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 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.

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.

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?).

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?

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".

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.

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.

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.

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.

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.

Not unlike certain posters I could name! — fishfry

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

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.

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?

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?

Godel had his ontological proof too — Gregory

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

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?

entertaining — fishfry

Sure, laughs at your expense.

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.

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?

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".

This is what I'm confused about with your objection to the extended reals (or integers, etc.) — fishfry

I have no objection to the system of extended reals. I explicitly said that it is rigorous. And, of course, it can be useful rotationally (and perhaps, I don't know, it is required for certain aspects of mathematics). I'll add also that it is interesting

And I don't claim that one can't define 'extended integer'.

For my thoughts further about all that, I refer to the posts I wrote.

I noted that the extended reals are essentially a notational convenience, and we could live without them. — fishfry

I don't know enough to say that there aren't contexts in which we do need them as not mere convenience. But, yes, part of my point is that they are not required for the simple context in this thread.

And you seemed to be arguing that because we COULD live without them, then we SHOULD live without them. — fishfry

That is not my argument. For my actual points, I refer to the posts I wrote.

The definition of limit is rather involved, at least for people first encountering it, involving epsilon and delta and universal and existential quantifiers and so forth. — fishfry

It's not really very involved, but my guess is that it is involved enough (it has a universal quantifier followed by an existential quantifier followed by a universal quantifier) that its mild convolution scares students in the first week of calculus. Though, aha!, if one has learned basic symbolic logic, the quantifiers are a snap.

By your logic (as I understand it), it would be parsimonious (a virtue, I gather, but I'm not sure why) to dispense with it, and do the raw epsilonics every time we want to mention a limit. — fishfry

Indeed, you don't understand what I said. Indeed, you are saying the exact opposite of what I said. My previous post was that we do not have to reiterate the definition of 'limit' each time. Indeed, I showed that we can boil the definiens down to just a 1-place operation symbol and a single argument appended to the operation symbol.

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

You object to the use of the extended reals as a notational convenience for infinite limits and limits at infinity — fishfry

No, I don't. I pointed out problems with a particular formulation you used for sequences. And later in the back and forth between us, I mentioned that, more generally, the extended reals can usually be dispensed. I didn't say use of them is generally incorrect.

I don't understand your point about the extended reals. — fishfry

Perhaps you would go back to the posts and tell me the first sentence I said that you don't understand and that is not explained in subsequent passages.

parsimonious — fishfry

I thought that in context 'parsimonious' would be understood in reference to the point I made that the reals are sufficient for the entities for basic analysis, so it is less parsimonious to rely on yet more entities (-inf and +inf). But perhaps I should have been explicit about that. And again, that is not to say that the system of extended reals is incorrect.

"0 = the limit of the function 1/x where x ranges over the positive natural numbers," what do you mean by limit? Do you mean that for all epsilon there exists a delta such that ... etc? Where does your definitional parsimony end? If I say, "Let X be a topological space," should I replace that with, "Let X be a set along with a collection of its subsets satisfying ..." and then repeat the definition of a topology? — fishfry

Yes, I mean the ordinary definition.

Your analogy works in my favor. Given that I already have a definition of 'a topology', I can just say 'a topology' without reciting again its definiens. Given that I already have a definition of 'the limit', I can just say 'the limit' without reciting again its definiens.

'lim' is a variable binding operator, but it can be reduced to a regular operation symbol:

Df. If g is a function from the set of positive natural numbers into the set of real numbers, and there exists a unique real number x such that for every positive real number y, there exists a positive natural number n such that |g(n) - x| < y, then Lg = the unique real number x such that for every positive real number y, there exists a positive natural number n such that |g(n) - x| < y

You see that 'L' is an operation symbol that takes only finitely many arguments - in this case, one argument.

The argument itself is an infinite set (an infinite sequence in this case), which is okay, because the operation symbol takes only finitely many arguments - in this case, one argument.

And we prove, regarding the function f we previously defined:

Lf = 0

In everyday parlance, "the limit of f is 0" [...] — TonesInDeepFreeze

So, we can boil it all down to a 1-place operation symbol, say 'L' and an argument, in this case 'f'.

Without the axiom of infinity, THERE IS NO SET containing all the natural numbers. — fishfry

That is a good one to get back to.

Dropping the axiom of infinity does not entail that there is not a set that has all the natural numbers as members.

Rather, dropping the axiom of infinity from ZF and adding the negation of the axiom of infinity entails that there is not a set that has all the natural numbers as members. (To be exact, adding the negation of the axiom of infinity entails that there is not a set that has all the natural numbers, whether or not we also drop the axiom of infinity, but to not drop the axiom of infinity and add the negation of the axiom of infinity entails all sentences in the language anyway.)

