Meanwhile, for a reality check, it would help to know that you recognize that to infer that there are no infinite sets requires not just dropping the axiom of infinity but moreover adopting the negation of the axiom of infinity. Merely dumping the axiom of infinity does not provide that there are no infinite sets. I wish I knew whether you see that now.

Also, that when one says "a true arithmetical sentence" one may take it for granted that, for full rigor, that would be understood as "a sentence in the language of arithmetic (the language for first order PA) that is true in the standard model for the language of arithmetic".

Your principle leads directly to a contradiction. The restriction needed to patch the problem is restricted comprehension, which would already require there to be an infinite set that's a superset of the infinite set you wish to conjure. Without the axiom of infinity, THERE IS NO SET containing all the natural numbers — fishfry

Every arithmetical statement is either true or false. There is a function that determines the truth or falsehood of every arithmetical statement. But, of course, it's not a computable function. The truth or falsehood of every arithmetical statement is determined, but there are arithmetical statements of which we could never find the determination. It's as if those statements and their determinations are "out there floating around" but I can't visualize what it means that they are true or false except that I know there is a function that determines them — TonesInDeepFreeze

Did not Godel and Cantor believe that once one sees Absolute Infinity he knows all (the whole story of mathematics)?

I am genuinely baffled why so often in this forum you get me completely backwards. — TonesInDeepFreeze

Leads to some spirited and entertaining conversations.

Did not Godel and Cantor believe that once one sees Absolute Infinity he knows all (the whole story of mathematics)? — Gregory

Cantor thought the absolute infinity was God. I don't know if he ever claimed it was the whole story of mathematics.

Don't recall reading whether Gödel had an opinion on the matter. I don't think the concept of absolute infinity was relevant by Gödel's time.

Did not Godel and Cantor believe that once one sees Absolute Infinity he knows all (the whole story of mathematics)? — Gregory

What passages by Cantor and Godel do you have in mind?

I don't know what relationship you have in mind between the quote of @fishfry (refuted by me) and the quote of me, especially since neither references absolute infinity.

entertaining — fishfry

Sure, laughs at your expense.

"
Cantor believed that God's infinity is the beginning and end of all other infinities, and that God's knowledge makes all infinity finite in some way.

Cantor believed that God put the concept of numbers into the human mind, and that the existence of numbers in God's mind was the basis for their existence in humans

Cantor's views on God were important to his defense of his theory of the transfinite, and he used his conception of God to motivate his conception of infinity in mathematics

Cantor believed that God put the concept of numbers into the human mind, and that the existence of numbers in God's mind was the basis for their existence in humans.

Cantor linked the absolute infinite, which is a number greater than any other quantity, with God"
Google AI

Cantor thought the absolute infinity was God. I don't know if he ever claimed it was the whole story of mathematics.

Don't recall reading whether Gödel had an opinion on the matter. I don't think the concept of absolute infinity was relevant by Gödel's time — fishfry

Well from what I've been reading from secondary sources God is the infinity model for all infinities for Cantor. Godel had his ontological proof too, but i would have to check Chatgpt for more info

I don't know what relationship you have in mind between the quote of fishfry (refuted by me) and the quote of me, especially since neither references absolute infinity — TonesInDeepFreeze

I thought fishfry was referencing the set of all sets and numbers, and you seemed to wonder how we can go without being able to prove mathematics as a total system. For Cantor google says God is beyond all mathematics (a 0 or a 1?) yet completes all of the theory of numbers and sets. This is very Pythagorean. Only a special infinity can subsume the whole of math

What passages do you refer to?

/

You said Cantor and Godel say "once one sees". I take 'one' to refer to humans, not to a god. But did Cantor or Godel say that any humans see absolute infinity?

Godel had his ontological proof too — Gregory

You do know that Godel's work in mathematics does not invoke his modal theological argument?

check Chatgpt for more info — Gregory

You're serious? You haven't caught on to the fact that such AI bots are so often horribly wrong and fabricate regularly?

you seemed to wonder how we can go without being able to prove mathematics as a total system. — Gregory

I don't know what you mean by a 'total system'. Do you mean a system that is negation complete?

Only a special infinity can subsume the whole of math — Gregory

I don't know what "the whole of math" is. But the existence of an inaccessible cardinal is sufficient to prove that ZFC has a model.

Only a special infinity can subsume the whole of math — Gregory

This sounds more theological than math foundational.

I thought fishfry was referencing the set of all sets and numbers, — Gregory

Cantor's absolute infinity was the collection of all ordinal numbers, which he called an "inconsistent multiplicity," since he recognized that the collection of all ordinals could not be an ordinal.

A quick web search turned up this interesting looking article. "Cantor, God, And
Inconsistent Multiplicities"

https://intapi.sciendo.com/pdf/10.1515/slgr-2016-0008

such AI bots are so often horribly wrong and fabricate regularly — TonesInDeepFreeze

Not unlike certain posters I could name! :rofl: :rofl: :rofl: :rofl: :rofl: :rofl:

Not unlike certain posters I could name! — fishfry

Yes, self-reference should be a snap for you.

You're serious? You haven't caught on to the fact that such AI bots are so often horribly wrong and fabricate regularly? — TonesInDeepFreeze

No because i hardly ever ever use them. I don't have original sources for quotes by them; I had learned a little about them from internet videos. I've never claimed to have other than secondary sources, but if you search quotes by Cantor on the internet, there are these:

"The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds."

