Dear Prof Dr Joel David Hamkins,
My name is Ellena Caudwell and I am a 3rd year Undergraduate Mathematics student from the University of Exeter, UK.
I am currently involved in a group project based around D.E.Knuth’s book ‘Surreal Numbers’, and we were hoping to find examples of where this book has been used as a tool for teaching mathematics. I found an old page on the University of Amsterdam’s website explaining a course you ran in the 2nd Semester 2004/05: Surreal Numbers (http://www.illc.uva.nl/MScLogic/courses/Projects-0405-IIc/Hamkins.html) and it appears this is exactly what we were looking for.
I understand this was a long time ago, and you must be very busy, but we would be very grateful if you could answer a couple of questions to help us in our work.

Firstly, what level were the students who took part in the course, and in what way were they assessed?
Second, D.E. Knuth states in the postscript of his book that he wasn’t really trying to teach the theory of surreal numbers, but to ‘provide some material that would help to overcome… the lack of training for research work’. Therefore it is questionable how well this book helps teach surreal numbers.
1. Do you feel this course showed this book can be used successfully to teach surreal numbers? Was this your aim?
2. Do you feel it effectively shows the process of exploration and discovery of mathematical proof?

Any other thoughts or ideas on the topic would be most useful.
Thank you for your time and I look forward to hearing from you.
Ellena Caudwell

The course was a lot of fun for all of us, and I count it as a success. I would definitely be interested in running such a course again, and I think the book works very well for this kind of course. Shorter than a regular semester course, the course was filled mostly with masters degree students, with a few PhD students, but there would be no problem running a similar course for much longer. It was a small class, with about 8 students, which I think was relevant for its success. Following the idea of Knuth that you mention, we used the book not only to learn about the surreals but also to illustrate the practice of mathematical research. In particular, the students themselves presented much of the material on a rotating schedule, filling out the ideas of the book. Thus, the practice was that they would read the next part of the book, master the material, including whatever proofs needed filling in (which is the nature of the book), and make a presentation on that topic, at which we all would ask further questions and figure things out together. In addition, as is my general practice, the students wrote term papers, on a topic chosen in discussion with me. Assessment was based on the presentations and the paper.

To answer your specific questions: (1) The book can definitely be used successfully to teach surreal numbers, and yes, this was a major part of my aim. But this book requires more work from the students than a regular textbook, laying out all the material, since it is more open-ended. (2) As a result, yes, I do feel that the book and the way we used it in that course gave a good introduction to the process of mathematical research, particularly at the masters degree level. I think it could also work well with advanced undergraduates, if they were very motivated.

Dear professor Hamkins,
I am looking for a reviewer to my paper in progress “$mathcal{L}_{omega_1omega}$ -transfer principle in algebraic geometry”. Are you or someone you know interested in reviewing my work?
Sincerely yours,
Ali Bleybel

Dear JD Hamkins,
I discovered your page “The global choice principle in Gödel-Bernays set theory”, that I found very interesting. And let me thank you for giving my name in this page. Let me add that, even if iI learnt mathematics with the immense Laurent Schwartz, I did not do a career as a professional mathematician, but as a statistician-economist finishing as the chief lawyer of the french national institute of statistics and economic studies. Being now retired, I am interested in set theory and reading the FOM list and Mathoverflow; I also have a modest participation in the splendid site Metamath of Norman Megill, that you may know.
I allow me to to ask some more questions related to “What happens for proper classes inGödel-Bernays set theory with AC and Without GCh”.
1-First of all, I would like to remark that by an answer of Sam Roberts on Mathoverflow today, we have another equivalence for GCh that is:
“If P(A) injects inton A then AV injects into A”
2- We saw in a model such that there exists a definable class of pairs of set having no choice function, (so that GCh fails), the proper class On (and all equivalent classes, that are the well-orders classes) strictly inject in the class P(On), that is linearly ordered and strictly injects into the universal class V. It seems intuitively difficult to think that some class can exist that is strtictly between On and P(On) or between P(On) and V ?
3- We also saw that we must have at least three non-equivalent proper classes, V that is the ultimate maximum because every class injects in V, On that is minimal in the sense that every proper class that injects in V must be bijective with V and the class W you defined that is incomparable with On. I would like to ask some questions about W.
(i) Is it true that W is minimal in the same sense as on ?
(ii) Are W and P(On) comparable ?
(iii) Is it possible that W strictly injects P(W) and P(W) strictly injects V ?

Yes, I have enjoyed your questions on MathOverflow, and thank you for writing here. Your final questions are good ones. I have thought at length about the case of having a class between Ord and P(Ord), but so far without success.

Dear JD Hamkins,
Let me thank you most heartily for giving such a positive answer to my message.
I wish you the best for the end of this year and the next year as well.
Bien cordialement.
Gérard Lang

TL;DR: Nice talk at Einstein Chair, left comment, want to learn Set Theory in some moment.

