Immanuel Kant was convinced that he knew it all

[godelian]

The English sentence is not authoritative. It is only the sentence in first-order logic that matters:

G ↔¬prov(⌜G⌝)

I really just can't parse that... Can you give me semantics for the "↔" operator.

I don't even understand why this is a binary operator when both the LHS and RHS are a function of G.

(G ↔¬prov(⌜G⌝)) ≡ ( (G ∧¬prov(⌜G⌝)) ∨(¬G ∧prov(⌜G⌝)) )

So, in general:

(S ↔ K) ≡ ( (S ∧K) ∨(¬S ∧ ¬K) )

ChatGPT: (G ↔¬prov(⌜G⌝)) ≡ ( (G ∧¬prov(⌜G⌝)) ∨(¬G ∧prov(⌜G⌝)) )

Yes, the logical equivalence you wrote is correct. It expresses the biconditional using its standard equivalent in classical logic. Here's a step-by-step verification:

Given:

Biconditional statement:

G↔¬prov(⌜G⌝)

This says:

“G is true if and only if G is not provable.”

You want to show it is logically equivalent to:

(G∧¬prov(⌜G⌝))∨(¬G∧prov(⌜G⌝))

General Law of Logic:

In classical propositional logic, the biconditional

P↔Q

is equivalent to:

(P∧Q)∨(¬P∧¬Q)

But you're using a biconditional with one side negated:

G↔¬prov(⌜G⌝)

Apply the same pattern with Q := ¬prov(⌜G⌝):

So:

G↔¬prov(⌜G⌝)≡(G∧¬prov(⌜G⌝))∨(¬G∧prov(⌜G⌝))

This is a valid transformation using the truth table for ↔.

Intuition:

This equivalence is important in Gödel’s first incompleteness theorem, where G is a self-referential sentence saying, roughly, “I am not provable.”

The equivalence shows:

If G is true and unprovable, the left side is true.
If G is false and provable, the right side is also true.
In all other cases, the biconditional fails.

Thus, it's a good way to analyze the fixed-point behavior of G.

[Skepdick]

(G ↔¬prov(⌜G⌝)) ≡ ( (G ∧¬prov(⌜G⌝)) ∨(¬G ∧prov(⌜G⌝)) )

So, in general:

(S ↔ K) ≡ ( (S ∧K) ∨(¬S ∧ ¬K) )

You are speaking Chinese to me.

Are these bound; or unbound variables?
Do they have types?

What are the semantics of ≡?

Thus, it's a good way to analyze the fixed-point behavior of G.

Was all of this about a fixed point?!?! f(x) = x?

You don't need that gibberish.

But seriously? Wall all that nonsense just so you can prove Godel's instantiation of Lawvere's general theorem?

https://en.wikipedia.org/wiki/Lawvere%2 ... nt_theorem

[godelian]

are speaking Chinese to me.

ChatGPT understands it perfectly fine.

Are these bound; or unbound variables?
Do they have types?

The variables represent logic sentences. The formulas accept natural numbers. G is bound by an existence quantifier. The context is Peano Arithmetic theory (PA).

What are the semantics of ≡?

The equivalence symbol has the same meaning as the biconditional symbol. It is often used to define the LHS.

Thus, it's a good way to analyze the fixed-point behavior of G.

Was all of this about a fixed point?!?! f(x) = x?
You don't need that gibberish.

That's just ChatGPT pointing out that Godel's incompleteness theorem is about the existence of a fixed point.

[Skepdick]

are speaking Chinese to me.

ChatGPT understands it perfectly fine.

How does that help anyone other than ChatGPT?

Are these bound; or unbound variables?
Do they have types?

The variables represent logic sentences.

So the variables represent logical sentences?
All the gibberish symbols weren't themselves logical sentences?

Were they illogical sentences?

The equivalence symbol has the same meaning as the biconditional symbol. It is often used to define the LHS.

That's a recursive/impredicative definition my guy. The definition of G (LHS) depends on the very thing being defined.

How do you get a Godel encoding of G until G is defined?

That's just ChatGPT pointing out that Godel's incompleteness theorem is about the existence of a fixed point.

That doesn't help me in any way.

Is the ≡ symbol defined?
Does its definition have a fixed point; or am I always one step behind the "final definition" ?

[godelian]

The English sentence is not authoritative. It is only the sentence in first-order logic that matters:

G ↔¬prov(⌜G⌝)

I really just can't parse that... Can you give me semantics for the "↔" operator.

are speaking Chinese to me.

