The Godel-sentence G is proven true in a meta-theory that is ordinary arithmetic. It is not at all controversial that in plain arithmetic the Godel-sentence is true.

It is completely a confused notion that G is false.

/

If the poser were sincere about discovery in mathematics, he'd use his own terminology and define it, rather than piling on confusion by conflation with standard terminology. Let alone that he'd sit himself down to learn the basics of the subject starting with a textbook in introductory symbolic logic rather than misconstruing and misrepresenting stray bits of poorly written and intellectually disorganized Wikipedia quotes.

And, just as I discussed, there we have the poster saying that he's a source of brand new ideas while all his interlocuters are wrong: the fantasy that he's a brilliant maverick math hero. And The Philosophy Forum is just one of the Internet forums that he uses to act out his fantasy.

But I applaud The Philosophy Forum for its toleration, allowing even the most incorrigible cranks to start thread after redundant thread, spewing disinformation like a crudely written bot.

The Godel-sentence G is proven true in a meta-theory that is ordinary arithmetic. It is not at all controversial that in plain arithmetic the Godel-sentence is true.

It is completely a confused notion that G is false. — TonesInDeepFreeze

I didn't even say that G was false.
According to the new foundation of True(L, x) that I provided in my original post
when neither G nor ~G can be proven in PA then G is neither true nor false in PA.

Diagonalization and and Gödelization are not any ordinary arithmetic what-so-ever.
Here is all that there is to PA https://en.wikipedia.org/wiki/Peano_axioms
There is no Diagonalization or Gödelization in PA.

The poster said that G is untrue. Now he says he did not say it is false.

/

Godel numbering and diagonalization use only arithmetic. Godel numbering and diagonalization use arithmetical operations in very complicated and surprising ways, but nothing beyond arithmetic is used. When one actually reads the proofs, one sees that each step is utterly unassailable mathematics, well within arithmetic even if a complicated and ingenious use of arithmetic. Indeed, the proof can be done in finitistic intuitionistically acceptable arithmetic. If there is a context that is epistemologically safer than that, then I'd like to know what it is.

The poster said that G is untrue. Now he says he did not say it is false. — TonesInDeepFreeze

To understand what I am saying requires knowledge of truth-maker maximalism that you seem to lack.

The poster says that understanding that a sentence can be both untrue but not false requires knowing about his own undefined "truth-maker maximalism".

More pertinent is that understanding this subject requires knowing, as the poster does not, at least the basics of symbolic logic with the basics of mathematical logic to follow.

But that is the way of the crank: The requirement that everyone else accept the crank's undefined terminology, impressionistic musings, and illogical arguments while the crank himself has no responsibility to learn even page one of an introductory textbook.

And the poster links to a Wikipedia specification of PA and points out that it does not include Godel-numbering or diagonalization.

As if that makes any point at all here!

A specification of PA itself does not include mention of all kinds of results in arithmetic, from the fact that there is no greatest prime to the fundamental theorem of arithmetic to the most advanced and complicated theorems about the natural numbers ... and to Godel-numbering and diagonalization. So what? It doesn't entail that those developments are not provided for, or pertain to PA.

Here's a pretty good analogy: The rules of chess make no mention of the incredibly complicated strategies of chess masters, but that does not entail that those strategies are not permitted by the rules nor that analysis of those strategies does not pertain to chess.

And as the poster touts the Wikipedia article, he conveniently omits including that the article itself DOES mention the incompleteness theorem!

What a seriously risible argument the poster makes! Really, the poster is as hopelessly ignorant, confused and irrational as they come. I've seen some that are more dishonest, but the poster ranks fairly high in dishonesty too, as just witnessed that he touts a Wikipedia article that actually shows the OPPOSITE of his own claim!

What a seriously risible argument the poster makes! Really, the poster is as hopelessly ignorant, confused and irrational as they come. I've seen some that are more dishonest, but the poster ranks fairly high in dishonesty too, as just witnessed that he touts a Wikipedia article that actually shows the OPPOSITE of his own claim! — TonesInDeepFreeze

All you have is ad hominem and cannot point out any actual errors in the essence of my reasoning.
Here is the essence of my reasoning:

Every expression of language X that is {true on the basis of its meaning} can only be verified as true on the basis of a connection to this meaning.

