Re: Undecidability based on epistemological antinomies V2 --Mendelson--

On 4/27/2024 3:18 AM, Mikko wrote:

On 2024-04-26 15:28:08 +0000, olcott said:
 

On 4/26/2024 3:42 AM, Mikko wrote:

On 2024-04-25 14:27:23 +0000, olcott said:
>

On 4/25/2024 3:26 AM, Mikko wrote:

epistemological antinomy

>
It <is> part of the current (thus incorrect) definition
of undecidability because expressions of language that
are neither true nor false (epistemological antinomies)
do prove undecidability even though these expressions
are not truth bearers thus not propositions.

>
That a definition is current does not mean that is incorrect.
>

>
...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)
>

An epistemological antinomy can only be an undecidable sentence
if it can be a sentence. What epistemological antinomies you
can find that can be expressed in, say, first order goup theory
or first order arithmetic or first order set tehory?
>

>
It only matters that they can be expressed in some formal system.
If they cannot be expressed in any formal system then Gödel is
wrong for a different reason.

 How is it relevant to the incompleteness of a theory whether an
epistemological antińomy can be expressed in some other formal
system?
 

When an expression of language cannot be proved in a formal system only
because it is contradictory in this formal system then the inability to
prove this expression does not place any actual limit on what can be
proven because formal system are not supposed to prove contradictions.
*Mendelson's statement of the bare essence of Gödel's proof*
    Observe that, in terms of the standard interpretation
    (∀x2) ¬𝒫𝑓 (x2, ⌜G⌝) says that there is no natural number
    that is the Gödel number of a proof in K of the wf G, which is
    equivalent to asserting that there is no proof in K of G. Hence,
    G is equivalent in K to an assertion that G is unprovable in K.
    In other words, G says “I am not provable in K”. This is an
    analogue of the liar paradox: “I am lying” (that is, “I am not
    true”). However, although the liar paradox leads to a contradiction,
    Gödel (1931) showed that G is an undecidable sentence of K.
    We shall refer to G as a Gödel sentence for K. (Mendelson: 2015:208)
G says “I am not provable in K”. (quoted from above)
Any proof of G in K requires a sequence of inference steps in K that
prove that they themselves do not exist. This is why there is no proof
of G in K.
Mendelson, Elliott 2015. Introduction to Mathematical Logic sixth
edition CRC Press Taylor & Francis Group Boca Raton, FL
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer