An infinite hierarchy of delusions

[Age]

'This one' now claims that letters believe things.

So, 'the delusions', here, continue.

ZFC is an abstract "thing".

Yet you claim 'it' does not just believe things, but insist that 'it' 'strongly believes' things.

In line with Platonism, I believe that it is an object in a metaphysical reality:

Gemini AI

Mathematical Platonism is the philosophical view that mathematical objects, like numbers and sets, exist independently of human minds and are abstract, eternal, and unchanging. It suggests that when mathematicians discover mathematical truths, they are uncovering pre-existing realities rather than creating them.

Here's a more detailed explanation:

Core Beliefs:

Realism:

Mathematical Platonism asserts that mathematical statements are true or false based on whether they accurately describe these independently existing mathematical entities.

Abstractness:

These entities are considered abstract, meaning they don't exist in space or time and are not physical objects.

Independence:

Mathematical objects exist independently of human thought, language, and any physical or mental activity.

Eternality and Unchangeability:

Mathematical entities are thought to be eternal and unchanging, existing outside of the temporal flow of the universe.

So, yes, if you don't believe in Platonism, it all sounds delusional.

[Eodnhoj7]

When you look at the natural numbers:

0, 1, 2, 3, ...

The total number of them is ℵ₀ (aleph zero), also called "countable infinity".

Technically, we can say that the cardinality of the set of natural numbers is ℵ₀.

Now we can create an rapidly growing infinite snowball by repeatedly exponentiating this already massive infinite number:

ℶ₀ = ℵ₀
ℶ₁ = 2^ℶ₀
ℶ₂ = 2^ℶ₁
ℶ₃ = 2^ℶ₂
...

This ℶ ("beth") sequence of infinities does not grow fast enough, however, to ever reach κ ("kappa"), the first inaccessible infinity ("inaccessible cardinal").

We are actually not even sure that κ exists, because we cannot reach such large infinity just by construction.

That is why we need to axiomatize the existence of κ. We simply adopt the belief that it exists. So, κ exists only by means of delusion.

You may ask yourself the question: Do we even need this nonsense?

Yes, we do.

ZFC set theory, which is the admiral ship of mathematics, cannot prove its own consistency.

According to Godel's second incompleteness theorem, if a theory proves its own consistency, then it is necessarily inconsistent.

So, we need a different metatheory for ZFC to prove its consistency.

The simplest approach to obtain such metatheory is to duly mutilate ZFC with a new axiom.

The best candidate turns out to be the axiomatic belief in the delusional existence of an inaccessible infinity.

So, we delude ourselves into believing that there exists an infinity that is monstrous enough that it is too big for ZFC to reach.

Next, we use our new metatheory, i.e. ZFC+inacc, to prove that ZFC does not contain contradictions.

How do we know that ZFC+inacc is itself consistent?

Well, we don't.

However, we can prove the consistency of ZFC+inacc by adopting an even bigger delusion.

Gemini: What new axiom allows us to construct a metatheory suitable for proving the consistency of ZFC+inacc?

Gödel's Second Incompleteness Theorem states that if a formal system is consistent, then it cannot prove its own consistency. This applies to ZFC. Furthermore, if a theory T proves the existence of a model for another theory S, then T implies the consistency of S.

Since ZFC + Inacc proves the consistency of ZFC, it follows from Gödel's Second Incompleteness Theorem that ZFC cannot prove the existence of an inaccessible cardinal (assuming ZFC is consistent).

To prove the consistency of ZFC + Inacc, you need a metatheory that is strictly stronger than ZFC + Inacc in terms of consistency strength.

You would need to move to a stronger large cardinal axiom. For example, if you assume the existence of a Mahlo cardinal, or a measurable cardinal, these axioms are strong enough to imply the existence of inaccessible cardinals, and thus provide a model for ZFC + Inacc.

So, while "there exists an inaccessible cardinal" is an axiom that strengthens ZFC and implies its consistency, to prove the consistency of ZFC + Inacc, you'd generally need to assume the existence of an even "larger" large cardinal. This process continues, forming a hierarchy of consistency strengths where each level can "prove" the consistency of the previous one.

So, the consistency of the entire body of mathematics is ultimately backed by an ever growing and open-ended hierarchy of delusions, produced by means of successive mutilation.

If delusions, distinctions (in this case number), are hierarchical than a constant form occurs by nature of linear progression and we are left with an absolute truth of illusion within a given context. This is more of an abstract metaphysical analysis...but number and metaphysics often overlap.

[puto]

Philosophy is about mathematics in the Enlightenment.