Dear Prof. Hamkins,
Recently I left a comment concerning your talk for general mathematical audience concerning CH; very nice indeed. I’m not really sure if the comment got its way since I didn’t get any acknowledging message from the server.

Though I’m working in a different area now, I would be deeply interested in learning set theory in the not-that-far future. No one in my department (and, AFAIK, no one in my country!) is working on ST. If, with a little luck, I gather some people to work on this, I would be eager to invite you to give some course in Argentina.

I need a word of advice (surely won’t be the last time!).

I’m preparing a course on Set Theory for next year, and I am expecting a mixed audience; ranging from graduate students that are starting to study the material (Kunen’s 1980) as of this writing, to perhaps some undergrads that will have seen basic (though fairly complete) courses on general topology and real variable. Most of them will only have a rudimentary knowledge of logic, and some of them will be seeing the definition of ordinal for the first time.

How much can one afford to compress in a 1-semester course? I’m just starting to prepare this, but it seems that I could go up to $L$, and perhaps a crash course on the basics of MA (I’m stealing this idea from some UCLA summer school notes by Justin Palumbo). But perhaps this is just too ambitious… On the other hand, the people reading Kunen right now are eager to “force”, at least a little bit.

Thank you for reading this, I hope it doesn’t bother you.

My preference in an introductory set theory course is to do some large cardinals before L or forcing. It is relatively easy to develop an analysis of V_kappa and H_kappa for inaccessible cardinals (or the second inaccessible, or Mahlo etc.) and so on, and this introduces the important idea of looking at various models of set theory, and then get to measurable cardinals and ultrapowers. After this, one can develop L and HOD. For forcing, I always start first with an extended analysis of MA, to get used to the concepts of density and genericity, but without any need yet for P-names. After this, one can develop P-names and the forcing extensions a bit more easily. But I’ve never been able to do ALL this in one course; it is simply too much. If they really are seeing ordinals for the first time, then I would be happy if they just get to know some large cardinals and the ultrapower construction, which is something that I think every mathematician should know.

Thank you very much for your quick reply. Searching for the topics you indicated, it seems to me that Drake is a good source; I also think Jech might be of use. Next year I’ll tell how things went through.

Is 2 a member of the set {6}? Of course the answer is No. But that thing in the braces isn’t just a squiggle, it’s the number 6, and if 6 MEANS anything, it surely means six of something. And if there are six of anything, there are at least two of those things. Can you clear this up for me?

Finally, is the answer to this question as uncontroversial as the answer to the question, what is two plus two?

Haven’t you cleared it up yourself? The number 2 is not an element of the set {6}, since that set has only one element, the number 6, and that number is different from the number 2. Meanwhile, 2 is less than an element of {6}, because 2 is less than 6 and 6 is an element of {6}. But being less than an element of a set is not the same as being an element of the set. The fact that my child is shorter than a student in my calculus class does not make him a student in my calculus class. The fact that I earn less money per year than at least one Wall Street investment banker does not make me a Wall Street investment banker. I am younger than some of the retired faculty at my college, but this does not make me a retired faculty member at my college.

Thanks very much for your prompt and clear answer. Yes I was thinking vaguely along those lines, and some other reading was beginning to sort it all out for me, but it’s nice to hear clarification from a professional. Cheers!

(I feel a little embarrassed by the fact that I wrote above, unblushingly, “Dear Joel”.)
I’d like to know if you received my email (sent to your GC address on Dec. 21st).

I saw you involved in the question”is there a class of all classes”on mathoverflow.It is obvious that the answer is NO.However,I saw some people agreed to call this such a big collection “2-classes”,and by recurrsion we can have n-classes.In my point of view,let me suppose that the collection of all classes is “2-classes”.Then I can conclude that this is the biggest collection.Since this is not a class.Therefore if the “2-classes” is contained in another collection U then it follows that the “2-classes” is a class.A contradiction.Hence I can,t understand how the “n-classes” can be generated.I am looking for your reply.Thanks!

Dear Joel:
Recently I am interested in the question such as “he set of all sets” and “the class of all classes”.And I did a lot of research on Internet including asked on “math stack exchange”.And finally I found that maybe the “Grothendieck universe” is what can answer my question.However,I think this is not directly related to my question.And I have a question which confused me:

1.Can you tell me the initial idea for the “Grothendieck set throry”.Is it just want to treat the question what I mentioned above?

2.In “Grothendieck set theory”,Is it legal to have “the set of all sets(i.e. the univesal set “V”) and “the class of all classe(I think this has several way to define as this is outstrip ZFC set theory’s boundary)” be a set?And this construction can be infinitely extended?

