What is religion ?

[Skepdick]

Soundness implies consistency.
Inconsistency implies unsoundness.
Unsoundness, however, does not imply anything else.
The ability to prove a falsehood from a theory makes it unsound.

In proof theory there's no such thing as proving truth or falsehood.

A theorem is either a consequence of some axioms; or it's not.
Either holds - or it doesn't.

Truth and falsehood are not well-defined notions.

As I originally said, PA could be unsound and therefore could possibly prove falsehoods.

No idea what that even means. If you can prove it - it's a theorem.

You could prove a negative statement as a theorem; or a positive statement as a theorem.

Soundness in formal languages amounts to nothing other than preserving validity. It has fuckall to do with truth.

[Skepdick]

Wrong.

False

Soundness is (or is not) a theorem in a particular axiomatic system.

Nonsense. https://en.wikipedia.org/wiki/Soundness

Soundness has a related meaning in mathematical logic, wherein a formal system of logic is sound if and only if every well-formed formula that can be proven in the system is logically valid with respect to the logical semantics of the system.[

The Mathematical notion of "soundness" is about validity.

No wonder you are confused. You don't even understand the difference between logic and mathematics.

The Soundness Theorem is the theorem that says that if Σ⊢σ in first-order logic, then Σ⊨σ, i.e. every structure making all sentences in Σ true also makes σ true.

In what axiomatic system are "Σ⊢σ" and "Σ⊨σ" valid sentences?

In your systemless approach, you fail to realize that the soundness of the system-wide premises of the theoretical context is a theorem of such theory. Hence, soundness theorem.

Dumb systemizer. Soundness in formal systems is nothing other than validity - the continuous preservation of properties under transformation

If ALL the sentences were false - they must remain false after transformation.
If ALL sentences were true - they must remain true after transformation.
If all sentences were cheesy - they must remain cheesy after transformation.

Now, you can construct semantics for the formal theory in a meta-theory, but guess what? The meta-theory is just anothe formal system.

So you can construct semantics for that... and you have a meta-meta theory. Which is another formal system....
What you end up with is a hierarchy of theories. Ad infinitum.

Yes, the soundness theorem can be provable within a particular formal system or theory, but it is not a given or inherent property of all theories. Whether the soundness theorem is provable depends on the logical system or formal framework you are working within.

Q.E.D

Any system which can prove its own soundness is necessarily unsound. Don't you know this?

[godelian]

In proof theory there's no such thing as proving truth or falsehood.

You are using the wrong tool for the wrong purpose. Proof theory does not deal with truth. It also does not deal with soundness. The correct tool for this job is model theory.

A theorem is either a consequence of some axioms; or it's not.
Either holds - or it doesn't.

If you look at proof theory alone, which is syntax only, it will give the impression that truth in mathematics is defined as a coherence theory:

https://en.m.wikipedia.org/wiki/Coheren ... y_of_truth

Coherence theories of truth characterize truth as a property of whole systems of propositions that can be ascribed to individual propositions only derivatively according to their coherence with the whole.

This is the wrong view on truth in mathematics. Model theory decisively defines truth in terms of a correspondence theory:

https://en.m.wikipedia.org/wiki/Corresp ... y_of_truth
In metaphysics and philosophy of language, the correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world and whether it accurately describes (i.e., corresponds with) that world.[1]

Correspondence theories claim that true beliefs and true statements correspond to the actual state of affairs.

Truth in mathematics is the correspondence between a theory and its model(s).

Truth and falsehood are not well-defined notions.

You write that because you are not familiar with Tarski's semantic theory of truth:

https://en.m.wikipedia.org/wiki/Semanti ... y_of_truth

The semantic conception of truth, which is related in different ways to both the correspondence and deflationary conceptions, is due to work by Polish logician Alfred Tarski.

A model is a collection of facts along with an interpretation of the non-logical symbols in the theory (the "signature ") that "interprets" the theory.

This collection of facts is the truth in mathematics:

https://en.m.wikipedia.org/wiki/Model_theory

In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).

For example, true arithmetic is the standard model for PA:

https://en.m.wikipedia.org/wiki/True_arithmetic

In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers.[1] This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms.

Hence, the truth about the natural numbers certainly exists. A statement about the natural numbers is true when it appears as a fact in the universe of true arithmetic.

