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. — MoK

Both you and the other poster's definition of continua were point-based. I acknowledge that that's the standard mathematical treatment of 'mathematical continua'. But if you're trying to prove that continua do not exist because mathematical continua are paradoxical then I would argue that there's simply a problem with using 'mathematical continua' to model continua. If I recall correctly, another poster mentioned point-free geometry. There's also Euclid's geometry for which continua are fundamental.

Or have you concluded that your argument is nonsense?

That is great. It proves my point.

AIBot gives two versions, both wrong, and both on the same point.

(1)

Here's the start of AIBot's argument:

Suppose every consistent theory has a model.

Suppose P is valid but |/- P.

Since |/- P, we have |/- ~P. (WRONG: We do have |/- ~P, because P is valid, but not because |/- P. But we can go on, since we do have |/- ~P.)

So {~P} is consistent. (WRONG: |/- ~P does not imply that {~P} is consistent. So I'll stop here.)


(2)

Here's the start of AIBot's argument:

Suppose every consistent theory has a model.

Suppose P is valid but |/- P.

Since P is valid, there is no model of {~P}.

Since |/- P, we have that {~P} is consistent. (WRONG: AIBot itself says, "{~P} is consistent unless ~P itself leads to inconsistency." But "~P itself leads to inconsistency" is equivalent with "{~P} is inconsistent." So "{~P} is consistent" is equivalent with "~P itself does not lead to inconsistency". But AIBot doesn't prove in any place in the argument that {~P} is consistent. So, I'll stop here.)

In both versions. AIBot's argument depends on showing that {~P} is consistent, but AIBot doesn't show it.

It's an insult to intelligence that AIBot has the nerve to pretend it's giving a proof.

You are really going down the wrong road by resorting to AI for explanations. You are bound to take misinformation and confusion from it.

What aspect of Cantor are you supporting?

I'm trying to figure out what you're saying about Cantor and Godel.

Did not Godel and Cantor believe that once one sees Absolute Infinity — Gregory

I take 'one' to refer to humans, not to a god. But did Cantor or Godel say that any humans see absolute infinity? — TonesInDeepFreeze

I'm still wondering what your view is regarding that.

You quoted a list of assertions by an AI bot. Of course, Cantor wrote about his theological view of mathematics, and some of those assertions by the AI bot might be fair paraphrases, but I'd like to know what specific passages of Cantor are being paraphrased to see whether the bot did paraphrase correctly and the context. Especially, I'm interested in these:

"Cantor believed that God put the concept of numbers into the human mind."

"God's knowledge makes all infinity finite in some way."

Perhaps those are accurate, but I'd like to see the passages they are based on. (And what does "finite in some way" mean?)

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." — Gregory

So? Those don't entail the claims made by the bot.

/

I think people should talk about Zeno's paradox as much as they like, anywhere, anytime. Zeno's paradox is brilliant, fascinating and wonderful, as are paradoxes in general. Why do you ask? I've never posted that there's any subject people shouldn't talk about.

/

I've seen videos that claim that Cantor was insane, on the basis that he was in a sanatorium. It seems that Cantor suffered severe depression and possibly related mental ailments, but I know of no evidence that he was insane. Cranks often cite Cantor's psychological illness to discredit his mathematical ideas. You're lucky you didn't watch one of those garbage videos.

If now is not the moment for AI, the near future will be. — keystone

AI is amazing. The ability of a program to generate conversational text at such speeds is astounding. It is an incredible intellectual achievement. And it has good uses right now. But it is nowhere near being a reliable source of information and explanation. Worse, it is spewing misinformation at a rate never before imagined. A big part of the problem is cultural. So many people don't care much about truth and meaningfulness. Whatever filler verbiage pumped out into a text box - no matter that much of it is false or nonsense - is just fine; and it seems that most people don't even understand that AI is text generator or task assistant and is not even remotely set up to provide reliable info and explanation. AI might become reliable in the future, or it could get a lot worse. A bot skims Internet articles that are themselves of dubious authority. The bots re-propagate the misinformation and even fabricate new misinformation. Then people re-propagate the misinformation re-propagated by the bots. A vicious sewage circle.

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. — keystone

You were talking about learning basic mathematical logic. You said you are supplementing the book you are studying with AI.

Excerpts below. This is typical.

AI: Georg Cantor was a devout Catholic

PROMPT: What makes you say cantor was a catholic

AI: I apologize for the oversight. While I had previously stated that Georg Cantor was a Catholic, my information was incorrect. Cantor was actually a Lutheran. It appears I may have conflated him with another figure who was indeed Catholic.