3.Based on my Q2,it doesn’t matter how you define such as “the class of all classes(I define this to be the collection including all the subclass of V and including “V” itself) and so on with the high level collection.This all can be set.So in general,given any universe of collection,we all can have a Grothendieck universe including that universe(especially contain the powerset of that universe–if that universe can be a set in Grothendieck set theory).Finally,Grothendieck universes hierarchy is the ultimate collection that cantains anything you want.

I teach an advanced high school elective on the history of mathematics, leading up to Godel’s Incompleteness Theorem. Last week we were learning about ordinals and I used your slides about Infinite Chess as a way to make the ordinals seems more concrete and more useful.

Several students followed up with me about some of the positions in your talk. Specifically, in Releasing the Hordes, the students wondered whether the position really had value omega^2. After the initial rook move, could white capture with a pawn from the right column instead, leaving a gap that the white king could then attempt to retreat into? We spent an hour or so after school trying to work it out and couldn’t get anywhere. Is there somewhere on the internet where we can enter infinite chess positions? Can you maybe personally explain to us why this either works or doesn’t work? My students think the value is less than omega^2.

That is a very interesting idea! Your students are very insightful. Let me say that these infinite chess positions are often very finicky that way, and many of them resulted from a series of modifications to earlier similar positions that had some such flaw or other of the kind you are mentioning.

Your student’s idea shows that one has to take care with the set up. Black cannot allow that white will hide his king away in a cave.

I want to think more about it, but I think that black can handle it. Black moves his rook up on the first move. If white takes from the right, which is the line you are proposing, then black will respond like this. First, move the black rook to prevent white access to the wall. Next, move the black king up to protect the hole. Keep the black king between the hole and the white king. During these moves, white may be moving pawns to bring the hole lower, but they will meet halfway, which is still good enough, since n/2 can be made as big as you like (that is, move the rook up twice as far as you had really wanted to on the first move). Now, we are in a situation where the black king is guarding the hole. Next, black should move his rook very far out to the right. In such a situation, the white king cannot hang around anywhere near the wall, since there will be perpetual checks coming from the black rook. Thus, black can force the white king to chase down the black rook, very far away (as far as one likes). White cannot gain advantage by trying to return to the wall, since we can do the same thing again.

Does it work? I will think more about it. Let me know if you see any issues. What this means is that in this line, we don’t get $\omega\cdot n$ if the original black rook moves up to height $n$, as I had thought, since we will have to burn a few pawn moves for each round of harrassment, rather than just one.

Meanwhile, it seems that we can address the problem also by changing the position to have a double row of white pawns on the outside wall. Alternatively, we could place the white king harrassment arena in a separate place, more isolated from the wall protecting the doorway. That is, have another separate vertical wall of white pawns, with the black rook and white king to the right of it, and the black king between them.

Thanks for your comment, and I’d like to hear back from you.

Here is another idea, which will be better if it works. After black moves his rook up on the first move, if white should takes from the right, as you suggest, then black will first of all use his rook to keep white away from the wall. Then, black will aim to capture a pawn in the right wall, between the door and the hole. Black aims to occupy this hold with his king, and then have his rook free. For example, black can position his king a knight’s move above and to the right of the desired place, capture the pawn with his black rook, and then move the king diagonally next to the rook (since the pawn is no longer protecting), and then exit the hole with check, giving time to occupy the hole with the king.

If this works, then white will have no line remaining leading to check, since the white wall pawns can no longer advance, and so the door will never open.

But one must consider the possibility that the white king attempts to interfere with the insertion of the black king into the hole. For example, if the white king stayed next to the wall, there would be no “exit with check”. For example, white could attempt to capture the rook by approaching it from below, staying next the wall. Indeed, I think that this idea may not work because of this counterplay, since black will be forced to move the king away because of zugzwang.

The idea would seem to work, however, if we give black a pawn on the right side, since then black can avoid the zugzwang, and simply occupy the hole with both the king and rook, moving his pawn up. There is no need for check harrassments in this case, since no pawns are advancing and so black aims for a full draw, instead of forestalling loss.

It seems best to pursue a combination of the two lines I mention. That is, using the first strategy we can get the we can aim to get the white king very far away from the wall, since otherwise there is perpetual check, with the black king prevent white king access to the cave. Then, we pursue the second strategy to aim at inserting the black king into the wall below the hole, which will black pawn advances below and cause a draw (good for black). The point is that if the white king is very far from the wall, then we can implement the exit-with-check part of the king insertion. So the line will lead to a situation where the white king is far away from the black king, and the black king has inserted itself into the wall. This prevents pawns from advancing, and there is no way to dislodge the king from that spot, if the black rook is free.

Alternatively, it may be good enough simply to use the black rook to keep the white king even just a few steps away from the wall as currently. Then black positions his king to a place readying for the king insertion, and then carries it out without interference.

For example, if we simply started with the white king a little farther from the wall, it would definitely work.

