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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: How a True(X) predicate can be defined for the set of analytic
 knowledge (HoTT)
Date: Sat, 22 Mar 2025 13:27:10 -0500
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On 3/22/2025 12:34 PM, Richard Damon wrote:
> On 3/22/25 12:11 PM, olcott wrote:
>> On 3/22/2025 8:37 AM, Richard Damon wrote:
>>> On 3/21/25 11:14 PM, olcott wrote:
>>>> On 3/21/2025 9:31 PM, Richard Damon wrote:
>>>>> On 3/21/25 8:47 PM, olcott wrote:
>>>>>> On 3/21/2025 6:49 PM, Richard Damon wrote:
>>>>>>> On 3/21/25 8:49 AM, olcott wrote:
>>>>>>>> On 3/21/2025 3:57 AM, Mikko wrote:
>>>>>>>>> On 2025-03-20 15:02:42 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>> On 3/20/2025 8:09 AM, Mikko wrote:
>>>>>>>>>>> On 2025-03-20 02:42:53 +0000, olcott said:
>>>>>>>>>>>
>>>>>>>>>>>> It is stipulated that analytic knowledge is limited to the
>>>>>>>>>>>> set of knowledge that can be expressed using language or
>>>>>>>>>>>> derived by applying truth preserving operations to elements
>>>>>>>>>>>> of this set.
>>>>>>>>>>>
>>>>>>>>>>> A simple example is the first order group theory.
>>>>>>>>>>>
>>>>>>>>>>>> When we begin with a set of basic facts and all inference
>>>>>>>>>>>> is limited to applying truth preserving operations to
>>>>>>>>>>>> elements of this set then a True(X) predicate cannot possibly
>>>>>>>>>>>> be thwarted.
>>>>>>>>>>>
>>>>>>>>>>> There is no computable predicate that tells whether a sentence
>>>>>>>>>>> of the first order group theory can be proven.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Likewise there currently does not exist any finite
>>>>>>>>>> proof that the Goldbach Conjecture is true or false
>>>>>>>>>> thus True(GC) is a type mismatch error.
>>>>>>>>>
>>>>>>>>> However, it is possible that someone finds a proof of the 
>>>>>>>>> conjecture
>>>>>>>>> or its negation. Then the predicate True is no longer complete.
>>>>>>>>>
>>>>>>>>
>>>>>>>> The set of all human general knowledge that can
>>>>>>>> be expressed using language gets updated.
>>>>>>>
>>>>>>> And thus your concept of truth breaks.
>>>>>>>
>>>>>>> Truth, by its definition is an immutable thing, but you just 
>>>>>>> defined it to be mutable.
>>>>>>>
>>>>>>> How often do we need to re-verify our truths?
>>>>>>>
>>>>>>>>
>>>>>>>>>> When we redefine logic systems such that they begin
>>>>>>>>>> with set of basic facts and are only allowed to
>>>>>>>>>> apply truth preserving operations to these basic
>>>>>>>>>> facts then every element of the system is provable
>>>>>>>>>> on the basis of these truth preserving operations.
>>>>>>>>>
>>>>>>>>> However, it is possible (and, for sufficiently powerful sysems, 
>>>>>>>>> certain)
>>>>>>>>> that the provability is not computable.
>>>>>>>>>
>>>>>>>>
>>>>>>>> When we begin with basic facts and only apply truth preserving
>>>>>>>> to the giant semantic tautology of the set of human knowledge
>>>>>>>> that can be expressed using language then every element in this
>>>>>>>> set is reachable by these same truth preserving operations.
>>>>>>>>
>>>>>>>
>>>>>>> But you aren't begining with basic facts, but with what has been 
>>>>>>> assumed to be the basic facts. 
>>>>>>
>>>>>> That is not what I stipulated.
>>>>>> When we begin with what actual are the set of basic
>>>>>> facts and are only allowed to apply truth preserving
>>>>>> operations to these basic facts then it is self-evident
>>>>>> that True(X) must always be correct.
>>>>>
>>>>> But you can't stipulate that you cant' get to things that you can 
>>>>> get to.
>>>>>
>>>>> If your system can define the Natural Numbers, then we get Godel 
>>>>> and Tarski, and you can't stop it.
>>>>>
>>>>
>>>> The entire semantics of G is defined in the body of human general
>>>> knowledge that can be expressed in language henceforth called (BOK).
>>>
>>> Yes, and that is that there does not exist a number that satifies a 
>>> particular involved Primative Recursive Relationship.
>>>
>>
>> That you provide reasonable replies that show good
>> insight some of the time seems to prove that you
>> are capable of having good insight.
> 
> So, you admit that I shows you something that breaks your claim?
> 

Not at all. What I said and you agreed with
it that G is provable in (GKEUL) in the same
way the G is provable in meta-math.

>>
>>>>
>>>> The whole language metalanguage thing is already taken care of
>>>> in a hierarchy of types that expresses multiple levels of logic
>>>> in the same formal system and formal language.
>>>
>>> Nope, it is clear you just don't understand what the metalanguage does, 
>>
>> The generic term meta-language is this:
>>    The truth definition itself was to be a definition of
>>    True in terms of the other expressions of the metalanguage.
>> https://plato.stanford.edu/entries/tarski-truth/#ObjLanMet
>>
>> I use Montague Grammar of natural language semantics to
>> denote the semantic meanings of other terms in this
>> same language using Rudolf Carnap Meaning postulates.
>> {cat} <is a type of> {Animal}
> 
> Which doesn't answer the question.
> 

A type hierarchy does accomplish the aspect of Tarki's
metalanguage that I quoted.

   "The truth definition itself was to be a definition of
   True in terms of the other expressions of the metalanguage."

>>
>>> it NUMBERS all the symbols and axioms of the system. A system can not 
>>> number itself, as the numbering creates axioms that would then need 
>>> to be numbered, and that makes the system infinite. This allows us to 
>>> convert ALL logic into mathematics
>>>
>>
>> The basic facts of the body of general knowledge that
>> can be expressed using language are the finite set of
>> all facts that cannot be derived from other facts.
> 
> fine.
> 

Great!

>>
>> Cats <are> {Animals} // basic fact
>> {Animals} <are> {Living Things} // basic fact
>> Therefore {cats} <are> {Living Things} // derived fact
> 
> Fine, you can answer simple questions, but you still can't handle the 
> tough ones where the truth is established by the infinite chain.
> 

The knowledge tree has no infinite chains.
Unless an infinite chain is algorithmically
compressed to a finite sequence it is not
an element of any set of knowledge.

>>
>>>>
>>>> https://en.wikipedia.org/wiki/History_of_type_theory#G%C3%B6del_1944
>>>>
>>>
>>> Which has nothing to do with this problem. Your problem is you don't 
>>> actually understand what this means, and have replaced words with 
>>> different meanings, and thus invalidated the truths in it.
>>>
>>
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