"A set is a Many that allows itself to be thought of as a One."

Again I don't know if these are in his original writings. I do most of my research from actual books. But I was trying to add something philosophical to the debate

.

Here is the argument:

D) By continuum I mean a set of distinct points without an abrupt change or gap between points — MoK

I like your argument but I would say that the conclusion that follows is that D is an inadequate definition of a continuum. A continuum cannot be completely described with points. I've had ramblings in another thread on an alternate view where continua are fundamental and points are derived but those conversations ended similar to your thread where others (rightfully) recommended I read more.

Well that's what I'm doing! I'm reading a book on the fundamentals of mathematical logic. I disagree with those here who say that ai chat bots are not helpful. As I'm working through the text and problem sets, gpt4o and now o1 have been instrumental in me understanding the material. It's like having a personal teaching assistant. It's not perfect but its benefits far outweigh it's drawbacks if you can't afford the time or money to focus on a PhD.

It's not ruled out that in certain circumstances a chatbot can provide more explanation. But one has to be very very careful. Aside from whatever correct information a chatbot gives, there is likely to be a lot that is plainly incorrect and misleading - degradation of knowledge and understanding. I've seen it time after time in various subjects, including mathematics. I mean ridiculously wrong. Chatbots don't have a conscience. They are not committed to fact. They generate text. They don't generate it conscientiously to deserve trust.

I am not gifted in logic and mathematics. So it surprises me that people I would guess to have a lot more natural ability than me get stuck on foundational basics. 95% of what I know about the subjects came from starting with the most basic textbooks and studying them carefully and exactly. Then composing systematic notes for myself. Then classroom instruction served mainly as a review, a way of further solidifying what I already had firmly understood from the books. Of course, people are different, so my personal route might not serve everyone. But I venture to say that if I had relied on chatbots to supplement the books, I would have been quite confused and misinformed. Try it yourself. Ask ChatGPT to prove that "every consistent theory has a model" entails "every validity is a theorem of the first order predicate calculus" (a key result of mathematical logic). See if you can get it to provide a proof that doesn't assume what is supposed to prove. (I just now tried it again, and it spewed nonsense. Junk. It just improvises verbiage to make a mere appearance of saying something meaningful.)

And the PhD remark would be taken in the spirit of casual hyperbole. Of course, one doesn't have to come within a million miles of a PhD just to learn basic mathematical logic.

internet videos — Gregory

Let me guess. Those are videos that are of the caliber of claiming that Cantor was a nutcase based on the fact that he was in sanitarium.

I like your argument but I would say that the conclusion that follows is that D is an inadequate definition of a continuum. A continuum cannot be completely described with points. — keystone

Yes, my definition of the continuum is not adequate. Another poster gave a definition continuum close to mine but it is correct. I can search the thread and find the definition for you if you are interested.

Those are videos that are of the caliber of claiming that Cantor was a nutcase based on the fact that he was in sanitarium.
6h — TonesInDeepFreeze

Ive never seen such a video. And i was supporting Cantor so i dont know what you are talking about. As for seeing beyond mathematics, when i pressed fishfry on Zeno, he said uh oh let's not discuss it. Do you have the same answer?

when i pressed fishfry on Zeno, he said uh oh let's not discuss it. — Gregory

Because it's miscategorized in this thread, not for any other reason. There's a lengthy and tedious supertask thread that was active a few weeks ago that I do not want to see reactivated here. That's what I was referring to.

Why would you attribute sentiments to me out of context like that?

You don't want me to start attributing you out of context, do you?

ps -- When you mentioned Zeno to me I responded on point here.

https://thephilosophyforum.com/discussion/comment/932896

I had no intention of misrepresenting you, but how many times over the years have we debated Zeno? Several for sure. It's not about supertasks. The cylinder simply lies there and the question of what color the top is after we notice it alternates from blue to purple is a basic arithematic geometric question that is so basic i suspect it has no answer. I'd gladly be proven wrong. But it seems the discussion always ends the same way. Is not the clash between string theory and loop quantum gravity largely about this? String theory says there the most basic thing is a zero dimensional string. Which is obviously a contradiction in terms. LQG says there is discrete space, but this suffers the exact same fate. We ARE the very union of finite and infinite, so we can not make sense of Zeno for this reason. So your response about supertasks was just another dodge in another year on a different, to my mind, with Zeno sitting firmly in place. Peaceful

See if you can get it to provide a proof that doesn't assume what is supposed to prove. — TonesInDeepFreeze

I don't know what chatbot you used and I'm not skilled enough at logic to evaluate the output of OpenAI's latest o1, but I'd be keen to know what you think:

Of course, one doesn't have to come within a million miles of a PhD just to learn basic mathematical logic. — TonesInDeepFreeze

I suppose it depends on what we're talking about. If we're talking about developing original ideas on continua I think more than a basic understanding of mathematical logic is required.

Of course, people are different, so my personal route might not serve everyone. But I venture to say that if I had relied on chatbots to supplement the books, I would have been quite confused and misinformed. — TonesInDeepFreeze

Not only are people different, but times are changing. If now is not the moment for AI, the near future will be.