Sure, let me know what you think. Currently I don’t think that the position needs to be modified and also that your line leads to a draw, so white won’t do that, since with the main line white will achieve checkmate.

math for kids is my new favorite website! i’d love to know your thoughts on zugzwangs in non-chance, perfect information games in general…

(i’ve made sure to post your Infinite Sudoku over there, so hopefully some of the other kids will be coming to check you work out. still a small community but holds great promise:)

Hello there! I’m an amateur-ish pure mathematician who specializes in set theory. A pressing matter is that it seems that Cantor’s Attic is down! There’s some sort of internal error going on, and it’s preventing me from adding to the website. For the functional part of this comment, I’m done! Just letting you know.

Also, I’d just like to say I contributed a lot to the website, being the first to add many fundamental large cardinal notions such as indescribable cardinals and silver indiscernibles. I feel a strong personal connection to it, as it was really the bridge that got me into higher mathematics, and eventually led me to finding the online math communities of which I am a part today. I’m very grateful that this place exists. Without it, I might’ve dropped what seemed to be a momentary infatuation with mathematics in favor of something else. This website is a big part of who I am today. So, thank you!

Oh don’t worry! I know you are a busy guy. It’ll get back up when it gets back up, and until then I can still even use the wayback machine to view pretty much any article I want, I just can’t make edits. Thank you for taking notice though, I really appreciate it.

I must commend you on the quality of communication you have nurtured in the comments board of your web-site.

Given your interest in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite, you might be interested in Thesis 1 in Section 1 of my forthcoming book (link (ii) below to the preprint version under final revision and indexing), which I hope to complete end-2020:

‘The Significance of Evidence-based Reasoning for Mathematics, Mathematics Education, Philosophy and the Natural Sciences’

where I seek to highlight what is known, and what ought not to be believed as definitive.

The argument is that we may need to recognise explicitly in our basic mathematical education that evidence-based reasoning:

(a) restricts the ability of highly expressive mathematical languages, such as the first-order Zermelo-Fraenkel Set Theory ZF, to categorically communicate abstract concepts which admit mathematically defined infinities, such as those involving Cantor’s first limit ordinal omega;

and:

(b) restricts the ability of eectively communicating mathematical languages, such as the first-order Peano Arithmetic PA, to well-define infinite concepts such as omega.

In other words, from an evidence-based perspective and, ideally, that of all disciplines which appeal to currently accepted scientific methods:

(i) although ZF admits unique, set-theoretical, definitions of—and allows us to unambiguously talk about the putative existence of—`ideal ‘ real numbers as the putative limits of Cauchy sequences of rational numbers, and their putative properties, in a mathematically defined, albeit Platonically conceived, universe,

(ii) only PA admits unique, algorithmically verifiable, number-theoretic definitions of—and allows us to unambiguously talk about the categorical existence of—specifiable real numbers, and their properties, which can be communicated as knowledge when describing the actual universe we inhabit.

Mathematics, therefore, needs to be treated as a sub-discipline of linguistics; and any ontological commitments associated with mathematical statements pertain not to the language per se, but to the conceptual metaphors that the language is intended to represent and communicate.

Further, the epistemological perspective of Thesis 1 is that logic, too, can be viewed as merely a methodological tool that seeks to formalise an intuitive human ability that pertains not to the language which seeks to express it formally, but to the cognitive sciences in which its study is rooted.

I shall be grateful if you could guide me as to any current work that reflects your present perspective, which you feel I might yet seek to present and/or reference in my book.

What has intrigued me particularly (in view of Theorem 18.1 in Section 18 on Goodstein’s Theorem in my book) is a suitable philosophical perspective from which to interpret the remarks, concerning putative arithmetical models of set theory in your paper ‘Satisfaction is not absolute’, that:

`On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure N={0,1,2,…} itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification.

… We make our argument in part by proving that different models of set theory can have a structure identically in common, even the natural numbers, yet disagree on the theory of truth for that structure’.

Sincerely,

Bhupinder Singh Anand
Mumbai

Links
=====
To (i) citeable archived initial draft of 28/04/2020, and (ii) current update (as of today) under final revision/editing/indexing:

Dear Prof Dr Joel David Hamkins,

My name is Ellena Caudwell and I am a 3rd year Undergraduate Mathematics student from the University of Exeter, UK.

I am currently involved in a group project based around D.E.Knuth’s book ‘Surreal Numbers’, and we were hoping to find examples of where this book has been used as a tool for teaching mathematics. I found an old page on the University of Amsterdam’s website explaining a course you ran in the 2nd Semester 2004/05: Surreal Numbers (http://www.illc.uva.nl/MScLogic/courses/Projects-0405-IIc/Hamkins.html) and it appears this is exactly what we were looking for.