As I originally said, PA could be unsound and therefore could possibly prove falsehoods.

No idea what that even means. If you can prove it - it's a theorem.

Because of Godel's incompleteness theorem, we cannot exclude that PA could prove a false theorem: Hence, your claim is utterly simplistic:

ChatGPT: Not assuming anything unprovable about the consistency of Peano Arithmetic, does the possibility exist that Peano Arithmetic proves a theorem that is false in all its models?

Yes, it is possible that Peano Arithmetic (PA) could prove a theorem that is false in all its models, assuming we do not assume the consistency of PA. This possibility arises from the fact that PA could be inconsistent, and in an inconsistent system, any sentence (including false sentences) can be proven.

Godel's incompleteness theorem proves that PA is inconsistent or incomplete, or possibly even both inconsistent and incomplete. That is why you can never exclude the possibility that PA proves a completely false theorem.

Soundness in formal languages amounts to nothing other than preserving validity. It has fuckall to do with truth.

Wrong!

The truth of a theory are the facts in its model.

Soundness is a property of the correspondence between a theory and its model.

You don't understand the truth in mathematics because you are unfamiliar with model theory, and especially, with Tarski's work on the matter, more precisely, Tarski's semantic theory of truth.

[godelian]

The Mathematical notion of "soundness" is about validity.

In proof theory, it is about validity. In model theory, it is about truth. Proof theory is about syntactic entailment only. It does not say anything about truth.

No wonder you are confused. You don't even understand the difference between logic and mathematics.

You are clearly ignorant of a core part of mathematical logic, i.e. model theory. Pretty much everything you write is in violation of Alfred Tarski's work on the matter.

In what axiomatic system are "Σ⊢σ" and "Σ⊨σ" valid sentences?

It is from a lecture on first-order logic. I repeat again: logic is itself an axiomatic system. You really seem to believe that systemless reasoning is possible, but seriously , it isn't. Welcome to the real world.

Dumb systemizer

Logic alone, is already a system. Get over it!

Any system which can prove its own soundness is necessarily unsound. Don't you know this?

Wrong!

Only when Godel's incompleteness theorems are provable from such system, it cannot prove its own consistency or its own soundness. Simpler systems do not have that problem.

ChatGPT: Can propositional logic prove its own consistency?

Yes, propositional logic can prove its own consistency. This is because propositional logic is much simpler and more limited compared to first-order logic (or systems that involve arithmetic).

ChatGPT: Can propositional logic prove its own soundness?

Yes, propositional logic can prove its own soundness. This is because propositional logic is simple and finite enough that its soundness can be demonstrated directly, without the complications that arise in more complex systems like first-order logic.

Propositional logic can prove its own consistency and its own soundness without therefore being inconsistent or unsound. That is already one counterexample to your claim about "any system".

[Skepdick]

In proof theory, it is about validity. In model theory, it is about truth. Proof theory is about syntactic entailment only. It does not say anything about truth.

No, in proof theory it's all about validity. In model theory it's all about validity-preserving transformation.

You know - because Mathematics has nothing to do with truth.

You are clearly ignorant of a core part of mathematical logic, i.e. model theory. Pretty much everything you write is in violation of Alfred Tarski's work on the matter.

It's in line with Tarski's work, not in violation. Tell me you have no idea what you are talking about without telling me.

It is from a lecture on first-order logic.

Q.E.D I asked you for a model-theoretic (Tarskian) formal model which gives meaning to these symbols and you couldn't even do that.

I repeat again: logic is itself an axiomatic system.

No, it's not. Logic is just a deductive aparatus. It's axioms that give you the inference rules.

I don't expect a dumb set theorist to understand this. Since set theory has no built-in deductive aparatus. The axioms of set theory interact (do you even understand the Geometry of Interaction) with the deductive aparatus.

it's a known fact (in homotopy type theory - which unifies deduction with inference) that set theory has no deductive aparatus itself and depends on first order logic. It has no commitment to higher order logic.

You really seem to believe that systemless reasoning is possible, but seriously , it isn't. Welcome to the real world.

You really seem to believe that just because you can't cope with systemless reasoning it must be impossible.

Logic alone, is already a system. Get over it!

Precisely. And the logician is a system-builder. Just just because I invent logic it doesn't mean I am a slave to it.

