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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: Mathematical incompleteness has always been a misconception ---
 Tarski
Date: Sun, 9 Feb 2025 08:56:29 -0600
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On 2/9/2025 4:33 AM, Mikko wrote:
> On 2025-02-08 15:32:00 +0000, olcott said:
> 
>> On 2/8/2025 4:45 AM, Mikko wrote:
>>> On 2025-02-07 16:21:01 +0000, olcott said:
>>>
>>>> On 2/7/2025 4:34 AM, Mikko wrote:
>>>>> On 2025-02-06 14:46:55 +0000, olcott said:
>>>>>
>>>>>> On 2/6/2025 2:02 AM, Mikko wrote:
>>>>>>> On 2025-02-05 16:03:21 +0000, olcott said:
>>>>>>>
>>>>>>>> On 2/5/2025 1:44 AM, Mikko wrote:
>>>>>>>>> On 2025-02-04 16:11:08 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>> On 2/4/2025 3:22 AM, Mikko wrote:
>>>>>>>>>>> On 2025-02-03 16:54:08 +0000, olcott said:
>>>>>>>>>>>
>>>>>>>>>>>> On 2/3/2025 9:07 AM, Mikko wrote:
>>>>>>>>>>>>> On 2025-02-03 03:30:46 +0000, olcott said:
>>>>>>>>>>>>>
>>>>>>>>>>>>>> On 2/2/2025 3:27 AM, Mikko wrote:
>>>>>>>>>>>>>>> On 2025-02-01 14:09:54 +0000, olcott said:
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> On 2/1/2025 3:19 AM, Mikko wrote:
>>>>>>>>>>>>>>>>> On 2025-01-31 13:57:02 +0000, olcott said:
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> On 1/31/2025 3:24 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>> On 2025-01-30 23:10:18 +0000, olcott said:
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Within the entire body of analytical truth any 
>>>>>>>>>>>>>>>>>>>> expression of language that has no sequence of 
>>>>>>>>>>>>>>>>>>>> formalized semantic deductive inference steps from 
>>>>>>>>>>>>>>>>>>>> the formalized semantic foundational truths of this 
>>>>>>>>>>>>>>>>>>>> system are simply untrue in this system. (Isomorphic 
>>>>>>>>>>>>>>>>>>>> to provable from axioms).
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> If there is a misconception then you have 
>>>>>>>>>>>>>>>>>>> misconceived something. It is well
>>>>>>>>>>>>>>>>>>> known that it is possible to construct a formal 
>>>>>>>>>>>>>>>>>>> theory where some formulas
>>>>>>>>>>>>>>>>>>> are neither provble nor disprovable.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> This is well known.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> And well undeerstood. The claim on the subject line is 
>>>>>>>>>>>>>>>>> false.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> a fact or piece of information that shows that something
>>>>>>>>>>>>>>>> exists or is true:
>>>>>>>>>>>>>>>> https://dictionary.cambridge.org/us/dictionary/english/ 
>>>>>>>>>>>>>>>> proof
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> We require that terms of art are used with their term-of- 
>>>>>>>>>>>>>>> art meaning and
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> The fundamental base meaning of Truth[0] itself remains 
>>>>>>>>>>>>>> the same
>>>>>>>>>>>>>> no matter what idiomatic meanings say.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Irrelevant as the subject line does not mention truth.
>>>>>>>>>>>>> Therefore, no need to revise my initial comment.
>>>>>>>>>>>>
>>>>>>>>>>>> The notion of truth is entailed by the subject line:
>>>>>>>>>>>> misconception means ~True.
>>>>>>>>>>>
>>>>>>>>>>> The title line means that something is misunderstood but that 
>>>>>>>>>>> something
>>>>>>>>>>> is not the meaning of "true".
>>>>>>>>>>
>>>>>>>>>> It is untrue because it is misunderstood.
>>>>>>>>>
>>>>>>>>> Mathematical incompleteness is not a claim so it cannot be untrue.
>>>>>>>>
>>>>>>>> That mathematical incompleteness coherently exists <is> claim.
>>>>>>>
>>>>>>> Yes, but you didn't claim that.
>>>>>>>
>>>>>>>> The closest that it can possibly be interpreted as true would
>>>>>>>> be that because key elements of proof[0] have been specified
>>>>>>>> as not existing in proof[math] math is intentionally made less
>>>>>>>> than complete.
>>>>>>>
>>>>>>> Math is not intentionally incomplete.
>>>>>>
>>>>>> You paraphrased what I said incorrectly.
>>>>>
>>>>> No, I did not paraphrase anything.
>>>>>
>>>>>> Proof[math] was defined to have less capability than Proof[0].
>>>>>
>>>>> That is not a part of the definition but it is a consequence of the
>>>>> definition. Much of the lost capability is about things that are
>>>>> outside of the scope of mathemiatics and formal theories.
>>>>>
>>>>
>>>> When one thinks of math as only pertaining to numbers then math
>>>> is inherently very limited.
>>>
>>> That's right. That limited area should be called "number theory",
>>> not "mathematics".
>>>
>>>> When one applies something like
>>>> Montague Grammar to formalize every detail of natural language
>>>> semantics then math takes on much more scope.
>>>
>>> It is not possible to specify every detail of a natural language.
>>> In order to do so one should know every detail of a natural language.
>>> While one is finding out the language changes so that the already
>>> aquired knowledge is invalid.
>>>
>>>> When we see this then we see "incompleteness" is a mere artificial
>>>> contrivance.
>>>
>>> Hallucinations are possible but only proofs count in mathematics.
>>>
>>>> True(x) always means that a connection to a semantic
>>>> truthmaker exists. When math does this differently it is simply
>>>> breaking the rules.
>>>
>>> Mathematics does not make anything about "True(x)". Some branches care
>>> about semantic connections, some don't. Much of logic is about comparing
>>> semantic connections to syntactic ones.
>>>
>>>>>>> Many theories are incomplete,
>>>>>>> intertionally or otherwise, but they don't restrict the rest of 
>>>>>>> math.
>>>>>>> But there are areas of matheimatics that are not yet studied.
>>>>>>>
>>>>>>>> When-so-ever any expression of formal or natural language X lacks
>>>>>>>> a connection to its truthmaker X remains untrue.
>>>>>>>
>>>>>>> An expresion can be true in one interpretation and false in another.
>>>>>>
>>>>>> I am integrating the semantics into the evaluation as its full 
>>>>>> context.
>>>>>
>>>>> Then you cannot have all the advantages of formal logic. In 
>>>>> particular,
>>>>> you need to be able to apply and verify formally invalid inferences.
>>>>
>>>> All of the rules of correct reasoning (correcting the errors of
>>>> formal logic) are merely semantic connections between finite strings:
>>>
>>> There are no semantic connections between uninterpreted strings.
>>> With different interpretations different connections can be found.
>>>
>>
>> When we do not break the evaluation of an expression of language
>> into its syntax and semantics such that these are evaluated
>> separately and use something like Montague Semantics to formalize
>> the semantics as relations between finite strings then
>>
>> it is clear that any expression of language that lacks a connection
>> through a truthmaker to the semantics that makes it true simply remains
>> untrue.
> 
> Of course, completness can be achieved if language is sufficiently
> restricted so that sufficiently many arithemtic truths become 
> inexpressible.
> 

This does not make any sense to me. It is not that truth remains 
inexpressible. We simply make the system expressible enough that
all of those truths made true through a truthmaker connection to
their formalized semantic meaning can reach this semantic meaning.

> It is far from clear that a theory of that kind can express all arithmetic
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