ChatGPT understands it perfectly fine.

How does that help anyone other than ChatGPT?

Well, it proves that the expression is perfectly parsable and that the semantics for the "↔" operator are well-defined:

ChatGPT: What are the semantics for the "↔" operator?

The "↔" operator is called the biconditional or logical equivalence operator in logic. Its semantics describe a relationship of if and only if between two propositions.

Formal Semantics:

For propositions P and Q:

P↔Q

is true if and only if P and Q have the same truth value. That is:
True when both P and Q are true, or both are false
False when one is true and the other is false

Truth Table:
P Q P ↔ Q
T T T
T F F
F T F
F F T

Equivalent Formulations:

The biconditional can also be expressed in other logical forms:
(P ∧ Q) ∨ (¬P ∧ ¬Q)
(P → Q) ∧ (Q → P)

These are logically equivalent to P ↔ Q.

I don't see what your problem is with the biconditional operator.

What are the semantics of ≡?

The "≡" operator is the somewhat meta-logical version of the "↔" operator. However, they express the same notion of logical equivalence. For example, the equivalence of two general formulas is typically expressed with the meta-logical equivalence operator:

(P ↔ Q) ≡ ( (P ∧ Q) ∨ (¬P ∧ ¬Q))

It means that this tautology is valid across propositional logic and not just in the particular problem at hand. In modular arithmetic it also has the meaning of congruence. Example: 17 ≡ 5 (mod 12).

[Skepdick]

Well, it proves that the expression is perfectly parsable and that the semantics for the "↔" operator are well-defined

It does? That's not well-defined means, but if you think so Implement and execute computations with it.

I don't see what your problem is with the biconditional operator.

I don't have a "problem". implement it and execute with it.

The "≡" operator is the somewhat meta-logical version of the "↔" operator. However, they express the same notion of logical equivalence. For example, the equivalence of two general formulas is typically expressed with the meta-logical equivalence operator:

(P ↔ Q) ≡ ( (P ∧ Q) ∨ (¬P ∧ ¬Q))

It means that this tautology is valid across propositional logic and not just in the particular problem at hand. In modular arithmetic it also has the meaning of congruence. Example: 17 ≡ 5 (mod 12).

Great... So by Godel's other (less famous) theorem you should be able to compute with it.

Implement it and execute with it.

[godelian]

It does? That's not well-defined means, but if you think so Implement and execute computations with it.

ChatGPT just did. It verified it against the rules of propositional logic.

[Skepdick]

It does? That's not well-defined means, but if you think so Implement and execute computations with it.

ChatGPT just did. It verified it against the rules of propositional logic.

That's not what execution is.

Ask chat GPT to explain what a first-class computational equality is.

[godelian]

It does? That's not well-defined means, but if you think so Implement and execute computations with it.

ChatGPT just did. It verified it against the rules of propositional logic.

That's not what execution is.

Ask chat GPT to explain what a first-class computational equality is.

Verification of an expression is a computation.

[Skepdick]

Verification of an expression is a computation.

yeah. So how do you verify a bi-conditional expression?

If you unpack another layer of the onion you might just recognize the truth-table of a bi-conditional is the negation of XOR.
And if you pay even more attention LEM is, in fact the XOR operator. Either P is true; or not-P is true but not both.

A biconditional is an alternative encoding of a LEM negation

How do you verify the negation of an axiom in classical logic?

[godelian]

Verification of an expression is a computation.

yeah. So how do you verify a bi-conditional expression?

Classical propositional logic uses truth tables as its primary semantic tool. Truth tables are defined using only two truth values, presupposing bivalence. Bivalence directly implies that the Law of the Excluded Middle (LEM) holds as a tautology.

The proof for Godel's incompleteness theorem as well as its theoretical context, PA, are deeply invested in classical logic and the LEM. If you reject the LEM, you need to design another proof, which may or may not be possible.

[Skepdick]

Verification of an expression is a computation.

yeah. So how do you verify a bi-conditional expression?

Classical propositional logic uses truth tables as its primary semantic tool. Truth tables are defined using only two truth values, presupposing bivalence. Bivalence directly implies that the Law of the Excluded Middle (LEM) holds as a tautology.

Please write down the truth table for the bivalence operator (otherwise known as XNOR); and recognize it as the truth table for "not( a XOR b)".
Please write down the truth table for A XOR ¬A (which is the proper encoding of excluded middle - see below) . And recognize it as a tautology.

