Philosophy Now Forum

Posted: **Mon Jul 07, 2025 3:43 am**

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.

Posted: **Mon Jul 07, 2025 7:57 am**

Some may well be asking "themselves", 'Do we need the above nonsense?'

And, the resounding blatantly obvious clear answer is, 'No'.

Posted: **Mon Jul 07, 2025 9:13 am**

Age wrote: ↑Mon Jul 07, 2025 7:57 am Some may well be asking "themselves", 'Do we need the above nonsense?'

And, the resounding blatantly obvious clear answer is, 'No'.

Well, if you ever use the fact that 3+4=7, which is provable in arithmetic (PA), you count on the idea that doing a thing like that will never go wrong.

You hope that arithmetic (PA) will never prove a falsehood.

Hence, you rely on the belief that arithmetic (PA) is consistent. But is it?

Set theory (ZFC) can prove that arithmetic (PA) is consistent.

However, is set theory itself (ZFC) consistent? Would it ever prove a falsehood?

Well, as mentioned previously, addressing this matter requires an open-ended hierarchy of delusions.

Posted: **Mon Jul 07, 2025 10:19 am**

godelian wrote: ↑Mon Jul 07, 2025 9:13 am
Age wrote: ↑Mon Jul 07, 2025 7:57 am Some may well be asking "themselves", 'Do we need the above nonsense?'

And, the resounding blatantly obvious clear answer is, 'No'.
Well, if you ever use the fact that 3+4=7, which is provable in arithmetic (PA), you count on the idea that doing a thing like that will never go wrong.

'you' have absolutely no idea what 'I' do, or do not, count on. So, please refrain from presuming 'you' know, and, from telling 'us' what 'you', in fact, do not yet even know, here.

godelian wrote: ↑Mon Jul 07, 2025 9:13 am You hope that arithmetic (PA) will never prove a falsehood.

Again, 'you' have absolutely no idea what 'I' do, or do not, hope.

godelian wrote: ↑Mon Jul 07, 2025 9:13 am Hence, you rely on the belief that arithmetic (PA) is consistent.

'I' do not. So, once again, 'your' belief, conclusion, and claim, here, are completely and utterly False, Wrong, Invalid, and unsound.

'I' will, again, suggest 'you' try again. And, 'I' suggest 'you' start at, from, and with 'Facts' only.

godelian wrote: ↑Mon Jul 07, 2025 9:13 am But is it?

Set theory (ZFC) can prove that arithmetic (PA) is consistent.

If 'you' say so, but who cares?

godelian wrote: ↑Mon Jul 07, 2025 9:13 am However, is set theory itself (ZFC) consistent?

Who cares, 'it' is obviously just a theory', anyway. And, in case 'you' have forgotten, 'I' do not do 'theory', in any way.

godelian wrote: ↑Mon Jul 07, 2025 9:13 am Would it ever prove a falsehood?

Falsehoods, like 'yours' above, here, can be and are shown, and/or proved, to be Falsehoods in many ways. As 'I' just showed, and proved, above, here.

godelian wrote: ↑Mon Jul 07, 2025 9:13 am Well, as mentioned previously, addressing this matter requires an open-ended hierarchy of delusions.

If 'this' is what 'you' want to believe and claim is irrefutably true, then so be it. Go on and do that.

Posted: **Mon Jul 07, 2025 12:32 pm**

Age wrote: ↑Mon Jul 07, 2025 10:19 am Who cares, 'it' is obviously just a theory', anyway. And, in case 'you' have forgotten, 'I' do not do 'theory', in any way.

My post is about the consistency strength of theories.

It is a hierarchy. If you want to prove the consistency of a weaker theory, you need a stronger theory. A theory is stronger if it believes in the existence of an even larger infinite cardinal.

ChatGPT: consistency strength

Consistency strength is a way of comparing formal theories (typically in mathematical logic and set theory) based on how powerful their consistency assumptions are. It provides a way to gauge how “strong” a theory is in terms of what kinds of mathematical truths it can prove.

Definition (Informal)

Theory A has greater consistency strength than theory B if:
The consistency of A implies the consistency of B,
But not vice versa (assuming both are consistent).
In symbols, if Con(A) ⊢ Con(B), but Con(B) ⊬ Con(A), then A is stronger in consistency strength than B.