Wrong!

Only when Godel's incompleteness theorems are provable from such system, it cannot prove its own consistency or its own soundness. Simpler systems do not have that problem.

You really don't seem to understand Tarski; or Godel's work...

ChatGPT: Can propositional logic prove its own consistency?

Yes, propositional logic can prove its own consistency. This is because propositional logic is much simpler and more limited compared to first-order logic (or systems that involve arithmetic).

ChatGPT: Can propositional logic prove its own soundness?

Yes, propositional logic can prove its own soundness. This is because propositional logic is simple and finite enough that its soundness can be demonstrated directly, without the complications that arise in more complex systems like first-order logic.

Propositional logic can prove its own consistency and its own soundness without therefore being inconsistent or unsound. That is already one counterexample to your claim about "any system".

You are so incapable of reasoning you'll believe anything ChatGPT tells you as long as it serves your confirmation bias.

Soundness is stronger than consistency - soundness means all provable statements are true, while consistency just means you can't prove both a statement and its negation. If propositional logic could prove its own soundness, it could also prove its own consistenc, yet by Gödel's Second Incompleteness Theorem, no sufficiently complex consistent formal system can prove its own consistency. Therefore, propositional logic cannot prove its own consistency. And it definitely can't prove anything stronger than consistency e.g soundness.

Look, ma! It gives different answers to different people!

Me: Can propositional logic prove its own consistency?

ChatGPT: No, propositional logic cannot prove its own consistency in the sense typically understood within formal logic. However, this is not due to the same limitations that apply to more expressive logical systems like first-order logic.

In propositional logic:

Limited Expressive Power: Propositional logic deals only with statements that are true or false and does not have the means to express statements about the system itself, such as "This system is consistent." Therefore, the notion of proving its own consistency isn't applicable within propositional logic because it lacks the resources to even state that it is consistent.

Any system lacking self-referentiality can't prove anything about itself. And if a system has self-referentiality it can't prove its consistency; or its soundness.

This is why we need a hierarchy of stronger and stronger systems in mathematics - each system's properties must be studied from "outside" in a stronger system. You can't bootstrap your way to proving your own foundations.

Come on, child! This is basic knowledge in programming language theory - compiler bootstrapping! It's the quintessential chicken-and-egg problem.

https://en.wikipedia.org/wiki/Bootstrapping_(compilers)

[godelian]

You know - because Mathematics has nothing to do with truth.

Really? What about the following:

https://en.wikipedia.org/wiki/Tarski%27 ... ty_theorem

Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic".[1]

The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.

So, in your opinion, Tarski's undefinability theorem is not a theorem in mathematics and/or it has nothing to do with truth in mathematics?

You really seem to believe that just because you can't cope with systemless reasoning it must be impossible.

Systemless reasoning is impossible because in that case you cannot use logic which means that you cannot conclude anything. I cannot cope with systemless reasoning exactly because it is impossible.

Therefore, propositional logic cannot prove its own consistency. And it definitely can't prove anything stronger than consistency e.g soundness.
Look, ma! It gives different answers to different people!
ChatGPT: Limited Expressive Power: Propositional logic deals only with statements that are true or false and does not have the means to express statements about the system itself, such as "This system is consistent." Therefore, the notion of proving its own consistency isn't applicable within propositional logic because it lacks the resources to even state that it is consistent.
Any system lacking self-referentiality can't prove anything about itself. And if a system has self-referentiality it can't prove its consistency; or its soundness.

I agree with the problem caused by the lack of self-referentiality in propositional logic. So, in a sense, it cannot prove anything about itself. It looks like its consistency is the result of analysis in a suitable meta-theory or other secondary theory:

ChatGPT: How can you prove the consistency of propositional logic?

Proving the consistency of propositional logic involves demonstrating that it is impossible to derive a contradiction from the axioms and rules of inference of the system. In other words, we need to show that there is no formula ϕϕ such that both ϕϕ and its negation ¬ϕ¬ϕ can be derived from the axioms and inference rules of the system.

Here are the main steps and approaches to prove the consistency of propositional logic:
1. Model-theoretic approach (Semantic proof)

A common way to prove consistency is through model theory. In this approach, you show that there exists at least one interpretation (model) under which the axioms of propositional logic are satisfied, and no contradiction can be derived.