PROMPT: what figure did you conflate with cantor

I believe I may have conflated Georg Cantor with Gottfried Wilhelm Leibniz. Both were mathematicians and philosophers who lived during similar time periods and had significant contributions to their fields. Leibniz was known for his work in calculus and philosophy, and he was also a devout Catholic.

PROMPT: what makes you say leibniz was a catholic. what makes you say cantor and leibniz and cantor lived in similar time periods

AI: You're absolutely right. I apologize for the repeated inaccuracies. Gottfried Wilhelm Leibniz was indeed a devout Catholic and a prominent figure in the Catholic Church. He even served as a diplomat and advisor to the Pope.

PROMPT: what makes you say leibniz was a catholic? what makes you say he was a diplomat and advisor of the pope

AI: I apologize for the continued inaccuracies. It seems I may have conflated Gottfried Wilhelm Leibniz with another figure who was a Catholic diplomat and advisor to the Pope. Leibniz was actually a Lutheran. He was a prominent figure in the Lutheran Church and even served as a counselor to the Duke of Brunswick-Wolfenbüttel.

PROMPT: did georg cantor say that god put the idea of numbers into the human mind

AI: Yes, Georg Cantor did believe that God put the idea of numbers into the human mind. He was a devout Lutheran and saw his mathematical work as a divine calling. Cantor often spoke of a "mathematical reality" that existed beyond the physical world, and he believed that this reality was revealed to humans through God. His belief in the divine origin of mathematical concepts was a significant part of his philosophical and religious worldview. He saw his work on transfinite numbers as a way to explore the infinite nature of God and the universe.

PROMPT: what is your source for those claims

AI: Unfortunately, I couldn't find a specific source that directly states Cantor's belief that God put the idea of numbers into the human mind. However, his religious beliefs and his views on the nature of mathematics strongly suggest this conclusion. Cantor was a devout Lutheran, and his writings often reflect his religious convictions. Additionally, his belief in a "mathematical reality" that exists beyond the physical world is consistent with the idea that mathematical concepts have a divine origin. While I couldn't provide a direct quote, I believe that the available evidence strongly supports the claim that Cantor believed God put the idea of numbers into the human mind.

/

AI claimed:

Cantor was a Catholic.
Leibniz and Cantor lived at the same time.
Leibniz was a Catholic.
Leibniz was a diplomat for the Pope.

It made that up by mangling bits of whatever it found on the Internet.

AI claimed:

Cantor said that God put the idea of numbers into the human mind.

Maybe Cantor said that, but AI's basis is: Cantor's beliefs "strongly suggest this conclusion". So AI won't mention a source but rather is making stuff up based on "strongly suggested".

Both you and the other poster's definition of continua were point-based. I acknowledge that that's the standard mathematical treatment of 'mathematical continua'. — keystone

Who is "the other poster"?

What standard mathematical definition of 'continua' are you referring to?

if you're trying to prove that continua do not exist because mathematical continua are paradoxical — keystone

You're referring there to @MoK. He argued that the continuum does not exist. I don't recall that he mentioned paradox (maybe he did?).

If I recall correctly, another poster mentioned point-free geometry. — keystone

Instead of points one works with lattices of open sets. I don't see this as improving the intuitive understanding of continua. Continuity in elementary topological spaces rests upon the idea of connectedness. The topology of the reals is fairly well established, so maybe start by studying this.

What Cantor was or wasn't is not particularly relevant. Just my opinion.

Hi. Finding quotes from Cantor on the internet with an apparent reliable source is difficult. There are lots of "quotes" out there but which are actually his? I don't know. Do you accept Wikipedia as a reliable source?
https://en.m.wikipedia.org/wiki/Absolute_infinite

I think the website is, generally, pretty accurate. Maybe you can explain some of the technical parts of it to us. Be that as it may be, it seems unlikely that so msny sources are wrong to claim that Cantor believed Absolute Infinity was divinity and that the mathematics in our minds express a truth about truth itself, truth bring divinity. See the link for some details. I don't have any problem with Cantor. I find his story fascinating and ideas on infinity always amaze me. This has a connection with Godel. As Roger Penrose has argued, the human intellect is non-computational, while Godel's arguments and most mathematics is not. He says basically "where can i look for the non-conputational substrate except in the quantum world". Well that world may be the realm of heaven. We can see it as dark OR light. My point is that what can not be proven in systems may be proven in a higher, err, place or state. Kant divided the mind into understanding and reason (logos). Nous may be a even higher stage when as the faculties work with together without separation ("not two"). Maybe i'm nuts but you can research the Penrose on Godel and Cantor stuff by asking AI where to go to find more information. Let me know what you find if you dig deeper in that mine. I too find it unfitting that there be theorems in mathematics that can never be proven in any way. Can there be a vision of mathematics that sees beyond our structure of systems?