Why It's Important
Consistency strength helps us:
Compare foundational systems (e.g., ZFC vs. PA).
Evaluate the impact of adding axioms, such as large cardinals, determinacy principles, etc.
Understand the limits of provability (due to Gödel's incompleteness theorems).

Examples
Theory Description Relative Strength
PA Peano Arithmetic Baseline
ZFC Zermelo-Fraenkel Set Theory + Choice Much stronger
ZFC + Inaccessible Adds existence of an inaccessible cardinal Stronger than ZFC
ZFC + Measurable Even stronger (requires larger cardinal) Even stronger
Each step upward adds assumptions that cannot be proved within the previous system, and whose consistency is strictly stronger.

Consistency Strength and Large Cardinals
Large cardinal axioms (e.g., inaccessible, measurable, Woodin) are used to measure the strength of theories:
The existence of an inaccessible cardinal has greater consistency strength than ZFC.
The existence of a measurable cardinal implies the existence of an inaccessible one, so it has even greater consistency strength.
This creates a hierarchy of theories, each one assuming more and proving more, but being harder to justify purely from weaker ones.

How Is It Used?
If we can't prove Con(ZFC) in ZFC itself (Gödel), we may:
Prove Con(ZFC) from ZFC + Inaccessible cardinal, thus showing relative consistency.
Compare set theories to see which is capable of proving more mathematics.

Let me know if you want a diagram of the consistency strength hierarchy, or if you're working with a specific theory (e.g., ZF, ZFC+AD, etc.).

As I have pointed out already, it is the delusion itself that makes the theory stronger. A more delusional theory can prove the consistency of a less delusional one. Hence, it has greater "consistency strength".

By the way, something interesting about ℶ ("beth"). If the ℶ ("beth") sequence has access to the inaccessible cardinal κ ("kappa"), it can use κ as an index ordinal to shoot beyond κ. That means that κ would actually become accessible. So, it is necessary to keep κ out of ZFC. It should only exist in ZFC+inacc.

The belief that the inaccessible cardinal κ really exists, is both delusional as amazingly powerful.

If you don't "do 'theory'", why are you replying to my post? My post is not just about theory. It is about delusional theory.

Posted: **Mon Jul 07, 2025 8:08 pm**

Age wrote: ↑Mon Jul 07, 2025 7:57 am Some may well be asking "themselves", 'Do we need the above nonsense?'

And, the resounding blatantly obvious clear answer is, 'No'.

Exactly what they said about the imaginary unit i, before it became indispensable in quantum physics and electronics.

Exactly what they said in the 1840s about Riemannian geometry, which shockingly violated Euclid's parallel postulate, until Einstein used it in 1915 to model gravity in general relativity.

Exactly what they said for two thousand years about the theory of factoring large numbers, till public key cryptography used advanced factoring theory to become the basis of all online commerce and security.

There is as yet no known practical use for higher set theory. But the the history of math and science suggests that we should not be so dismissive of what might become useful in the future.

Indeed, one might ask why humans have an intuition of the infinite, if it doesn't mean something or suggest something important, if only we can figure it out. Cosmologists think the universe might be infinite, or that there might be infinitely many parallel universes. They are already starting to consider the problem of how to assign probabilities in physically infinite situations. This is called the measure problem (not to be confused with the measurement problem of quantum physics).

https://en.wikipedia.org/wiki/Measure_p ... cosmology)

If you want to bet that higher set theory (large cardinals, etc) won't be practical in the next ten years, that's a pretty good bet. But if you look out a couple of centuries, I don't think it's a good bet at all. Obscure and counterintuitive math has a long track record of eventually becoming an indispensable part of physical science.

Posted: **Tue Jul 08, 2025 3:37 am**

wtf wrote: ↑Mon Jul 07, 2025 8:08 pm If you want to bet that higher set theory (large cardinals, etc) won't be practical in the next ten years, that's a pretty good bet. But if you look out a couple of centuries, I don't think it's a good bet at all.

In my opinion, large cardinals already do something really useful. They contribute to shedding light on the consistency of mathematical theories.

I am currently reading up on the connection between reflection principles and large cardinals. I wasn't aware of the fact that the models for ZFC were so different from the ones for PA. These things are so surprising. It's definitely an entertaining hobby. I don't care if it will ever be useful.