In propositional logic, a model is a truth assignment to the propositional variables (i.e., each propositional variable is assigned either true or false). An interpretation is a mapping from propositional variables to truth values.

Axioms and rules of inference: We assume the standard axioms and inference rules of propositional logic (such as modus ponens, truth tables, etc.).

Proving consistency via a model: If you can show that a model exists where all the axioms hold and no contradictions arise, then propositional logic is consistent. For example, the truth table approach is often used to show that there is no interpretation in which a contradiction (e.g., ϕϕ and ¬ϕ¬ϕ) holds. Since propositional logic is a sound and complete system, this means that no contradiction can be derived from its axioms.

2. Proof-theoretic approach (Formal proof)

Another approach to prove the consistency of a logical system is proof-theoretic, where you attempt to build a formal proof that no contradictions can be derived.

Use of a consistency proof: A consistency proof in proof theory usually shows that no derivation leads to a contradiction. For propositional logic, one common strategy is to show that a contradiction would lead to a violation of the basic principles of the system, such as the law of excluded middle or the principle of non-contradiction.

Inductive proof: A more formal method might involve using an inductive proof to show that no contradiction can arise from the axioms and inference rules of propositional logic.

3. Gödel's Second Incompleteness Theorem

In formal systems like arithmetic or set theory, proving consistency within the system itself becomes impossible due to Gödel's Second Incompleteness Theorem. However, propositional logic is a much simpler system, and it is consistent in the sense that there is no proof within the system of a contradiction (i.e., a derivation of both ϕϕ and ¬ϕ¬ϕ).

In propositional logic, it is not necessary to use Gödel’s Second Incompleteness Theorem, because propositional logic is a relatively weak formal system, and it does not lead to the same kind of self-referential issues that more complex systems (such as Peano Arithmetic) encounter.
4. Soundness and Completeness Theorem

The soundness of propositional logic states that if a formula can be derived using the axioms and inference rules of propositional logic, it is true under every interpretation (model). The completeness theorem states that if a formula is true in every model, it can be derived using the axioms and inference rules.

Since propositional logic is sound and complete, the existence of a model where the axioms hold (i.e., there is no contradiction in any interpretation) implies the consistency of propositional logic. If a contradiction were derivable, the system would not be sound, but since we know it is sound, we can conclude that no contradiction can be derived.

Conclusion

Propositional logic is consistent because:

Model-theoretically, there are interpretations where all axioms hold and no contradictions arise.
Proof-theoretically, there are no derivations of contradictions from the axioms and inference rules.
Soundness and completeness guarantee that the system cannot prove contradictions and that the truth of the system is reflected in its derivations.

This is a basic outline of how consistency is typically shown for propositional logic.

The problem is that proving the consistency of propositional logic requires the use of a second theory, of which the consistency is equally in doubt, leading to a problem of infinite regress. Propositional logic is in fact merely equiconsistent with some other theory.

The situation is clearly not better than for PA. Gentzen's consistency proof does not prove the absolute consistency of PA but merely the equiconsistency with PRA (Primitive Recursive Arithmetic):

https://en.wikipedia.org/wiki/Gentzen%2 ... ency_proof

Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradiction (i.e. are "consistent"), as long as a certain other system used in the proof does not contain any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0", is neither weaker nor stronger than the system of Peano axioms. Gentzen argued that it avoids the questionable modes of inference contained in Peano arithmetic and that its consistency is therefore less controversial.

So, ok. I admit that while ChatGPT is right to mention that on the one hand Gödel's second incompleteness theorem does not apply to propositional calculus, it indeed fails to mention that on the other hand this also means that propositional calculus is not strong enough to talk about itself and its own consistency. This omission gives the wrong impression that it can successfully prove its own consistency, which it cannot, because it lacks the tools to do so.

[godelian]

You know - because Mathematics has nothing to do with truth.

You are clearly ignorant of a core part of mathematical logic, i.e. model theory. Pretty much everything you write is in violation of Alfred Tarski's work on the matter.

It's in line with Tarski's work, not in violation. Tell me you have no idea what you are talking about without telling me.