Not a full time job, but a bit of labor it is.

At this point I'm just trolling you. — fishfry

A confession long overdue.

/

"I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle)." - TonesInDeepFreeze

The words "with some restrictions on that principle" explicitly declare that the idea is not unrestricted comprehension.

the required restriction would negate your point. — fishfry

The required restriction is that we don't claim that every property determines a set. That doesn't entail that there are not many properties that do entail a set. One of them may be the property of being a natural number. And (1) informally, in non-axiomatic reasoning, such as was my context, one may take the property of being a natural number to determine a set, without contradiction, and (2) Even formally, in ZF\AxInf, we have that AxInf is equivalent with "there exists the set of all and only the natural numbers". So even there, one may have the background of ZF\AxInf and also consider the property of being a natural number and think, "You know what, I think that property determines a set; in fact, I feel so strongly about it that I'm going to make it an axiom" and that in no way commits one to unrestricted comprehension.

If in one's mind, one views the property of being a natural number as suitable for determining a set, then one hasn't thereby committed to unrestricted comprehension. And, more generally, if one looks favorably on the notion that properties determine sets, albeit with some restrictions, one hasn't thereby committed to unrestricted comprehension. And this doesn't even have to be formal. I consider the property of being a book on my bookshelf and regard that property as determining a set.

I am not committing contradiction when I think "The property of being a natural number. Yeah, I think that is one of the properties to which the restrictions don't need to be applied, so, yeah, I do see that property as determining a set" while also being aware, "The property of being a member of oneself. Yeah, I've learned that that I better not allow that that property determines a set".


/

And Dedekind himself gave an argument for the existence of infinite sets without reference to axioms.

Was it not perfectly clear the other day that the usual order on the integers carries over to the extended integers — fishfry

Your question is answered by reading what I posted about that.

But today you were flat out wrong to claim that "reason" shows that there are infinite sets. — fishfry

You skipped what I said about that.

Likewise your invocation of unrestricted comprehension to justify that falsehood. — fishfry

Whoa. I did not at all invoke unrestricted comprehension. Not only is invoking unrestricted comprehension not in my words, not derivable from my words, but you would well know from many exchanges that you've seen that I have myself explained to other posters that unrestricted comprehension is contradictory. And I even explicitly wrote "with some restrictions".

I couldn't find any reasonable liberty there so I corrected your material errors. — fishfry

My error was to say 'formal'.

It's a full time job, let me tell you. — fishfry

You claim.

/

By the way, Dedekind himself gives an intuitive argument for the existence of infinite sets unrelated to axioms.

I don't get it. Or maybe adducing that quote is just your way of saying "yeah yeah" ironically. If so, whatever.

You are the one who attacks every little technical inaccuracy anyone makes, even if their overall meaning is obvious. But you don't hold yourself to that same standard. — fishfry

(1) It's hyperbole that I "attack" in all cases. Rather, most often I just plainly state the correction. But often when it's a crank who continues to ignore the correction, then I do comment personally.

(2) It's hyperbole that I take exception to "every little" inaccuracy. I quite understand that in the confines of posting boxes, one can't always cross every technical 'i' and dot every technical 't' (spoonerism intended). I've remarked about that a few times. So I comment on lapses or liberties taken when (a) They are material, (b) They are slight but still the subject is markedly better represented by corrections. (c) I would like to suggest notation I think is sharper or significantly more elegant.

(3) I do hold myself to a standard. Indeed, when I post technical stuff, I almost always regret that my formulations and explanations still could be more rigorous. But, as mentioned, posting does not always allow such perfection. And when I have make a significant material error, I either correct myself in edit (stating that an edit has been made if the correction is consequential enough) or correct myself when brought to my attention. Indeed, in this instance, it was not even technical, but still material, that I erred by saying "formal" when what I meant is 'deductive' or 'discursive' even if not formal, and I posted that I recognized that error.

[Added in edit:]

even if their overall meaning is obvious — fishfry

Even if the gravamen is correct, there still can be points along the way that bear correction.

/

There is no simple formula for when corrections or suggestions for sharper formulations are due. So one has to "play it by ear" and use one's best judgment. I hope that I am corrected when appropriate but also hope that some reasonable liberty is granted, just as I hope that that is how I comment on others. And, as said, I virtually always kinda rue that it would be unwieldy for me to even make every formulation pinpoint given the context of posting. I understand that others might sometimes feel similarly about their own posts too.

Main thing is: If someone replies to me with a plain and correct correction, I do not (or at least I hope I do not) take that personally. And I think it's a lot better when others don't take it personally. Cranks, though, are a whole other ballgame.

