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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Mathematical incompleteness has always been a misconception ---
 Tarski
Date: Sat, 8 Feb 2025 22:31:05 -0500
Organization: i2pn2 (i2pn.org)
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On 2/8/25 9:45 PM, olcott wrote:
> On 2/8/2025 4:28 PM, Richard Damon wrote:
>> On 2/8/25 10:32 AM, olcott wrote:
>>> 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.
>>
>> But no one has been claiming that, so you are just fighting strawmen.
>>
>> The problem is these links can be infinite, and proofs must be finite.
>>
> 
> Math is only incomplete when it is intentionally defined
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