983. Remarks on the Foundations of Mathematics (Appendix III), 118-119. Cambridge, Massachusetts and London, England: The MIT Press http://www.liarparadox.org/Wittgenstein.pdfModel theory
A model of a theory is a structure (e.g. an interpretation)
that satisfies the sentences of that theory.
Defining Gödel Incompleteness Away
https://www.researchgate.net/publicatio ... eness_Away
Copyright 2020 PL Olcott