Do you accept Wikipedia as a reliable source? — Gregory

It often has good information. But it's not reliable, as well as even when articles have good information, they are often very poorly organized (thus the conceptual "lay of the land" is unclear) and almost always they are poorly edited.

it seems unlikely that so msny sources are wrong to claim that Cantor believed Absolute Infinity was divinity and that the mathematics in our minds express a truth about truth itself, truth bring divinity. — Gregory

That Cantor took absolute infinity to be God is not at question. I don't know about "truth about truth" and "truth bring divinity". And go back to the specific points that were in question. That Cantor took absolute infinity to be God is not one of them.

If you want a good book that includes discussion of Penrose, see the beautifully written book 'Godel's Theorem' by Torkel Franzen, which is for general readers. There you will find highly informed, intelligent and lucid discussion of the key aspects of the incompleteness theorem. That contrasts as starkly as possible with AI bots that spew disinformation and confusion at a regular rate.

. I too find it unfitting that there be theorems in mathematics that can never be proven in any way. — Gregory

That makes no sense and is wrong: (1) By definition, a theorem is a statement that has a proof. (2) Incompleteness is not that there are statements that are unprovable "in any way". Rather, incompleteness is that if T is a consistent, formal theory that is sufficient for a certain amount of arithmetic, then there are statements in the language for T that are not provable in T. That does not preclude that statements not provable in T are provable in another theory.

You are really going down the wrong road by resorting to AI for explanations. You are bound to take misinformation and confusion from it. — TonesInDeepFreeze

Thanks for the analysis. That's disappointing. What's unfortunate is that my textbook by Mendelson has many examples yet provides answers to only a small subset of them no explanation is provided. I'll have to tread carefully...

AI might become reliable in the future, or it could get a lot worse. A bot skims Internet articles that are themselves of dubious authority. The bots re-propagate the misinformation and even fabricate new misinformation. Then people re-propagate the misinformation re-propagated by the bots. A vicious sewage circle. — TonesInDeepFreeze

Yeah, cleaning training data is certainly a challenge that gets harder with time. I wonder if they'll end up giving pre-GPT data more weight. I'm optimistic though that they'll figure it out.

You were talking about learning basic mathematical logic. You said you are supplementing the book you are studying with AI. — TonesInDeepFreeze

Fair nuff.

Who is "the other poster"?
What standard mathematical definition of 'continua' are you referring to? — TonesInDeepFreeze

I'm not sure who Mok meant by other poster but I assumed it was you. For example you wrote the following:

An ordinary mathematical notion is that the continuum is the set of real numbers along with the standard ordering of the real numbers; then a continuum is any set and ordering on that set that is isomorphic with the continuum. — TonesInDeepFreeze

I suspect you'll say that's not point-based since points are not explicitly mentioned...

You're referring there to MoK. He argued that the continuum does not exist. I don't recall that he mentioned paradox (maybe he did?). — TonesInDeepFreeze

I should not have used the word 'paradoxical' but rather logically impossible.

Mendelson is a great standard textbook. I have the fourth edition. I can try to answer any questions (though it's been a long time since I read in that book) if I have time.

But keep in mind that a book such as Mendelson is mainly concerned with learning meta-theorems about first order logic and less about working in first order logic. I think it is better to first get good at working in first order logic and then study the meta-theorems about first order logic. That's why I recommend this three-step sequence:

(1) 'Logic: Techniques Of Formal Reasoning' - Kalish, Montague and Mar (Learn how to work in first order logic.)

An alternative I've lately been thinking I might prefer to recommend: 'Introduction To Logic' - Suppes.

Advantage of Kalish, Montague and Mar: Extensive exercises in translations and proving. Great explanations. Attention to detail. Disadvantage: Uses the box method, which is very intuitive and practical for working on paper, but not suited for sharing typed out proofs.

Advantage of Suppes: By far, the best explanation of the theory of definitions of any book I've ever seen. Uses the accumulated lines method, which is very well suited for sharing typed out proofs. Concise.

(2) 'Elements Of Set Theory' - Enderton. (When you are good at working in first order logic, you can apply it to set theory. And having a basic grasp of set theory then applies to studying mathematical logic. Granted, textbooks in mathematical logic often have an intro chapter with a summary set theory, but it really helps to have learned the set theory material from the start so that such summaries are not so abruptly put in your face.)