Posted: **Tue Jul 08, 2025 3:52 am**

godelian wrote: ↑Mon Jul 07, 2025 12:32 pm
Age wrote: ↑Mon Jul 07, 2025 10:19 am Who cares, 'it' is obviously just a theory', anyway. And, in case 'you' have forgotten, 'I' do not do 'theory', in any way.
My post is about the consistency strength of theories.

No one knows what the so-called 'consistency strength' of 'theories' (with an 's') is.

And, to even imagine or suggest that there is some so-called 'consistency strength' among 'all theories' is just absurd and irrational to say the least.

Now, and once again, any theory is just a presumption, guess, or an imagined 'what might be'. Which, well to me anyway, is what you people refer to as 'just a waste of time'. And, why any and all 'theories' are 'just a waste of time' is because 'what actually is' can be uncovered anyway.

godelian wrote: ↑Mon Jul 07, 2025 12:32 pm It is a hierarchy. If you want to prove the consistency of a weaker theory, you need a stronger theory. A theory is stronger if it believes in the existence of an even larger infinite cardinal.

Once again, EVERY theory is just a guess.

And, also once again, why do you human beings, in the days when this is being written, even bother with guesses, or in other words, 'what could be's'?

Why not just focus on what are 'actual Truths' only, instead?

To me it is 'delusional thinking' to concentrate on 'what if's', especially when the actual 'what is' is blatantly obvious, and as some of you might say, 'staring directly at you' and/or is 'staring at you' and 'staring you in the face'?

godelian wrote: ↑Mon Jul 07, 2025 12:32 pm
ChatGPT: consistency strength

Consistency strength is a way of comparing formal theories (typically in mathematical logic and set theory) based on how powerful their consistency assumptions are. It provides a way to gauge how “strong” a theory is in terms of what kinds of mathematical truths it can prove.

Definition (Informal)

Theory A has greater consistency strength than theory B if:
The consistency of A implies the consistency of B,
But not vice versa (assuming both are consistent).
In symbols, if Con(A) ⊢ Con(B), but Con(B) ⊬ Con(A), then A is stronger in consistency strength than B.

Why It's Important
Consistency strength helps us:
Compare foundational systems (e.g., ZFC vs. PA).
Evaluate the impact of adding axioms, such as large cardinals, determinacy principles, etc.
Understand the limits of provability (due to Gödel's incompleteness theorems).

Examples
Theory Description Relative Strength
PA Peano Arithmetic Baseline
ZFC Zermelo-Fraenkel Set Theory + Choice Much stronger
ZFC + Inaccessible Adds existence of an inaccessible cardinal Stronger than ZFC
ZFC + Measurable Even stronger (requires larger cardinal) Even stronger
Each step upward adds assumptions that cannot be proved within the previous system, and whose consistency is strictly stronger.

Consistency Strength and Large Cardinals
Large cardinal axioms (e.g., inaccessible, measurable, Woodin) are used to measure the strength of theories:
The existence of an inaccessible cardinal has greater consistency strength than ZFC.
The existence of a measurable cardinal implies the existence of an inaccessible one, so it has even greater consistency strength.
This creates a hierarchy of theories, each one assuming more and proving more, but being harder to justify purely from weaker ones.

How Is It Used?
If we can't prove Con(ZFC) in ZFC itself (Gödel), we may:
Prove Con(ZFC) from ZFC + Inaccessible cardinal, thus showing relative consistency.
Compare set theories to see which is capable of proving more mathematics.

Let me know if you want a diagram of the consistency strength hierarchy, or if you're working with a specific theory (e.g., ZF, ZFC+AD, etc.).
As I have pointed out already, it is the delusion itself that makes the theory stronger. A more delusional theory can prove the consistency of a less delusional one. Hence, it has greater "consistency strength".

By the way, something interesting about ℶ ("beth"). If the ℶ ("beth") sequence has access to the inaccessible cardinal κ ("kappa"), it can use κ as an index ordinal to shoot beyond κ. That means that κ would actually become accessible. So, it is necessary to keep κ out of ZFC. It should only exist in ZFC+inacc.

The belief that the inaccessible cardinal κ really exists, is both delusional as amazingly powerful.

If you don't "do 'theory'", why are you replying to my post? My post is not just about theory. It is about delusional theory.