Ah, another fallacy from the poster. The fact that personal comments are added to substantive comments does not entail that the substantive comments have not been amply given.

I've given exact detailed technical explanations of how the poster has been wrong in so many ways. Discussing also that the poster is a crank does not erase the technical explanations.

The poster is flat out lying that I have not shown errors in his reasoning. My refutations have been copious.

And notice that the poster yet again evades dealing with the very quote of mine that he posts. To wit:

The poster's reasoning includes giving evidence that Wikipedia's article on the Peano axioms does not mention Godel numbering and diagonalization. But (1) The axioms don't mention a lot of things, not even such things as the fundamental theorem of arithmetic. The axioms don't mention Godel numbering and diagonalization but that doesn't entail that they are not basically applications of arithmetic. Indeed, finitistic arithmetic. (2) The Wikipedia article actually DOES discuss the incompleteness theorem regarding PA.

And that is just the latest in a long chain of evasions and misrepresentations by the poster, not even counting all the other redundant threads he's propagated on these subjects.

And his latest message misses the point also:

The question is not ascertaining what is true. Mathematical logic has a rigorous definition of 'true' in which sentences are evaluated first with their atomic components such that an atomic sentence is true if and only if it corresponds to the given state of affairs.

Rather, the question has been about entailment. And the principle of explosion doesn't say that the conclusion is true, only that a contradiction entails any statement. That is, since there are no states of affairs in which a contradiction is true, there are no states of affairs in which a contradiction and any other sentence are together true. Regardless of how we reckon the truth of atomic statements, there are no circumstances in which a contradiction is true thus no circumstances in which both a contradiction and any conclusion are both true.

/

Getting back to the key point that started this part of the discussion:

The poster claims that classical logic is not truth preserving. But it is, as I have explained (and it is proven). And the fact that the poster prefers his own vague, undefined and confused outlook on logic does not entail that classical logic is not truth preserving.

Truth preservation is: If the premises are true then the conclusion is true. And that is PROVABLY upheld by classical logic.

Truth preservation is: If the premises are true then the conclusion is true. And that is PROVABLY upheld by classical logic. — TonesInDeepFreeze

That part is correct yet simply ignores the actual point

Every expression of language X that is {true on the basis of its meaning} can only be verified as true on the basis of a connection to this meaning. — PL Olcott

Therefore:
True(PA, G) == false
True(PA, ~G) == false.

The poster responds yet again to refutations by merely reposting his pet statements. Either the poster suffers from repetition compulsion or the poster is a bot.

And True(PA, G) has no apparent meaning.

A sentence is true or not per a model, not per a theory. Though, of course, a sentence may be true in all models of a theory, which reduces to the sentence being a theorem of the theory.

The poster has to be bot. — TonesInDeepFreeze

In other words you believe that there are are sequence of truth
preserving operations from the axioms of PA to G or to ~G.

When I prove my point ALL YOU HAVE IS AD HOMINEN.

The poster writes two sentences that are lies.

The poster lies that I believe that PA proves G or it proves ~G. It is the opposite. PA proves neither G nor ~G. That is the very statement of incompleteness.

The poster not only fails to desist from (let alone retract) his previous lie that I have not addressed the subject substantively, but he reposts it in all capital letters.

The poster lies that I believe that PA proves G or it proves ~G. It is the opposite. PA proves neither G nor ~G. That is the very statement of incompleteness. — TonesInDeepFreeze

I am establishing a brand new foundation for analytical truth and simply ignoring that I
am doing this is no actual rebuttal at all.

Whatever else the poster thinks he is doing, he claimed that classical logic is not truth preserving. I explained why that is false. The poster still refuses to understand the matter. To explain that classical logic is truth preserving, one doesn't have to affirm the poster's grandiose vision that from the mountain top he is bringing to the world the stone tablets of a new foundation.

Whatever else the poster thinks he is doing, he claimed that classical logic is not truth preserving. I explained why that is false. The poster still refuses to understand the matter. — TonesInDeepFreeze

Can you manage to stay focused on the point at hand?