I don't know the relevance you intend with that quote.

"When caught in a material error, I just claim I didn't really mean it that way." — fishfry

I said that I should not have said 'formally'. Recognizing an error in wording is not a bad thing. I don't blame you if you took my remarks not in the spirit I hope they would be received; but I have corrected a certain wording that I recognize now to be not what I meant. Actually, it occurs to me that an even better word is 'discursively' in the sense of 'analyzing'. If another poster did that in conversation with me, then I would welcome that as quite reasonable.

Again, my point was that I do see a difference in conceiving discursively and visualizing. And that, personally, I conceive of infinite sets discursively even though I can't wholly visualize them.

It's a sidenote in this discussion inspired by what I thought was an apt and well said comment by another poster. My take is not even remotely a technical or even philosophical thing.

Too tedious to likewise mock and debunk every single point you made in this post. I stand by my previous remarks. — fishfry

"When caught in a material error, I just claim that I was not in error".

/

(1) Rejecting the axiom of infinity does not entail that there are no infinite sets. Rather, both rejecting the axiom of infinity and adopting the negation of the axiom of infinity entails that there are no infinite sets. (This pertains to ZF.)

(2) It is not precluded that one may provide an axiomatization that proves that there are infinite sets other than by adopting the axiom of infinity.

(3) It is utterly ordinary that mathematicians and philosophers say "true statement of arithmetic" as implicit for "true in the standard model". If you don't believe me, then you could do an Internet search on 'true arithmetic' and see an article at your favorite go-to Internet encyclopedia. Moreover, you have seen many posts by me in which I was emphatic in explaining that, formally, truth is always per models.

Reason is utterly insufficient to determine if there are infinite sets.

The only thing that guarantees the existence of an infinite set is the axiom of infinity. — fishfry

I should not have said 'formally'. I meant informal deduction. My point was to contrast informal deduction with "mind's eye" visualization. I reason from the notion that there is the property of being a natural number. And I don't claim that reasoning is as rigorous in avoiding paradox as axioms. My point was at a very broad level of generality, not necessarily responsible to formal contradictions. I should have counted on someone not recognizing that my remarks are merely in the spirit of personal reflection. Moreover, I am not advocating such reflections as philosophical conclusions that others necessarily should adopt.

If we adopt the axiom of infinity, there are infinite sets. If we reject the axiom of infinity, there are no infinite sets. It's as simple as that. — fishfry

It's not as a simple as that, and it is not correct:

(1) To reject the axiom of infinity is to not include it in the axioms. Not including the axiom of infinity does not entail that there are no infinite sets. Rather, to entail that there are no infinite sets requires both not including the axiom of infinity and also adopting the negation of the axiom of infinity.

(2) The axiom of infinity is used for many axiomatizations. It is not precluded that one may axiomatize differently and still derive the theorem that there exist infinite sets. It is fine to say that with ZFC, for example that the axiom of infinity is required to prove the existence of an infinite set; but my remarks were not specific to axiom systems.

The existence of an infinite set is purely a matter of accepting or rejecting the assumption that says there is an infinite set. — fishfry

Again, I should not have said 'formal'. I meant informal deduction.

It's easy: I understand the property of being a natural number; and I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle).
— TonesInDeepFreeze

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. — fishfry

(1) The axiom of infinity provides that there is a successor-inductive set. The axiom of separation provides deriving the set of natural numbers from the fact that there exists a successor-inductive set. w is the unique set that is a subset of all successor-inductive sets. There is no particular proper superset mentioned. Only that there is a successor-inductive set (whether or not a proper superset of any other successor-inductive sets). Then, after we have proven that there is a unique set that is a subset (notice: not even presuming that it is a proper subset) of all successor-inductive sets, we prove that there are proper supersets of w that are successor-inductive (e.g. w+w).

(2) I only described (subjective) reasoning, not even associated with any particular axioms. I mentioned that the principle "for every property there is a set" would not be taken without restriction (if I were to expand on that remark, it would be that such properties as "is not a member of itself" or more generally, unrestricted comprehension don't fly). Taking the property "is a natural number" to provide the set of natural numbers is not contradictory.

One problem though does haunt me: Every arithmetical statement is either true or false.
— TonesInDeepFreeze

No. Just no. Given an arithmetical statement (a syntactic entity) AND AN INTERPRETATION of the symbols of that statement (semantics) then the statement becomes true or false. — fishfry

Of course truth is relative to models. But mathematicians and philosophers most often take the liberty of saying "true" or "true in arithmetic" or "true statement of arithmetic" implicitly to mean "true in the standard model for PA". When a mathematician or philosopher says "it's a true statement of arithmetic", it is implicit that more formally that would be "is true in the standard model of PA".