If you reject the notion of truth in mathematics, then you effectively reject Tarski's semantic theory of truth, and then everything you say, is in violation of Tarski's work on the matter:

https://iep.utm.edu/s-truth/

The semantic theory of truth (STT, hereafter) was developed by Alfred Tarski in the 1930s. The theory has two separate, although interconnected, aspects. First, it is a formal mathematical theory of truth as a central concept of model theory, one of the most important branches of mathematical logic. Second, it is also a philosophical doctrine which elaborates the notion of truth investigated by philosophers since antiquity.

How can you reasonably deny that your view, "Mathematics has nothing to do with truth", is in violation with Tarski's work?

[Belinda]

Belinda's answer to Immanuel Can's post:

True. Also true, religious people and also non-religious may be indoctrinated and both religious and non-religious frequently are indoctrinated. Religious doctrines must be combined with liberal education which is both high quality and available to all throughout life.There must be as much autocracy as befits a liberal legal system.

Religions must be democratic and have moral codes that depend upon agape- love not fear. Many Christian , Muslim, and Eastern religious myth is supported by old codes of deference towards kings and priests--you probably know the hymn 'All Things Bright and Beautiful' 'The rich man in his castle, the poor man at his gate'. God is sung as 'ordering their estate'. When religions support castes the religions have become cultish.

True, established religions have been the only carriers of satisfactory moral codes, as historians will verify. This is why the saints and the seers of all great religious traditions are held to be saints and seers, because they have singlehandedly taught pagans more civilised ways to live together. I include of course, not only Jesus but also Muhammad , Mandela, and Confucius.

Most people at least would agree cultishness is a matter of scale, but not always. Established religions have more followers.The worst of cults is that they idolise a person who does not merit trust. Jesus ,Confucius, the Buddha, and Muhammad for instance merited absolute trust because they helped people to cooperate instead of fighting each other. Constantine by contrast merited trust only insofar as his politics were a main vehicle for the spread of Christianity (with all its technological and peaceful benefits ) across much of Europe.

Mariolatry seems harmless enough despite that it's a hangover from paganism.

Superstition! Yes that's a problem. To believe God is a person who intervenes at will in his own creation is superstition and reverts to magical thinking when people believe that rituals can and do influence God to intervene.

[Skepdick]

You know - because Mathematics has nothing to do with truth.

Really? What about the following:

https://en.wikipedia.org/wiki/Tarski%27 ... ty_theorem

What about it? You can't define truth. It's a negative result.

So, in your opinion, Tarski's undefinability theorem is not a theorem in mathematics and/or it has nothing to do with truth in mathematics?

Can you even read/understand what Traski says? He's literally telling you that you CAN'T define "arithmetical truth" (whatever the hell that is) in arithmetic! Why is 1+1=2 true?

More broadly - you can't define Mathematical truth in Mathematics.The theorem is literally about the NON-truth of Mathematics.

Systemless reasoning is impossible because in that case you cannot use logic which means that you cannot conclude anything. I cannot cope with systemless reasoning exactly because it is impossible.

Impossible for you doesn't mean it's actually impossible. You are just arguing from ignorance.

You literally don't know how to reason without system. My son doesn't know how to ride a bycicle without training wheels.

Same thing realy.

The problem is that proving the consistency of propositional logic requires the use of a second theory, of which the consistency is equally in doubt, leading to a problem of infinite regress. Propositional logic is in fact merely equiconsistent with some other theory.

Using a word "like equiconsistent" when you can't determine the consistency of either theory is silly.

The situation is clearly not better than for PA. Gentzen's consistency proof does not prove the absolute consistency of PA but merely the equiconsistency with PRA (Primitive Recursive Arithmetic):

https://en.wikipedia.org/wiki/Gentzen%2 ... ency_proof

You are just kicking the can down the road without addressing the issue.

[Skepdick]

If you reject the notion of truth in mathematics, then you effectively reject Tarski's semantic theory of truth, and then everything you say, is in violation of Tarski's work on the matter:

https://iep.utm.edu/s-truth/

The semantic theory of truth (STT, hereafter) was developed by Alfred Tarski in the 1930s. The theory has two separate, although interconnected, aspects. First, it is a formal mathematical theory of truth as a central concept of model theory, one of the most important branches of mathematical logic. Second, it is also a philosophical doctrine which elaborates the notion of truth investigated by philosophers since antiquity.