Every expression of language x that is {true on the basis of its meaning}
can only be verified as true on the basis of a connection to this meaning.
This does enable a True(L, x) predicate to be defined where L is a formal
language of a formal system.

I am precisely focused on the central point from which this discussion is pursuant:

Contrary to the poster's false claim, classical logic is truth preserving.

And these points that the poster lacks focus to understand:

Contrary to the poster's confusion, the principle of explosion accords with truth preservation.

Contrary to the poster's ignorance, classical logic handles the notion of truth in terms of correspondence with states of affairs.

Contrary to the poster's addlement, truth and entailment are different notions. A set of premises entails a conclusion if and only if there are no states of affairs in which the premises are all true and the conclusion is false. Whether or not the conclusion is or is not false is a different question.

Contrary to the poster's ineducation and lack of intellectual curiosity, there is an approach to logic called 'relevance logic' in which the conditional is handled with regard to the contentual features of the antecedent and consequent, as that study is, unlike the poster's pronouncements, informed and rigorous.

Contrary to the poster's intellectual recalcitrance, merely to refute the poster's falsehoods about classical logic, one is not required to indulge the poster in his confused notions about some claimed new foundation he thinks he has created.

That the poster is monomaniacal about some pet idea of his own doesn't entail that anyone else who points out his copious errors about classical logic lacks focus for not nodding in acceptance of his spammy gibberish.

I am precisely focused on the central point from which this discussion is pursuant: — TonesInDeepFreeze

Can you manage to stay focused on the point at hand?

Every expression of language x that is {true on the basis of its meaning}
can only be verified as true on the basis of a connection to this meaning.
This does enable a True(L, x) predicate to be defined where L is a formal
language of a formal system.

Whatever the shortcomings of the poster, at least he demonstrates skill in the use of Ctrl-C and Ctrl-V.

This isn't a particularly productive discussion.

This isn't a particularly productive discussion. — fdrake

It is not my fault that people want to change the subject away from the original post.
I have found that allowing people to do this to play Trollish head games is not
productive. The strawman deception of changing the subject as a form of fake
rebuttal never works with me.

It is not my fault.... — PL Olcott

It is exactly your fault, Olcott. By my count through at least 450 posts in good will and good faith made in an attempt to gain any clarity about what you are talking about, you have dodged, evaded, and avoided every attempt, content to make and repeat nonsense claims, and when pressed to change the subject.
Not a good look for you, and to my way of thinking making it impossible to have any respect for you. Sympathy? Maybe. Respect - which also implies trust - no.

As to these topics, your ideas are useless for being nonsensical, and useless to engage with you because you are simply nonresponsive.

It is exactly your fault, Olcott. By my count through at least 450 posts in good will and good faith made in an attempt to gain any clarity about what you are talking about, you have dodged, evaded, and avoided every attempt, content to make and repeat nonsense claims, and when pressed to change the subject.
Not a good look for you, and to my way of thinking making it impossible to have any respect for you. Sympathy? Maybe. Respect - which also implies trust - no. — tim wood

I just told you what I want to talk about so we can skip all of the other posts
I simplified what I want to talk about so that it will be easier for you to focus
your attention on this one single idea the follows:

Every expression of language x that is {true on the basis of its meaning}
can only be verified as true on the basis of a connection to this meaning.
This does enable a True(L, x) predicate to be defined where L is a formal
language of a formal system.

This isn't a particularly productive discussion. — fdrake

I agree.

But sometimes a tree looks nice to bang your head into. I would be very thankful if someone gave their time to answer my ideas as @TonesInDeepFreeze and @tim wood have given to @PL Olcott on this thread.

Every expression of language x that is {true on the basis of its meaning}
can only be verified as true on the basis of a connection to this meaning.
This does enable a True(L, x) predicate to be defined where L is a formal
language of a formal system. — PL Olcott

Let's debate this from another angle.

In mathematics, do you think there can be true, but unprovable statements?

Do you think that all true statements are also provable?

And finally, can an indirect reductio ad absurdum proof prove something?

Yes on No answers would be appreciated (of course with reasoning too).

In mathematics, do you think there can be true, but unprovable statements? — ssu

When I provide a simple yes/no answer all that I get is ad hominem attacks
without anyone even looking at what I said. So I encode my yes/no answer
in the reasoning used to derive that yes/no answer.

