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From: Keith Thompson <Keith.S.Thompson+u@gmail.com>
Newsgroups: comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics
Subject: Re: Simply defining =?utf-8?Q?G=C3=B6del?= Incompleteness and Tarski Undefinability away V24 (Are we there yet?)
Date: Sun, 12 Jul 2020 19:32:50 -0700
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olcott <NoOne@NoWhere.com> writes:
> On 7/12/2020 7:22 PM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>> On 7/12/2020 4:04 PM, Keith Thompson wrote:
[...]
>>>> Robinson Arithmetic cannot prove or disprove commutativity
>>>> of addition.  We can construct a consistent system based on
>>>> Robinson Arithmetic in which addition is provably commutative.
>>>
>>> Sure just add an axiom: ∀x ∈ ℕ ∃y ∈ ℕ (x + y = y + x)
>>>
>>>> Can we construct a consistent system based on Robinson Arithmetic
>>>> in which addition is provably *not* commutative?
>>>
>>> Not within the conventional semantics of the meaning of those terms.
>>
>> OK.  Can you prove that?
>
> Nothing can possibly be disproved that is true by definition.

I suppose that's a true statement, but how is it relevant?  What
"definition" implies that no consistent system based on Robinson
Arithmetic can have non-commutative addition?

Please don't bother posting a reply that doesn't actually answer my
question..

-- 
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */