Pete Olcott

2017-06-17 18:41:35 UTC

Thus the LHS nickname is to be considered one-and-the-same-thing as the expression on the RHS, as if macro substitution was to take place.

The primary purpose of this syntax is to provide the means for an expression to directly refer to itself. This is not at all the same convention that Tarski provided where a quoted expression is considered to be the name of the expression so that an expression can refer to its name.

With Tarski's convention the true nature of pathological self-reference cannot be expressed, it slips through the cracks of expressibility without ever being seen.

We use: L ∈ Formal_Systems

instead of: ∃L ∈ Formal_Systems

because L ∈ Formal_Systems is an axiom and not an assertion, Formal_Systems is a predefined constant that is defined to have at least one element.

Why do you introduce Γ? Isn't it true that if Γ ⊢ X then, forThe primary purpose of this syntax is to provide the means for an expression to directly refer to itself. This is not at all the same convention that Tarski provided where a quoted expression is considered to be the name of the expression so that an expression can refer to its name.

With Tarski's convention the true nature of pathological self-reference cannot be expressed, it slips through the cracks of expressibility without ever being seen.

We use: L ∈ Formal_Systems

instead of: ∃L ∈ Formal_Systems

because L ∈ Formal_Systems is an axiom and not an assertion, Formal_Systems is a predefined constant that is defined to have at least one element.

any G such that Γ ⊂ G, G ⊢ X? It appears all you need is

Not, I fear, very enlightening.

The above expression named G is enormously simpler than the 1931 Incompleteness Theorem:

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."

The above expression named G is neither provable nor refutable in any formal system what-so-ever.

If we stopped our analysis right there we would have an enormous simplification of the 1931 GIT that is much broader in scope than the 1931 GIT in that it simultaneously applies to every possible formal system of logic.

--

(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)

(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)