How can you reasonably deny that your view, "Mathematics has nothing to do with truth", is in violation with Tarski's work?

We've clearly established that being able to read things; an copy-paste wikipedia or ChatGPT amounts to nothing if you don't understand what it is youare reading.

Tarski's theory of truth is circular: "P" is true if, and only if, P. "1+1=2" is true if, and only if 1+1=2 is true. Ok - is it true; and why?

So exactly in line with Tarski's work I can merely repeat myself till I'm blue in the face THAT 1+1=2 is true ... because its true, but I can't fucking tell you why it's true; or what this truth amounts to. I can also tell you THAT 1+1=2 follows from some other rules e.g I can tell you that I can axiomatize it.

But I can also axiomatize 1+1=1. I can even axiomatize 1=0. So axiomatization on its own is worthless, because it trivially follows from Godel's completeness theorem which equates truth with provability in 1st order logic. So provable = true?!?! But then that's proof theory, not model theory.

Mathematics has nothing to do with truth. 1+1=2 is an axiomatized assertion.

Is it true? I want to know WHY it's true; not THAT it's true.

[godelian]

Can you even read/understand what Traski says? He's literally telling you that you CAN'T define "arithmetical truth" (whatever the hell that is) in arithmetic! Why is 1+1=2 true?

It is again and obviously you who fail to understand Taski!

Tarski's undefinability theorem proves that it is not possible to define in arithmetic a predicate true(⌜s⌝) that will be able to compute the truth of any arbitrary sentence s. There does not exist a predicate φ(n) as such that ∀s ( φ(⌜s⌝) ⟺ s ).

More broadly - you can't define Mathematical truth in Mathematics.The theorem is literally about the NON-truth of Mathematics.

Tarski's undefinability theorem is about the impossibility to define the truth as a predicate. The truth can only be defined as a model.

Using a word "like equiconsistent" when you can't determine the consistency of either theory is silly.

Equiconsistency is a standard term in mathematics and means that either both theories are consistent or both theories are inconsistent:

https://en.wikipedia.org/wiki/Equiconsistency

In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other".

In general, it is not possible to prove the absolute consistency of a theory T. Instead we usually take a theory S, believed to be consistent, and try to prove the weaker statement that if S is consistent then T must also be consistent—if we can do this we say that T is consistent relative to S. If S is also consistent relative to T then we say that S and T are equiconsistent.

[Skepdick]

It is again and obviously you who fail to understand Taski!

You can keep lying to yourself about that...

Your commentary server as a great demonstration of confirmation bias. Even in Mathematics.

You are using Tarski's negative result to jump to a positive conclusion that confirms your preexisting belief about mathematical truth.

Tarski's undefinability theorem proves that it is not possible to define in arithmetic a predicate true(⌜s⌝) that will be able to compute the truth of any arbitrary sentence s. There does not exist a predicate φ(n) as such that ∀s ( φ(⌜s⌝) ⟺ s ).

Exactly! This tells you the ABSENCE of a predicate to determine truth.

So how can somethign LACKING a truth-predicate be true ?!?

Tarski's undefinability theorem is about the impossibility to define the truth as a predicate.

PRECISELY YOUR FUCKING PROBLEM.

Tarski's theorem tells us where truth CANNOT be defined (within the system), but offers no positive account of where truth CAN be defined, HOW truth can be defined, WHY any mathematical statement is actually true or what mathematical truth actually IS.

The truth can only be defined as a model.

Non-sequitur....

Tarski's theorem tells us where truth CANNOT be defined (within the system), but offers no positive account of where truth CAN be defined, HOW truth can be defined, WHY any mathematical statement is actually true or what mathematical truth actually IS.

It means that either both theories are consistent or both theories are inconsistent.

https://en.wikipedia.org/wiki/Equiconsistency

In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other".

So you can call them equiinconsistent - just the same. They stand or fall together.

This has nothing to do with proving they actually stand.

In general, it is not possible to prove the absolute consistency of a theory T. Instead we usually take a theory S, believed to be consistent, and try to prove the weaker statement that if S is consistent then T must also be consistent—if we can do this we say that T is consistent relative to S. If S is also consistent relative to T then we say that S and T are equiconsistent.

Yeah... you working on the premise that the absence of evidence for inconsistency is as good as consistency.