To show and prove how and why 'doing theories' is keeping you human beings so far behind, and has been and, still, is stopping and preventing you human beings from moving forward.

Why would any sane person 'look for', 'what might be the case', when, 'what is the case', is HERE, for all to 'look at' and 'see', instead?

Why even discuss over 'things' that are essentially just assumptions and guesses, which obviously could be False and/or Wrong, when 'what is' actually True and Right, and irrefutably so can be 'looked at', and discussed over as well, and instead?

Posted: **Tue Jul 08, 2025 4:00 am**

Age wrote: ↑Tue Jul 08, 2025 3:52 am Why even discuss over 'things' that are essentially just assumptions and guesses, which obviously could be False and/or Wrong, when 'what is' actually True and Right, and irrefutably so can be 'looked at', and discussed over as well, and instead?

Okay, you don't like math. You are definitely not alone. Fine, but in that case why are you posting comments in "the philosophy of mathematics" section? Why do you want to philosophize about something that you do not even like? Why don't you philosophize about something that you do like?

Posted: **Tue Jul 08, 2025 5:15 am**

godelian wrote: ↑Tue Jul 08, 2025 3:37 am In my opinion, large cardinals already do something really useful. They contribute to shedding light on the consistency of mathematical theories.

Correct, though it's more the large countable ordinals that measure proof strength of theories. I was going to mention that to you, but I didn't want to engage with someone calling higher set theory a "delusion."

You confused me a little. You kept calling higher set theory a delusion, so I thought you were just railing against mathematical abstraction for its own sake. Now you seem to be saying you are interested in the concepts.

I would like to suggest that you lay off ChatGPT when trying to get insight into these matters, but suggesting that is like tilting at windmills these days. Still, LLMs are bad at math and are long on facts but often short on insight.

Wikipedia has some decent articles on the subject.

Posted: **Tue Jul 08, 2025 5:32 am**

wtf wrote: ↑Tue Jul 08, 2025 5:15 am Correct, though it's more the large countable ordinals that measure proof strength of theories. I was going to mention that to you, but I didn't want to engage with someone calling higher set theory a "delusion."

It is exactly because it is obviously delusional that I like it so much. It is pure genius.

You see, math is a hobby of mine. It's undoubtedly my favorite hobby.

Pure math is pure abstraction. That is why it is by design meaningless. Consequently, it systematically aims to be useless. However, it still happens to be incredibly surprising. Hence, its only redeeming quality is that it is absolutely ridiculous.

In my opinion, there is no better intellectual hobby than pure math.

Pure math is challenging to understand, but that is exactly what I like about it. When I do understand it -- which is not always the case -- it gives me tremendous satisfaction. It is definitely my favorite literature. Not all math is "good", though. It is truly the amount of surprise that matters. In my opinion, "good" math is fantastically ridiculous.

Posted: **Tue Jul 08, 2025 10:18 am**

wtf wrote: ↑Mon Jul 07, 2025 8:08 pm
Age wrote: ↑Mon Jul 07, 2025 7:57 am Some may well be asking "themselves", 'Do we need the above nonsense?'

And, the resounding blatantly obvious clear answer is, 'No'.
Exactly what they said about the imaginary unit i, before it became indispensable in quantum physics and electronics.

Exactly what they said in the 1840s about Riemannian geometry, which shockingly violated Euclid's parallel postulate, until Einstein used it in 1915 to model gravity in general relativity.

you seemed to have confused the word 'need' with something else, here.

When I said and wrote the word, 'need', above, here, what do you imagine or envision this was in relation to, exactly, then will you 'now'?

wtf wrote: ↑Mon Jul 07, 2025 8:08 pm Exactly what they said for two thousand years about the theory of factoring large numbers, till public key cryptography used advanced factoring theory to become the basis of all online commerce and security.

There is as yet no known practical use for higher set theory. But the the history of math and science suggests that we should not be so dismissive of what might become useful in the future.

If you have not already answered, what did you think the word, 'need', above, here, is in reference to, exactly?

wtf wrote: ↑Mon Jul 07, 2025 8:08 pm Indeed, one might ask why humans have an intuition of the infinite, if it doesn't mean something or suggest something important, if only we can figure it out.