Indeed, at least a few times in the past, I've mentioned to you that in greater formality, of course, sentences are only true or false per a given model or per a class of models (by the way, 'class' also is used informally (but such that with other locutions we can avoid 'class') there since, at least in Z set theories, there does not exist the proper class of all the models of a theory.

So when I say "true statement of arithmetic" you may always regard it at the usual shorthand for "true in the standard model of PA".

That was Zeno's scam. He conned people into thinking that Tortoise had just as good a chance as Achilles, then he took people's bets on Tortoise. It's talked about in the books "Conundrums For Dummies" and "The Complete Idiot's Guide to Greek Antinomies".

Only reason can understand the infinite, not imagination (it would seem). — Gregory

That seems well put.

I use reason to formally conceive that there are infinite sets. It's easy: I understand the property of being a natural number; and I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle).

But I can't informally see in my "mind's eye" an infinite set. Fortunately, though, I don't study mathematics with a dictate that I must reject formal, deductive understandings merely on the basis that my mind's eye doesn't visualize them.

One problem though does haunt me: 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.

Can we calculate the contribution of the w-th term to the series? — MoK

What series? A series is an infinite summation, which is the limit of a sequence of finite sums; ordinarily the domain is w, so there is no w-th involved; w is not a member of w, so w is not in the domain of a sequence whose domain is w. There are sequences in which w is in the domain (such as the domain being wu{w}), but I don't know of one that is a series (maybe there is such a thing?).

The system of extended reals is rigorous. And we can define '-inf' and '+inf' in a way to instantiate the system. Then, one may wish to define such notation as 'lim[n -> inf] f(n)' where 'inf' does denote and is not merely a contextual notation. That's all fine.

But there's another approach, in which we don't need to have the system of extended reals for most of ordinary analysis, but rather we take the notations '-inf' and '+inf' as not denoting but rather as merely contextual.

And we can use either approach as suits the discourse.

I don't see a need for disagreement here. You can notate as you wish; and I can say why I also use that notation but like to point out that 'inf' is dispensable when we unpack, which has a pedagogical purpose: Many people, such as in Internet threads, are clueless about axiomatic, rigorous mathematics, so they have the misconception that 'inf' must name a certain entity. So it is instructive to explain that it is merely contextual notation that does not invoke any entities other than real numbers and a function.

Limiting ourselves for the sake of convenience to just Achilles and the tortoise, what exactly is there to debate about? — tim wood

What should be the payoff if you bet 1000 euros on Achilles.

Personally, I don't take the first one as involving the extended reals.

I take 'inf' in

lim[x -> inf] 1/x = 0

as notation that unpacks as:

0 = the limit of the function 1/x where x ranges over the positive natural numbers

So there is no need to involve an object named 'inf'. It's a more parsimonious approach.

I'm not sure offhand how I would unpack lim[x -> 0] 1/x = inf; but my hunch is that I could do it.

Back to matters at hand:

Hopefully you might now at least be beginning to understand the answers to your question:

There is the denumerable sequence that pairs each step in the staircase with a natural number. But there is also a proper superset of that sequence that has the set of natural numbers (w) itself as an argument and you may define whatever you wish to be the value of the sequence at that argument.

For example (starting with 1 instead of 0):

1 maps to 1
2 maps to 1/2
3 maps to 1/4
...
w maps to 0

I take from @fishfry (as an analogy in my thinking) that the extended reals have a point of positive infinity (inf) that is like w in this sense: inf is not itself an integer, but it is an extension that comes after all the integers. And w is not itself a natural number, but it is an ordinal that comes after all the natural numbers.

ave correctly said that the poster is ignorant and confused about mathematics.
— TonesInDeepFreeze — Metaphysician Undercover

The quintessential crank* Metaphysician Undercover misquotes me. I didn't post any sentence that begins with "ave". But this is a true sentence I did post:

After many posts where I made no personal comments, I have correctly said that the poster is ignorant and confused about mathematics. — TonesInDeepFreeze

If I have an "MO" it is providing an ignorant and confused poster with information and explanations (the poster himself this time makes a point of repeatedly saying that the information and explanations have been helpful to him) but also remarking on the emerging pertinent point that the poster persists in ignorance and confusion.

But now the crank, himself ignorant and confused about mathematics, emerges with a false, pointless, substanceless, sophomoric, clueless attempt at wit, including even botching his quote of me. He has nothing to say other than to spit an inane, gratuitous attempt at a snipe. Actually, it's good that he doesn't say anything about the subject of discussion, since virtually everything he spouts about mathematics is irrational disinformation.

* Being a crank includes sophistry and more.