When you encode LEM as A XOR ¬A (one or the other but not both) you now have a stronger system, because you no longer require the non-contradiction axiom.

The non-contradiction axiom is only necessary when you use the inclusive OR operator (A OR not-A); because the case of both operands being true is not automatically handled.

This isn't about the foundations of mathematics. This is about the foundations of logic.

Classical logic with LEM + LNC is redundant.
The XOR formulation A ⊕ ¬A captures the intended meaning directly

What this gives you isn't "If you encounter a contradiction then its truth value is False".
What this gives you IS "Contradictions are impossible to exist! Period."

(A and not-A) is false ?!?!? Which component makes it false? A; or not-A?
What you get (instead of contradictions) is proofs of self-negation.

Proofs of semantic vacuity. You have a continuous transformation from A to not-A. Because it's empty of any semantic content.

Syntax is not even true. It's just symbol manipulation.

The proof for Godel's incompleteness theorem as well as its theoretical context, PA, are deeply invested in classical logic and the LEM. If you reject the LEM, you need to design another proof, which may or may not be possible.

Please further recognize Godel's theorem as a direct invocation of XNOR - a negation of XOR. A negation of excluded middle.

Godel is the birth of constructive Mathematics.

G ↔¬prov(⌜G⌝) is an implicit rejection of LEM!
G is its own witness, unless; or until you prove it - this captures the provisional, constructive nature of mathematical truth:

Which is to say - G is provisionally true unless, and until you construct a computation for it.
When you construct a computation (a proof!) for G you negate the statement ¬prov(⌜G⌝).
G is no longer unprovable (uncomputable).

To read classical Mathematics with Godelian spectacles (literally and figuratively) is to misunderstand most of it.
Gödel's construction is fundamentally constructivist and anti-classical.

[godelian]

Syntax is not even true. It's just symbol manipulation.

Yes, that is why we have model theory and Tarski's semantic theory of truth.

Godel is the birth of constructive Mathematics.
Gödel's construction is fundamentally constructivist and anti-classical.

Not so sure about that. Gödel acknowledged some of the constructivist concerns but he wasn't a constructivist and neither was his work.

ChatGPT: Was Gödel a constructive mathematician?

Kurt Gödel was not a constructive mathematician in the strict sense of the term. He worked primarily within classical logic and classical mathematics, which accepts principles that constructivists reject—such as the Law of the Excluded Middle (LEM) and non-constructive existence proofs.

However, his relationship with constructivism is nuanced:

Classical Orientation:
Gödel’s incompleteness theorems (1931) were proved within the framework of classical first-order logic and Peano Arithmetic—systems that are not constructive. He used non-constructive methods, including diagonalization and arithmetization of syntax, both of which rely on classical assumptions.

Interest in Constructivism:
Despite not being a constructivist himself, Gödel showed deep interest in intuitionistic logic, especially the work of Brouwer and Heyting.
In 1933, he gave a famous embedding of intuitionistic logic into modal logic (now known as the Gödel–Gentzen translation), showing that intuitionistic logic can be interpreted within a modal system. This shows philosophical sympathy and serious engagement with constructive ideas.
He was also interested in constructive type theories and finitism, especially in the context of Hilbert’s program.

Summary:
Gödel was not a constructive mathematician, but he respected and engaged with constructive and intuitionistic approaches, especially from a foundational and philosophical perspective. His work stands firmly within the classical tradition, even as he explored constructive ideas critically and analytically.

Gödel wasn't anti-classical at all.

[Skepdick]

Not so sure about that. Gödel acknowledged some of the constructivist concerns but he wasn't a constructivist and neither was his work.
...
Gödel wasn't anti-classical at all.

Way to miss the point.

His Godel sentence is an implicit rejection of LEM! Fancy notation - identical semantics!
This is sub-structurally true (as in the Tarski sense!) whether you agree or not.

By copy-pasting text from ChatGPT you are demonstrating precisely that which I keep pointing out: you don't actually understand.
You parrot.

[godelian]

By copy-pasting text from ChatGPT you are demonstrating precisely that which I keep pointing out: you don't actually understand.
You parrot.

I am just a reader of mathematical literature. I do not write any. At best, I interpret and paraphrase what I have read. That is in fact also what ChatGPT does. I have never published any original theorem in mathematics. That is simply not my intention. It is not because someone does not write original theorems in mathematics that he does not understand the original work that someone else has published.