If, and when, you human beings have an intuition of the infinite, then this is just because of the, irrefutable, existence of 'the infinite', itself.

wtf wrote: ↑Mon Jul 07, 2025 8:08 pm Cosmologists think the universe might be infinite, or that there might be infinitely many parallel universes.

What any group of people, in any day and age, 'think' never ever necessarily has any bearing at all on what is actually True, and Right, in Life.

Also, are you suggesting that absolutely all, in the past, 'currently', or always in the future, in the group of human beings who have been placed under the label "cosmologists" all think either the Universe 'could be' infinite, or there 'might be' infinitely many parallel universes?

Either way, there is One irrefutable Truth, here. Which, by the way, is already known, by 'us' non labelled "cosmologists".

wtf wrote: ↑Mon Jul 07, 2025 8:08 pm They are already starting to consider the problem of how to assign probabilities in physically infinite situations. This is called the measure problem (not to be confused with the measurement problem of quantum physics).

Okay. But, and again, 'I' do not see absolutely any so-called 'problem' at all, here.

The 'Universe', Itself, by definition, can only be the One Thing, which 'It' is.

And, as always, if absolutely any one would like to have a discussion about what the Universe actually, and irrefutably, is, then let 'us' have a discussion.

wtf wrote: ↑Mon Jul 07, 2025 8:08 pm https://en.wikipedia.org/wiki/Measure_p ... cosmology)

If you want to bet that higher set theory (large cardinals, etc) won't be practical in the next ten years, that's a pretty good bet. But if you look out a couple of centuries, I don't think it's a good bet at all.

What made 'you' even begin to consider, let alone begin to think along the lines of 'practicality'?

What 'I' actually said, and actually meant, is in 'the words' above, here.

Again, 'I' will suggest that 'you' human beings read only 'the words' that 'I' write, and mean, here, and just stop 'assuming' what 'I' 'could be' meaning. Once more, if 'you' did this, instead, then 'you' will not get so far off track and thus so far astray.

wtf wrote: ↑Mon Jul 07, 2025 8:08 pm Obscure and counterintuitive math has a long track record of eventually becoming an indispensable part of physical science.

Okay. But again, absolutely nothing at all to what I was actually saying, referring to, and talking about.

Posted: **Tue Jul 08, 2025 10:29 am**

godelian wrote: ↑Tue Jul 08, 2025 4:00 am
Age wrote: ↑Tue Jul 08, 2025 3:52 am Why even discuss over 'things' that are essentially just assumptions and guesses, which obviously could be False and/or Wrong, when 'what is' actually True and Right, and irrefutably so can be 'looked at', and discussed over as well, and instead?
Okay, you don't like math.

What 'we' have, here, now is another Truly absurd and completely irrational presumption of yours.

you have, once again, completely and utterly missed and misunderstood 'the point' I have been making, here.

And, as such, you arrived at a conclusion that could not be more False, more Wrong, more Inaccurate, now more Incorrect.

Do you not feel embarrassed at all making so many False, and Wrong, assumptions and conclusions on a publicly open website?

godelian wrote: ↑Tue Jul 08, 2025 4:00 am You are definitely not alone. Fine, but in that case why are you posting comments in "the philosophy of mathematics" section?

1. Because of delusional claims like, 'an infinite hierarchy of delusions'.

2. To reinforce that looking at and discussing theories has been, and continues to be, what is called a 'complete waste of time'. Continue to do so only continues to hold you human beings from moving forward and advancing.

godelian wrote: ↑Tue Jul 08, 2025 4:00 am Why do you want to philosophize about something that you do not even like?

Here is a great example of how and when one starts, or begins, from a False assumption, then they are led completely astray.

'This one' is now asking 'me' a question, which has absolutely no bearing at all on what is actually True, in Life.

godelian wrote: ↑Tue Jul 08, 2025 4:00 am Why don't you philosophize about something that you do like?

Are you able to tell me what I do like, considering the Fact that you believe that you already do know what I do not like?

Posted: **Tue Jul 08, 2025 10:38 am**

wtf wrote: ↑Tue Jul 08, 2025 5:15 am
godelian wrote: ↑Tue Jul 08, 2025 3:37 am In my opinion, large cardinals already do something really useful. They contribute to shedding light on the consistency of mathematical theories.

Correct, though it's more the large countable ordinals that measure proof strength of theories.