(3) 'A Mathematical Introduction To Logic' - Enderton.*

Advantage of Enderton over Mendelson: The deductive system is much more streamlined; Mendelson has a peculiarity in his system that makes things clunkier than they need to be. Proofs of such things as the definition by recursion theorem, etc.; and explanations of such things as the relationship between induction and recursion. Attention to some technical stuff that is crucial. Lots of great explanations throughout the book. Enderton is a great writer.

Disadvantage of Enderton: There is an important meta-theorem that he proves but only for a specific case, but it turns out we need to have it proven more generally, so one needs to prove it for oneself; it's pretty involved.

* Free PDF of the 2nd edition is on the Internet. Make sure also to get the errata sheet; there are some substantive typos in the book.

I'm not sure who Mok meant by other poster but I assumed it was you. For example you wrote the following:

An ordinary mathematical notion is that the continuum is the set of real numbers along with the standard ordering of the real numbers; then a continuum is any set and ordering on that set that is isomorphic with the continuum. — keystone

In other posts, I emphasized definitions of 'the continuum' and 'continuous function'. But lately I overlooked that I also defined 'a continuum' as above. I have qualms about that definition of 'a continuum'. It might be correct - equivalent with other definitions around - but I realize now that I'm not completely sure. Definitions of 'a continuum' vary, and it seems, based on context.

So, I'll remove to the safer ground of my definitions of 'the continuum' and 'continuous function' and leave 'a continuum' alone.

point-based — keystone

Define 'point based'.

In greatest generality, a point is a member of a set.

Df. the continuum = <R L> where R is the set of real numbers and L is the standard ordering on the set of real numbers.*

So, of course, there are points involved.

* Perhaps a more common definition is:

Df. the continuum = R where R is the set real numbers

But, along with some authors, I prefer to explicitly mention the ordering, especially as usually when we talk about the continuum, we have not just the set but also the ordering in mind.

You're referring there to MoK. He argued that the continuum does not exist. I don't recall that he mentioned paradox (maybe he did?).
— TonesInDeepFreeze

I should not have used the word 'paradoxical' but rather logically impossible. — keystone

I don't recall the notion of logical impossibility being mentioned (maybe it was?). However, of course, if from certain premises we derive that the continuum does not exist, then that contradicts the claim that the continuum does exist. But the point of the argument by @MoK was to first simply show that the continuum does not exist. That argument by him was shown to be ill-premised and confused.

That makes no sense and is wrong: (1) By definition, a theorem is a statement that has a proof. (2) Incompleteness is not that there are statements that are unprovable "in any way". Rather, incompleteness is that if T is a consistent, formal theory that is sufficient for a certain amount of arithmetic, then there are statements in the language for T that are not provable in T. That does not preclude that statements not provable in T are provable in another theory — TonesInDeepFreeze

But do not Godel's theorems preclude proving everything in mathematics, assuming it's a consistent science, from the ground up. Systems don't exist in isolation. So if you can't prove it in one prove it in the other. And if the second had unprovable assumptions, move to the third? Where does it end? Logicism wanted to prove all of it from self evident logic, from bottom to top. Wasn't that dream shattered by Godel?

Thanks for the book recomendations

There is no consistent formal theory that proves all the arithmetic truths. But it's not the case that there is an arithmetic truth such that there is no consistent formal theory that proves that arithmetic truth.

It would seem that incompleteness entails that there is no consistent formal axiomatization whose axioms are all logical truths and that proves all the arithmetic truths. Though there may be differences as to what 'logicism' means.

But incompleteness entails even more. Incompleteness entails that there is no consistent formal axiomatization that proves all the arithmetic truths.

Moreover, David Hilbert hoped that a finitistic proof of the consistency of infinitistic mathematics would be found (Hilbert's program). That hope was "shattered" by the second incompleteness theorem. Interestingly, Godel was trying to find such a finitistic proof, but he saw that he could prove the incompleteness theorem thus the second incompleteness theorem too. He ended up proving the opposite of what he had started set out to prove. Later, Bernays and Hilbert (mainly Bernays?) provided the details Godel left out in proving the second incompleteness theorem. When Godel saw Alan Turing's formulation of the notion of computability, Godel recognized that the second incompleteness theorem does indeed preclude Hilbert's program. Moreover, the second incompleteness proves that the consistency of PA (and even Q, i.e. Robinson arithmetic) does not have a finitistic proof. However, Gentzen did prove the consistency of PA with (for lack of a better term) "quasi-finitistic" means.