I understand this was a long time ago, and you must be very busy, but we would be very grateful if you could answer a couple of questions to help us in our work.

Firstly, what level were the students who took part in the course, and in what way were they assessed?

Second, D.E. Knuth states in the postscript of his book that he wasn’t really trying to teach the theory of surreal numbers, but to ‘provide some material that would help to overcome… the lack of training for research work’. Therefore it is questionable how well this book helps teach surreal numbers.

1. Do you feel this course showed this book can be used successfully to teach surreal numbers? Was this your aim?

2. Do you feel it effectively shows the process of exploration and discovery of mathematical proof?

Any other thoughts or ideas on the topic would be most useful.

Thank you for your time and I look forward to hearing from you.

Ellena Caudwell

The course was a lot of fun for all of us, and I count it as a success. I would definitely be interested in running such a course again, and I think the book works very well for this kind of course. Shorter than a regular semester course, the course was filled mostly with masters degree students, with a few PhD students, but there would be no problem running a similar course for much longer. It was a small class, with about 8 students, which I think was relevant for its success. Following the idea of Knuth that you mention, we used the book not only to learn about the surreals but also to illustrate the practice of mathematical research. In particular, the students themselves presented much of the material on a rotating schedule, filling out the ideas of the book. Thus, the practice was that they would read the next part of the book, master the material, including whatever proofs needed filling in (which is the nature of the book), and make a presentation on that topic, at which we all would ask further questions and figure things out together. In addition, as is my general practice, the students wrote term papers, on a topic chosen in discussion with me. Assessment was based on the presentations and the paper.

To answer your specific questions: (1) The book can definitely be used successfully to teach surreal numbers, and yes, this was a major part of my aim. But this book requires more work from the students than a regular textbook, laying out all the material, since it is more open-ended. (2) As a result, yes, I do feel that the book and the way we used it in that course gave a good introduction to the process of mathematical research, particularly at the masters degree level. I think it could also work well with advanced undergraduates, if they were very motivated.

Dear professor Hamkins,

I am looking for a reviewer to my paper in progress “$mathcal{L}_{omega_1omega}$ -transfer principle in algebraic geometry”. Are you or someone you know interested in reviewing my work?

Sincerely yours,

Ali Bleybel

Sure, I’d be happy to take a look at it, but not sure whether I might find anything useful to say. Kindly send it to me at jhamkins@gc.cuny.edu.

Dear JD Hamkins,

I discovered your page “The global choice principle in Gödel-Bernays set theory”, that I found very interesting. And let me thank you for giving my name in this page. Let me add that, even if iI learnt mathematics with the immense Laurent Schwartz, I did not do a career as a professional mathematician, but as a statistician-economist finishing as the chief lawyer of the french national institute of statistics and economic studies. Being now retired, I am interested in set theory and reading the FOM list and Mathoverflow; I also have a modest participation in the splendid site Metamath of Norman Megill, that you may know.

I allow me to to ask some more questions related to “What happens for proper classes inGödel-Bernays set theory with AC and Without GCh”.

1-First of all, I would like to remark that by an answer of Sam Roberts on Mathoverflow today, we have another equivalence for GCh that is:

“If P(A) injects inton A then AV injects into A”

2- We saw in a model such that there exists a definable class of pairs of set having no choice function, (so that GCh fails), the proper class On (and all equivalent classes, that are the well-orders classes) strictly inject in the class P(On), that is linearly ordered and strictly injects into the universal class V. It seems intuitively difficult to think that some class can exist that is strtictly between On and P(On) or between P(On) and V ?

3- We also saw that we must have at least three non-equivalent proper classes, V that is the ultimate maximum because every class injects in V, On that is minimal in the sense that every proper class that injects in V must be bijective with V and the class W you defined that is incomparable with On. I would like to ask some questions about W.

(i) Is it true that W is minimal in the same sense as on ?

(ii) Are W and P(On) comparable ?

(iii) Is it possible that W strictly injects P(W) and P(W) strictly injects V ?

Bien cordialement.

Gérard Lang

Yes, I have enjoyed your questions on MathOverflow, and thank you for writing here. Your final questions are good ones. I have thought at length about the case of having a class between Ord and P(Ord), but so far without success.

Dear JD Hamkins,

Let me thank you most heartily for giving such a positive answer to my message.

I wish you the best for the end of this year and the next year as well.

Bien cordialement.

Gérard Lang

TL;DR: Nice talk at Einstein Chair, left comment, want to learn Set Theory in some moment.

Dear Prof. Hamkins,

Recently I left a comment concerning your talk for general mathematical audience concerning CH; very nice indeed. I’m not really sure if the comment got its way since I didn’t get any acknowledging message from the server.

Though I’m working in a different area now, I would be deeply interested in learning set theory in the not-that-far future. No one in my department (and, AFAIK, no one in my country!) is working on ST. If, with a little luck, I gather some people to work on this, I would be eager to invite you to give some course in Argentina.