Again, 'the proof', or 'the lack of proof', in 'theories' can very easily and very simply be found. And, without all of the unnecessary complications you adult human beings, here, added in I will add.

Now, and again, the making up of, looking at, and discussing theories is only holding you people back from discovering and uncovering what the actual irrefutable of Truths is, exactly.

And, if absolutely any one of you would like the 'actual proof' of this, then just provide an actual example of an actual theory, which you, still, are unsure if it is Right, Accurate, or Correct, or not.

Can I make this any simpler and easier for you human beings, here?

wtf wrote: ↑Tue Jul 08, 2025 5:15 am I was going to mention that to you, but I didn't want to engage with someone calling higher set theory a "delusion."

When some one, here, makes a claim I like to test them to see if they can prove the claim True, Right, Accurate, and/or Correct, or not.

Instead of just dismissing them, as though 'i' know what the actual Truth is, I like to see if they can actually back up and support their claim, irrefutably, or not.

wtf wrote: ↑Tue Jul 08, 2025 5:15 am You confused me a little. You kept calling higher set theory a delusion, so I thought you were just railing against mathematical abstraction for its own sake. Now you seem to be saying you are interested in the concepts.

I would like to suggest that you lay off ChatGPT when trying to get insight into these matters, but suggesting that is like tilting at windmills these days. Still, LLMs are bad at math and are long on facts but often short on insight.

Wikipedia has some decent articles on the subject.

Posted: **Tue Jul 08, 2025 10:49 am**

godelian wrote: ↑Tue Jul 08, 2025 5:32 am
wtf wrote: ↑Tue Jul 08, 2025 5:15 am Correct, though it's more the large countable ordinals that measure proof strength of theories. I was going to mention that to you, but I didn't want to engage with someone calling higher set theory a "delusion."
It is exactly because it is obviously delusional that I like it so much. It is pure genius.

Is there any possibility in the whole Universe that what is you claim is, so-called, 'obviously delusional', and/or 'an infinite hierarchy of delusions', is not actually delusional at all, and, that what is actually 'delusional', itself, is 'the way' you are 'looking at' and 'seeing' things, here?

Or, is this not a possibility at all?

godelian wrote: ↑Tue Jul 08, 2025 5:32 am You see, math is a hobby of mine. It's undoubtedly my favorite hobby.

Just because one has a 'favorite hobby', over other hobbies, this never ever means that one is actually 'good at' their 'favorite hobby'.

godelian wrote: ↑Tue Jul 08, 2025 5:32 am Pure math is pure abstraction. That is why it is by design meaningless.

Now 'this' is delusional.

If some thing is 'pure abstraction', then 'it' 'must be', by design, meaningless.

'I want to create and live in a Truly peaceful world', is 'pure abstraction', it could be said and argued. But, is 'this', by design or not, meaningless?

If yes, then how, and why, exactly?

Also, if some thing is 'by design', then 'that thing' was 'designed', and every thing 'designed' must have been 'designed' by something else. So, who and/or what was 'it', exactly, which 'designed' 'pure abstraction' to be meaningless?

And, why did 'it' design some things to be meaningful and other things to be meaningless?

godelian wrote: ↑Tue Jul 08, 2025 5:32 am Consequently, it systematically aims to be useless. However, it still happens to be incredibly surprising. Hence, its only redeeming quality is that it is absolutely ridiculous.

What is, actually, absolutely ridiculous are some of your assumptions, beliefs, and claims, there, throughout this forum.

godelian wrote: ↑Tue Jul 08, 2025 5:32 am In my opinion, there is no better intellectual hobby than pure math.

Who cares?

godelian wrote: ↑Tue Jul 08, 2025 5:32 am Pure math is challenging to understand, but that is exactly what I like about it.

Why are 'you' even so 'challenged' 'to understand', from the beginning?

godelian wrote: ↑Tue Jul 08, 2025 5:32 am When I do understand it -- which is not always the case -- it gives me tremendous satisfaction. It is definitely my favorite literature. Not all math is "good", though. It is truly the amount of surprise that matters. In my opinion, "good" math is fantastically ridiculous.

With 'this kind of belief' of yourself, here, there is no wonder how and why you also arrived at 'the conclusion', and belief, that 'higher set theory is a delusion'.

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