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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: Mathematical incompleteness has always been a misconception ---
 Tarski
Date: Sat, 8 Feb 2025 21:39:53 -0600
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On 2/8/2025 9:31 PM, Richard Damon wrote:
> 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.
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