Best regards,

Pedro.-

Dear Joel,

I need a word of advice (surely won’t be the last time!).

I’m preparing a course on Set Theory for next year, and I am expecting a mixed audience; ranging from graduate students that are starting to study the material (Kunen’s 1980) as of this writing, to perhaps some undergrads that will have seen basic (though fairly complete) courses on general topology and real variable. Most of them will only have a rudimentary knowledge of logic, and some of them will be seeing the definition of ordinal for the first time.

How much can one afford to compress in a 1-semester course? I’m just starting to prepare this, but it seems that I could go up to $L$, and perhaps a crash course on the basics of MA (I’m stealing this idea from some UCLA summer school notes by Justin Palumbo). But perhaps this is just too ambitious… On the other hand, the people reading Kunen right now are eager to “force”, at least a little bit.

Thank you for reading this, I hope it doesn’t bother you.

Best wishes,

PST.-

My preference in an introductory set theory course is to do some large cardinals before L or forcing. It is relatively easy to develop an analysis of V_kappa and H_kappa for inaccessible cardinals (or the second inaccessible, or Mahlo etc.) and so on, and this introduces the important idea of looking at various models of set theory, and then get to measurable cardinals and ultrapowers. After this, one can develop L and HOD. For forcing, I always start first with an extended analysis of MA, to get used to the concepts of density and genericity, but without any need yet for P-names. After this, one can develop P-names and the forcing extensions a bit more easily. But I’ve never been able to do ALL this in one course; it is simply too much. If they really are seeing ordinals for the first time, then I would be happy if they just get to know some large cardinals and the ultrapower construction, which is something that I think every mathematician should know.

Dear Joel,

Thank you very much for your quick reply. Searching for the topics you indicated, it seems to me that Drake is a good source; I also think Jech might be of use. Next year I’ll tell how things went through.

Thanks again and best wishes,

PST.-

Dear Prof. Hamkins:

I have a foundational question about set theory:

2 is a member of the set {2, 4, 6}.

Is 2 a member of the set {6}? Of course the answer is No. But that thing in the braces isn’t just a squiggle, it’s the number 6, and if 6 MEANS anything, it surely means six of something. And if there are six of anything, there are at least two of those things. Can you clear this up for me?

Finally, is the answer to this question as uncontroversial as the answer to the question, what is two plus two?

Cheers

Max

Haven’t you cleared it up yourself? The number 2 is not an element of the set {6}, since that set has only one element, the number 6, and that number is different from the number 2. Meanwhile, 2 is less than an element of {6}, because 2 is less than 6 and 6 is an element of {6}. But being less than an element of a set is not the same as being an element of the set. The fact that my child is shorter than a student in my calculus class does not make him a student in my calculus class. The fact that I earn less money per year than at least one Wall Street investment banker does not make me a Wall Street investment banker. I am younger than some of the retired faculty at my college, but this does not make me a retired faculty member at my college.

Thanks very much for your prompt and clear answer. Yes I was thinking vaguely along those lines, and some other reading was beginning to sort it all out for me, but it’s nice to hear clarification from a professional. Cheers!

Dear Prof. Hamkins,

(I feel a little embarrassed by the fact that I wrote above, unblushingly, “Dear Joel”.)

I’d like to know if you received my email (sent to your GC address on Dec. 21st).

Best wishes for this New Year,

PST.-

I saw you involved in the question”is there a class of all classes”on mathoverflow.It is obvious that the answer is NO.However,I saw some people agreed to call this such a big collection “2-classes”,and by recurrsion we can have n-classes.In my point of view,let me suppose that the collection of all classes is “2-classes”.Then I can conclude that this is the biggest collection.Since this is not a class.Therefore if the “2-classes” is contained in another collection U then it follows that the “2-classes” is a class.A contradiction.Hence I can,t understand how the “n-classes” can be generated.I am looking for your reply.Thanks!

Dear Joel:

Recently I am interested in the question such as “he set of all sets” and “the class of all classes”.And I did a lot of research on Internet including asked on “math stack exchange”.And finally I found that maybe the “Grothendieck universe” is what can answer my question.However,I think this is not directly related to my question.And I have a question which confused me:

1.Can you tell me the initial idea for the “Grothendieck set throry”.Is it just want to treat the question what I mentioned above?

2.In “Grothendieck set theory”,Is it legal to have “the set of all sets(i.e. the univesal set “V”) and “the class of all classe(I think this has several way to define as this is outstrip ZFC set theory’s boundary)” be a set?And this construction can be infinitely extended?

