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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic,comp.theory
Subject: Re: Concise rebuttal of incompleteness and undecidability
Date: Sun, 2 Jun 2024 14:03:41 -0400
Organization: i2pn2 (i2pn.org)
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On 6/2/24 1:36 PM, olcott wrote:
> Because of Quine's paper: https://www.ditext.com/quine/quine.html most
> philosophers have been confused into believing that there is no such
> thing as expressions of language that are {true on the basis of their
> meaning}.

Except that, in FORMAL LOGIC SYSTEMS, the ONLY definition of "meaning" 
is what is derived from the formal definitions and axioms of the system.

> 
> The unique contribution I have made to this is that the semantic meaning
> of these expressions is always specified by other expressions. When we
> can derive x or ~x by applying truth preserving operations to a set of
> semantic meanings then this perfectly aligns with Wittgenstein's concise
> critique of Gödel: https://www.liarparadox.org/Wittgenstein.pdf
> 
> Unless P or ~P has been proved in Russell's system P has no truth value
> and thus cannot be a proposition according to the law of the excluded
> middle.
> 
> As Richard keeps pointing out:
> Sometimes this "proof" may require an infinite sequence of steps.
> 


But the problem is that if it takes an "infinite sequence of steps" to 
make it true, that set of steps is NOT a PROOF, as proof is defined as a 
FINITE number of steps.

Thus, there exists statement that are TRUE (by being established by an 
infinite sequence of steps) but can not be PROVEN (which requires 
finding a finite number of steps to show it)