That's a non-seqitur.

Tarski's undefinability theorem proves that it is not possible to define in arithmetic a predicate true(⌜s⌝) that will be able to compute the truth of any arbitrary sentence s. There does not exist a predicate φ(n) as such that ∀s ( φ(⌜s⌝) ⟺ s ).

Exactly! That's a negative result!

And yet here you are giving positive account of Mathematical truths. Despite Tarski proving that's an undefinable notion.

Tarski's undefinability theorem is about the impossibility to define the truth as a predicate.

PRECISELY YOUR FUCKING PROBLEM.

Tarski's theorem tells us where truth CANNOT be defined (within the system), but offers no positive account of where truth CAN be defined, HOW truth can be defined, WHY any mathematical statement is actually true or what mathematical truth actually IS.

The truth can only be defined as a model.

Non-sequitur....

Tarski's theorem tells us where truth CANNOT be defined (within the system), but offers no positive account of where truth CAN be defined, HOW truth can be defined, WHY any mathematical statement is actually true or what mathematical truth actually IS.

It means that either both theories are consistent or both theories are inconsistent.

https://en.wikipedia.org/wiki/Equiconsistency

In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other".

So you can call them equiinconsistent - just the same. They stand or fall together.

This has nothing to do with proving they actually stand.

In general, it is not possible to prove the absolute consistency of a theory T. Instead we usually take a theory S, believed to be consistent, and try to prove the weaker statement that if S is consistent then T must also be consistent—if we can do this we say that T is consistent relative to S. If S is also consistent relative to T then we say that S and T are equiconsistent.

Yeah... you working on the premise that the absence of evidence for inconsistency is as good as consistency.

That's a non-seqitur.

[godelian]

If you reject the notion of truth in mathematics, then you effectively reject Tarski's semantic theory of truth, and then everything you say, is in violation of Tarski's work on the matter:

https://iep.utm.edu/s-truth/

The semantic theory of truth (STT, hereafter) was developed by Alfred Tarski in the 1930s. The theory has two separate, although interconnected, aspects. First, it is a formal mathematical theory of truth as a central concept of model theory, one of the most important branches of mathematical logic. Second, it is also a philosophical doctrine which elaborates the notion of truth investigated by philosophers since antiquity.

How can you reasonably deny that your view, "Mathematics has nothing to do with truth", is in violation with Tarski's work?

We've clearly established that being able to read things; an copy-paste wikipedia or ChatGPT amounts to nothing if you don't understand what it is youare reading.

Tarski's theory of truth is circular: "P" is true if, and only if, P. "1+1=2" is true if, and only if 1+1=2 is true. Ok - is it true; and why?

So exactly in line with Tarski's work I can merely repeat myself till I'm blue in the face THAT 1+1=2 is true ... because its true, but I can't fucking tell you why it's true; or what this truth amounts to. I can also tell you THAT 1+1=2 follows from some other rules e.g I can tell you that I can axiomatize it.

But I can also axiomatize 1+1=1. I can even axiomatize 1=0. So axiomatization on its own is worthless, because it trivially follows from Godel's completeness theorem which equates truth with provability in 1st order logic. So provable = true?!?! But then that's proof theory, not model theory.

Mathematics has nothing to do with truth. 1+1=2 is an axiomatized assertion.

Is it true? I want to know WHY it's true; not THAT it's true.

You originally wrote that your views are in line with Tarski's work, and not in violation of it, but now you write that "Tarski's theory of truth is circular". You keep moving the goalpost.

[godelian]

Tarski's theorem tells us where truth CANNOT be defined (within the system), but offers no positive account of where truth CAN be defined, HOW truth can be defined, WHY any mathematical statement is actually true or what mathematical truth actually IS.

The truth of PA is defined by its model. You can find it in the true arithmetic set of sentences:

https://en.wikipedia.org/wiki/True_arithmetic

In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers.[1] This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms.

The truth of a theory is the collection of sentences that are true in its standard model.

[Skepdick]

You originally wrote that your views are in line with Tarski's work, and not in violation of it, but now you write that "Tarski's theory of truth is circular". You keep moving the goalpost.

Those two sentences are not mutually incompatible. What is it that you don't understand?

"1+1=2" is true is 1+1=2 is true.

This fails to address WHY it's true.