3.Based on my Q2,it doesn’t matter how you define such as “the class of all classes(I define this to be the collection including all the subclass of V and including “V” itself) and so on with the high level collection.This all can be set.So in general,given any universe of collection,we all can have a Grothendieck universe including that universe(especially contain the powerset of that universe–if that universe can be a set in Grothendieck set theory).Finally,Grothendieck universes hierarchy is the ultimate collection that cantains anything you want.

I hope that you can answer my question.Thanks!

Best Regards!

Andrew Song

Prof. Hamkins,

I teach an advanced high school elective on the history of mathematics, leading up to Godel’s Incompleteness Theorem. Last week we were learning about ordinals and I used your slides about Infinite Chess as a way to make the ordinals seems more concrete and more useful.

Several students followed up with me about some of the positions in your talk. Specifically, in Releasing the Hordes, the students wondered whether the position really had value omega^2. After the initial rook move, could white capture with a pawn from the right column instead, leaving a gap that the white king could then attempt to retreat into? We spent an hour or so after school trying to work it out and couldn’t get anywhere. Is there somewhere on the internet where we can enter infinite chess positions? Can you maybe personally explain to us why this either works or doesn’t work? My students think the value is less than omega^2.

Thanks!

Will Rose

Dear Will,

That is a very interesting idea! Your students are very insightful. Let me say that these infinite chess positions are often very finicky that way, and many of them resulted from a series of modifications to earlier similar positions that had some such flaw or other of the kind you are mentioning.

Here is the position: Releasing the Hordes

Your student’s idea shows that one has to take care with the set up. Black cannot allow that white will hide his king away in a cave.

I want to think more about it, but I think that black can handle it. Black moves his rook up on the first move. If white takes from the right, which is the line you are proposing, then black will respond like this. First, move the black rook to prevent white access to the wall. Next, move the black king up to protect the hole. Keep the black king between the hole and the white king. During these moves, white may be moving pawns to bring the hole lower, but they will meet halfway, which is still good enough, since n/2 can be made as big as you like (that is, move the rook up twice as far as you had really wanted to on the first move). Now, we are in a situation where the black king is guarding the hole. Next, black should move his rook very far out to the right. In such a situation, the white king cannot hang around anywhere near the wall, since there will be perpetual checks coming from the black rook. Thus, black can force the white king to chase down the black rook, very far away (as far as one likes). White cannot gain advantage by trying to return to the wall, since we can do the same thing again.

Does it work? I will think more about it. Let me know if you see any issues. What this means is that in this line, we don’t get $\omega\cdot n$ if the original black rook moves up to height $n$, as I had thought, since we will have to burn a few pawn moves for each round of harrassment, rather than just one.

Meanwhile, it seems that we can address the problem also by changing the position to have a double row of white pawns on the outside wall. Alternatively, we could place the white king harrassment arena in a separate place, more isolated from the wall protecting the doorway. That is, have another separate vertical wall of white pawns, with the black rook and white king to the right of it, and the black king between them.

Thanks for your comment, and I’d like to hear back from you.

Here is another idea, which will be better if it works. After black moves his rook up on the first move, if white should takes from the right, as you suggest, then black will first of all use his rook to keep white away from the wall. Then, black will aim to capture a pawn in the right wall, between the door and the hole. Black aims to occupy this hold with his king, and then have his rook free. For example, black can position his king a knight’s move above and to the right of the desired place, capture the pawn with his black rook, and then move the king diagonally next to the rook (since the pawn is no longer protecting), and then exit the hole with check, giving time to occupy the hole with the king.

If this works, then white will have no line remaining leading to check, since the white wall pawns can no longer advance, and so the door will never open.

But one must consider the possibility that the white king attempts to interfere with the insertion of the black king into the hole. For example, if the white king stayed next to the wall, there would be no “exit with check”. For example, white could attempt to capture the rook by approaching it from below, staying next the wall. Indeed, I think that this idea may not work because of this counterplay, since black will be forced to move the king away because of zugzwang.

The idea would seem to work, however, if we give black a pawn on the right side, since then black can avoid the zugzwang, and simply occupy the hole with both the king and rook, moving his pawn up. There is no need for check harrassments in this case, since no pawns are advancing and so black aims for a full draw, instead of forestalling loss.

It seems best to pursue a combination of the two lines I mention. That is, using the first strategy we can get the we can aim to get the white king very far away from the wall, since otherwise there is perpetual check, with the black king prevent white king access to the cave. Then, we pursue the second strategy to aim at inserting the black king into the wall below the hole, which will black pawn advances below and cause a draw (good for black). The point is that if the white king is very far from the wall, then we can implement the exit-with-check part of the king insertion. So the line will lead to a situation where the white king is far away from the black king, and the black king has inserted itself into the wall. This prevents pawns from advancing, and there is no way to dislodge the king from that spot, if the black rook is free.

Alternatively, it may be good enough simply to use the black rook to keep the white king even just a few steps away from the wall as currently. Then black positions his king to a place readying for the king insertion, and then carries it out without interference.