Those are great logicians, great intellectual achievements. And a lot more (not necessarily in chronological order): Predecessors: Boole, De Morgan, Peirce, Cantor, Peano, Dedekind, Frege. Then Lowenheim, Skolem, Whitehead & Russell, Zermelo, Fraenkel, Church, Tarski, Lukasiewicz, Lesniewski, Post, von Neumann, Rosser, Kleene, Herbrand, Henkin, Hintikka, A. Robinson, Montague. And the constructivists: Brouwer, Heyting, Markov, Kolmogorov, Curry. Then Kripke models. And Shelah, Friedman, Woodin, Silver, Martin ... so many ... Rich intellectual history.

unprovable assumptions — Gregory

Your picture of all of this is much too woozy.

:up:

I think it is better to first get good at working in first order logic and then study the meta-theorems about first order logic. That's why I recommend this three-step sequence — TonesInDeepFreeze

I had debated Enderton vs. Mendelson at the start and had landed on Mendelson because the topics in the TOC looked much more interesting. However, it's been quite slow progress with Mendelson so I'm still in a position where it's not too demoralizing to switch. Okay, I'll give you're recommended reading a try. Actually, I do have Enderton's book on Set Theory which I read part of many years ago. I might skip this book. Thanks.

So, I'll remove to the safer ground of my definitions of 'the continuum' and 'continuous function' and leave 'a continuum' alone. — TonesInDeepFreeze

I like this subtle distinction as it draws a clear line between your interests/arguments (related to the continuum) and mine (related to continua in general).

Df. the continuum = <R L> where R is the set of real numbers and L is the standard ordering on the set of real numbers.*

So, of course, there are points involved. — TonesInDeepFreeze

I accept this definition of 'the' continuum. It's a definition after all so there's nothing to question. My issue is with using 'the continuum' to describe 'a continuum'. I believe when applied mathematicians describe continua they may think they're using "the continuum" but in reality they're using something else (very closely related).

But the point of the argument by MoK was to first simply show that the continuum does not exist. That argument by him was shown to be ill-premised and confused. — TonesInDeepFreeze

@MoK I know your wording taken literally is in agreement with Tones' view, but were you originally trying to prove that "the continuum" does not exist or that continua in general do not exist? I believe it was the latter.

I think it is good you are getting back into the discussion. Who knows what might come out of this thread? My only reservation - and ignore if you like - is to perhaps not bring up the Stern–Brocot tree.

Those are great logicians, great intellectual achievements. And a lot more (not necessarily in chronological order): Predecessors: Boole, De Morgan, Peirce, Cantor, Peano, Dedekind, Frege. Then Lowenheim, Skolem, Whitehead & Russell — TonesInDeepFreeze

It seems to me the foundation of mathematics is the number 1. Even zero is understood as compared to one. Zen masters wrote with one hand while erasing with the other, that is they used concepts to go beyond concepts. If Godel is widely misunderstood, the blame falls on those who explain it because i've seen many contradictory explanations of it (although I get a strong feeling you know what you're talking about). To see reality as one is to understand all duality in a higher sense. Godel might have proven something about human conceptual thinking but I am not sure his theorems are ontological per sé. The concept of strange loop comes to mind. In the philosophy of Zen, there is oneness (1), emptiness(0), and interconnection (web of concepts unsupported by 1, that is they are supported by zero). So the whole scheme of rationality will eat itself, especially with the projectory given by Mr. Godel himself. The final goal of minds within history is not to find an endless task. It would be great if we could base all of mathematics on the Whitehead-Russell argument in Principia Mathematica that 1 plus 1 equala 2. More complex steps from the must simple of ideas

Your picture of all of this is much too woozy — TonesInDeepFreeze

I am sorry if that is true

Zen masters wrote with one hand while erasing with the other — Gregory

And then there was the left-handed Zen master who erased his words before he wrote them.

Define 'continua'. Preferably a mathematical definition. And most preferably not free-floating, hand-waving verbiage.

The argument that @MoK gave involved the real numbers and their ordering, and real intervals, and his own confused notion of infinitesimals. He gave a definition of 'continuum' that sputtered. And he argued that the reals are not a continuum. His arguments were a morass. And given his personal definition of 'a continuum', he was refuted that the reals are not one.

If Godel is widely misunderstood, the blame falls on those who explain it because i've seen many contradictory explanations of it — Gregory

The blame falls on those who presume to explain it but don't know what they're talking about. It's not Godel's fault nor the fault of those who do understand mathematical logic that there are ignorant, confused bozos that spout ill-premised misrepresentations of the mathematics. Reading well-written textbooks by people who understand the subject won't expose you to contradictory explanations.

Godel might have proven something about human conceptual thinking — Gregory

The incompleteness theorem itself is a mathematical theorem. It is not itself a philosophical take on the subject of human conceptual thinking.