XPost: comp.theory, sci.logic, sci.math
From: 046-301-5902@kylheku.com

On 2025-11-29, olcott  wrote:
> On 11/29/2025 3:39 PM, Kaz Kylheku wrote:
>> On 2025-11-29, olcott  wrote:
>>> On 11/29/2025 2:23 PM, Kaz Kylheku wrote:
>>>> On 2025-11-29, olcott  wrote:
>>>>> On 11/29/2025 11:53 AM, Kaz Kylheku wrote:
>>>>>> On 2025-11-29, olcott  wrote:
>>>>>>> Any expression of language that is proven true entirely
>>>>>>> on the basis of its meaning expressed in language is
>>>>>>> a semantic tautology.
>>>>>>
>>>>>> A tautology is an expression of logic which is true for all
>>>>>> combinations of the truth values of its variables and propositions,
>>>>>> which is, of course, regardless of what they mean/represent.
>>>>>
>>>>> I did not say tautology. I said semantic tautology.
>>>>> I am defining a new thing under the Sun.
>>>>
>>>> The existing tautology is already semantic. You have to know the
>>>> semantics (the truth tables of the logical operators used in the
>>>> formula, and the workings of quantifiers and whatnot) to be able to
>>>> conclude whether a formula is a tautology.
>>>>
>>>
>>> Try and show how Gödel incompleteness can be
>>> specified in a language that can directly encode
>>> self-reference and has its own provability operator
>>> without hiding the actual semantics using Gödel numbers.
>>
>> The numbers are essential, because Gödel Incompleteness is
>> about number theory.
>>
>
> The generalization Gödel incompleteness applies to
> every formal system that has arithmetic or better.

And there you are, trying to take the numbers out of it.

>> The Gödel Theorem involves a proof in which a certain number,
>> the "Gödel number" that may be called G, is asserted to have
>> a number-theoretical property.
>>
>
> G := (F ⊬ G) // G says of itself that it is unprovable in F

No, it doesn't; that is an outside interpretation of what it is saying.
Gödel's sentence says that a certain number isn't a theorem-number.

The interpretation that the number is the Gödel number of
that very sentence is made externally to the sentence.

Is there any part of your understanding that is accurate?

>> An example of a number-theoretical property is "25 is a perfect
>> square". Except we need it in more formal language.
>>
>> Gödel discovered that you can encode statements of number theory as
>> integers, and manipulate them (e.g. do derivation) by arithmetic.
>>
>
> That simply abstracts away the underlying semantics.
> G is unprovable in F because G is semantically unsound,

G is semantically sound, and can be adopted as an axiom.

> We can't see that with Gödel numbers.

A Gödel number can be decoded to recover the syntas of the formula.

In the case of the Gödel sentence, we don't need to do that; we
already know that the Gödel number decodes to that sentence.

>> Then it became obvious that whether or not a formula is a theorem
>> is a property of its Gödel number: a number-theoretical property.
>>
>> There are theorem-numbers and non-theorem-numbrers.
>>
>> The Gödel sentence says somethng like "The Gödel number
>> calculated by the expression G is not a theorem-number."
>>
>> But G turns out to be the Gödel number of that very sentence
>> itself.
>>>
>>>> Pick another word. Since only dimwitted crackpots like yourself will
>>>> want to discuss anything using that word, keep the syllable count low
>>>> and make sure there aren't too many off-centre vowels.
>>>
>>> Ad hominem the first choice of losers.
>>
>> I'm not making an argument; I'm suggesting a way of choosing
>> an alternative word, since "tautology" is taken.
>>
>>>>> *Semantic tautology is stipulated to mean*
>>>>
>>>> Reject; call it something else.
>>>>
>>>>> Any expression of language that is proven true entirely
>>>>> on the basis of its meaning expressed in language.
>>>>
>>>> You are gonna need to supply an example.
>>>
>>> The key is that a counter-example is categorically
>>> impossible.
>>
>> So you are saying every expression in a certain language
>> is proven true, so that its syntax admits no false sentences?
>
> It syntax admits anything that any human can
> say in any language comprised of symbols.

But that could be false. It is baffling by what you mean
bhy "counter-example is categorically impossible"; at ths point
it seems like a dodge from giving an example of sentence
that is proven true entierly on the basis of its meaning
expressed in language.

>> What language is that, and what are examples? What happens
>> when you try to make a false sentence?
>
> English, Second Order Predicate logic, C++...

How does C++ express a sentence that is proven entirely true
on the basis of its meaning expressed in a language;
do you need templates or Boost?

>> Is it possible to utter conjectures which later turn out false;
>> and if so, then what happens?
>
> Conjectures are not elements of the body of knowledge.

Some eventually are; but their syntax and meaning doesn't change.

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