True and unprovable never means EXACTLY what it says:
We know that X is true and have no way what-so-ever to know
that X is true yet we know it is true anyway, as if by majick.

When I provide a simple yes/no answer all that I get is ad hominem attacks
without anyone even looking at what I said. — PL Olcott

I've not made any ad hominem attacks. Please understand that just repeating the same thing will get tempers to rise. Always true to really think what the other one is trying to say.

True and unprovable never means EXACTLY what it says:
We know that X is true and have no way what-so-ever to know
that X is true yet we know it is true anyway, as if by majick. — PL Olcott

But we do! We can give an indirect proof.

Do you understand how an reductio ad absurdum proof goes? That's why I asked the third question.

This is quite crucial here in my view.

But we do! We can give an indirect proof. — ssu

Then is never really was literally unprovable.
True yet cannot possibly be proved in any way what-so-ever
does not allow indirect proof.

A sequence of inference steps in PA that do not derive G only says G
cannot be proved in PA it does not say that G cannot be proved.

Then is never really was literally unprovable.
True yet cannot possibly be proved in any way what-so-ever
does not allow indirect proof. — PL Olcott

OK, it seems where the problem lies and just why you had this long argument with @TonesInDeepFreeze and @tim wood (both or one, some pages ago). This is very important to understand here. Giving a direct proof and giving an indirect proof aren't the same thing. Also proving something and a mathematical statement being true aren't exactly the same thing.

Because let's assume the following statement S

S = This statement is unprovable

So how can you prove this? Well, you prove it by reductio ad absurdum. So let's assume the opposite is true and hence statement S is provable and then go and prove that this cannot be. . Did you give a direct proof? No. You didn't prove S. You proved that not-S is false.

Hence I'll put again to you the last question:

Can an indirect reductio ad absurdum proof prove something in your view?

(Cantor's diagonalization is the easiest way in my view to understand this: with diagonalization we have shown that not all reals would be in the list, but of course this opens up questions like the Continuum Hypothesis.)

So how can you prove this? Well, you prove it by reductio ad absurdum. So let's assume the opposite is true and hence statement S is provable and then go and prove that this cannot be. . Did you give a direct proof? No. You didn't prove S. You proved that not-S is false. — ssu

We must start with the common lack of sufficient precision of your first statement. Most everyone makes tis same mistake.

In mathematics, do you think there can be true, but unprovable statements? — ssu

There cannot possibly be any expression of language that is true and does not have a truth-maker making it true. When it is said that G is true and unprovable it never means EXACTLY what it says.

If we don't start from the exact same common ground then we never get to mutual agreement, thus we must first agree that true and not provable by any means what-so-ever is contradictory.

Either a mathematical statement S is true or not-S (the negation of S) is true. S and not-S cannot be both true in mathematics. (Either the statement 1+1=3 is true or then 1+1=3 is false is true. That 1+1=3 is false is true.)

A direct proof would be simply to prove S. And indirect proof is to prove not-S is false. Since we assume that mathematics (or logic) is consistent, we do admit the indirect proof and say that S is true if not-S is false. But it's not a direct proof.

There cannot possibly be any expression of language that is true and does not have a truth-maker making it true. — PL Olcott

Truth-maker making it true means that there's a proof that it is true? Sorry, with negative self reference you can easily do that.

Like try to give an answer here that you don't give here. Are there those kinds of answers? Obviously yes. Can you give them? No.

When it is said that G is true and unprovable it never means EXACTLY what it says. — PL Olcott

What do you mean by this? Again, the ability to give a direct proof and something to be true are two different things.

The logic behind negative self reference is quite easy to understand.

When it is said that G is true and unprovable it never means EXACTLY what it says.
— PL Olcott
What do you mean by this? Again, the ability to give a direct proof and something to be true are two different things. — ssu

If there is no possible way to know that expression X is true then we can't possibly know
that expression X is true. AKA when X lacks a truth-maker then X is not true.

We must get through this key point first because it is the core foundation of everything
that I am saying. I oversimplified this a little bit so that you can get the gist of what I am saying.