For example, if we simply started with the white king a little farther from the wall, it would definitely work.

Wow. We’ll get back to you in a few days after we get a chance to sit down and look into this. Thanks for taking the time to reply!

Sure, let me know what you think. Currently I don’t think that the position needs to be modified and also that your line leads to a draw, so white won’t do that, since with the main line white will achieve checkmate.

math for kids is my new favorite website! i’d love to know your thoughts on zugzwangs in non-chance, perfect information games in general…

https://www.reddit.com/r/abstractgames/comments/8egu6n/games_that_allow_zugzwang/

(i’ve made sure to post your Infinite Sudoku over there, so hopefully some of the other kids will be coming to check you work out. still a small community but holds great promise:)

Hello there! I’m an amateur-ish pure mathematician who specializes in set theory. A pressing matter is that it seems that Cantor’s Attic is down! There’s some sort of internal error going on, and it’s preventing me from adding to the website. For the functional part of this comment, I’m done! Just letting you know.

Also, I’d just like to say I contributed a lot to the website, being the first to add many fundamental large cardinal notions such as indescribable cardinals and silver indiscernibles. I feel a strong personal connection to it, as it was really the bridge that got me into higher mathematics, and eventually led me to finding the online math communities of which I am a part today. I’m very grateful that this place exists. Without it, I might’ve dropped what seemed to be a momentary infatuation with mathematics in favor of something else. This website is a big part of who I am today. So, thank you!

I’m so sorry, and I’ll try to get it back up as soon as possible.

Oh don’t worry! I know you are a busy guy. It’ll get back up when it gets back up, and until then I can still even use the wayback machine to view pretty much any article I want, I just can’t make edits. Thank you for taking notice though, I really appreciate it.

Dear Joel,

I must commend you on the quality of communication you have nurtured in the comments board of your web-site.

Given your interest in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite, you might be interested in Thesis 1 in Section 1 of my forthcoming book (link (ii) below to the preprint version under final revision and indexing), which I hope to complete end-2020:

‘The Significance of Evidence-based Reasoning for Mathematics, Mathematics Education, Philosophy and the Natural Sciences’

where I seek to highlight what is known, and what ought not to be believed as definitive.

The argument is that we may need to recognise explicitly in our basic mathematical education that evidence-based reasoning:

(a) restricts the ability of highly expressive mathematical languages, such as the first-order Zermelo-Fraenkel Set Theory ZF, to categorically communicate abstract concepts which admit mathematically defined infinities, such as those involving Cantor’s first limit ordinal omega;

and:

(b) restricts the ability of eectively communicating mathematical languages, such as the first-order Peano Arithmetic PA, to well-define infinite concepts such as omega.

In other words, from an evidence-based perspective and, ideally, that of all disciplines which appeal to currently accepted scientific methods:

(i) although ZF admits unique, set-theoretical, definitions of—and allows us to unambiguously talk about the putative existence of—`ideal ‘ real numbers as the putative limits of Cauchy sequences of rational numbers, and their putative properties, in a mathematically defined, albeit Platonically conceived, universe,

(ii) only PA admits unique, algorithmically verifiable, number-theoretic definitions of—and allows us to unambiguously talk about the categorical existence of—specifiable real numbers, and their properties, which can be communicated as knowledge when describing the actual universe we inhabit.

Mathematics, therefore, needs to be treated as a sub-discipline of linguistics; and any ontological commitments associated with mathematical statements pertain not to the language per se, but to the conceptual metaphors that the language is intended to represent and communicate.

Further, the epistemological perspective of Thesis 1 is that logic, too, can be viewed as merely a methodological tool that seeks to formalise an intuitive human ability that pertains not to the language which seeks to express it formally, but to the cognitive sciences in which its study is rooted.

I shall be grateful if you could guide me as to any current work that reflects your present perspective, which you feel I might yet seek to present and/or reference in my book.

What has intrigued me particularly (in view of Theorem 18.1 in Section 18 on Goodstein’s Theorem in my book) is a suitable philosophical perspective from which to interpret the remarks, concerning putative arithmetical models of set theory in your paper ‘Satisfaction is not absolute’, that:

`On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure N={0,1,2,…} itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification.

… We make our argument in part by proving that different models of set theory can have a structure identically in common, even the natural numbers, yet disagree on the theory of truth for that structure’.

Sincerely,

Bhupinder Singh Anand

Mumbai

Links

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To (i) citeable archived initial draft of 28/04/2020, and (ii) current update (as of today) under final revision/editing/indexing:

(i) PhilPapers (28/04/2020): https://philpapers.org/rec/ANATSO-4

(ii) Dropbox update (as of now):

https://www.dropbox.com/s/gd6ffwf9wssak86/16_Anand_Dogmas_Submission_Update_3.pdf?dl=0