Randall Holmes

Welcome!

Hello Randall Holmes, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers:

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I hope you enjoy editing here and being a Wikipedian! Please sign your name on talk pages using four tildes (~~~~); this will automatically produce your name and the date. If you have any questions, check out Wikipedia:Where to ask a question or ask me on my talk page. Again, welcome!  HolyRomanEmperor 21:17, 14 December 2005 (UTC)

Style remarks

Hello and welcome from me too. Since you are new, I thought I would share a few Wikipedia tips. First is that it is good if one uses an edit summary when one contributes. It helps others understand what you change; one can think of it as the "Subject:" line in an email. Second, one should not insert newlines or too many empty lines in text. Unlike TeX or HTML, the Wiki markup system really keeps track of that, and it does not look good if somebody edits the same text in a textbox of a different size, and the wikipedia "Diff" feature (compare different versions) does not work too well. Thanks, and I hope you like it here. Cheers, Oleg Alexandrov (talk) 22:18, 15 December 2005 (UTC)

I replied on my talk page. Oleg Alexandrov (talk) 16:04, 16 December 2005 (UTC)

Welcome

Hi Randall. Welcome to Wikipedia. I've noticed your good work on several set theory articles. Thought you might like to know about Wikipedia's Mathematics Project, and might want add yourself to Wikipedia:WikiProject Mathematics/Participants. Also much of Wikipedia's mathematical conversation takes place here: Wikipedia talk:WikiProject Mathematics so you might want to put that page on your watchlist. Again welcome. Regards, Paul August ☎ 04:57, 19 December 2005 (UTC)

Comment

Hi Randall, and thank you for the good work. But I would like to reiterate my request that you do not use linebreaks when you edit. Take a look at this diff and you may see why. Thank you, Oleg Alexandrov (talk) 01:18, 20 December 2005 (UTC)

I know what you are talking about; it seems that I make errors reflexively. I have caught some of them with the previewer, but evidently not all... Randall Holmes 01:22, 20 December 2005 (UTC)

No problem. I just wanted to make use you are aware of the issue. :) Oleg Alexandrov (talk) 01:25, 20 December 2005 (UTC)

From Dean

Randall - Many thanks for your kind words about Zermelo set theory. And a very warm welcome to Wikipedia. We need as many good people like yourself as we can get!

I take a look at your site from time to time, though mathematical logic is not my area of expertise.

Would you be interested in becoming involved with the logic project? The objective is to improve the standard of logic articles in Wikipedia. We are beginning with Logic, where User: Chalst and I are working on a draft here: User:Dbuckner/logic Dbuckner 18:43, 28 December 2005 (UTC)

I am privately doing something about the logic articles I encounter; take a look at first-order logic, for example. It's a question of how much time is involved... Randall Holmes 21:23, 28 December 2005 (UTC)

And a very good edit too. Much improved. People tend to be scared of subtracting, so they add their own bit, and the result is a disconnected mess. Glad to see you got the pruning shears out! Dean Dbuckner 19:31, 29 December 2005 (UTC)

more about second order

So the least upper bound property of real numbers is not expressible in a first order language with parameters for numbers only, and there exist models of real closed fields which do not satisfy it. Maybe the sort of thing that the author had in mind when he said that topology is difficult to express in first order language. It doesn't mean much though, since real numbers are typically constructed within ZFC. Maybe that's the distinction you had in mind when you said "classical mathematics"? -lethe ^talk 03:56, 3 January 2006 (UTC)

but of course the least upper bound property is expressible in practice in (first-order) set theory. Moreover, all actual reasoning about the least upper bound property and similar things which invite second order treatment is actually first-order reasoning about sets. Randall Holmes 05:26, 3 January 2006 (UTC)

Well, yeah, I know that the lub property of the reals is expressible in FOL in the language of sets, sure. But the point is, there are certain things that require second-order. I edited an article to suggest that such things exist, and then you challenged me to tell you one. I actually don't know too much about it, so I'm hoping maybe you can tell me. As I understand it, one such thing is the lub property in a language of real numbers (instead of sets of real numbers). Is that not an example? If not, why not? -lethe ^talk 07:30, 3 January 2006 (UTC)

Everything without exception that can be expressed in second-order logic can be expressed in the multi-sorted first-order theory adding sorts for type(s) of sets of the elements of the domain(s) of the original theory. Everything without exception that can be proved in practice in second-order logic is in fact proved in said multi-sorted first-order theories. The only sense in which second-order logic can "express" anything that first-order logic cannot is that its semantics are defined differently: it is declared that the domains of predicates in models of a second-order theory must be the full power sets of the domains of objects of which they are predicates. So for example the second-order theory of the natural numbers or of the reals is categorical. But the only practical implementation of the Peano axioms is the first-order Peano arithmetic (or its embedding in more powerful first-order theories which allow reasoning about sets of natural numbers as well), and similarly for the theory of the reals and any other second-order theory. These first-order theories are of course not categorical. We note that the second-order theories of the reals and the natural numbers are categorical, but we cannot actually use this information to determine what is true about the reals or the natural numbers, because we have no access to the true power sets involved. Randall Holmes 14:01, 3 January 2006 (UTC)

Hold on a moment: there are (more than) two notions of second-order going about. In the first place there are the special sorts of multi-sorted FOL you talk about, and then there is the unaxiomatisable notion that Quine so loathed, that you could have been hinting at with your talk of structures we cannot get at. You need to talk about both or you will confuse people who have sort of heard of the latter. --- Charles Stewart 23:12, 3 January 2006 (UTC)

I'm still a bit unclear here. I've just skimmed Enderton's chapter on second-order logic, so I think I know what that stuff about sorts is that you mentioned. I gather "categorical" means that all models of the theory are isomorphic? -lethe ^talk 01:13, 4 January 2006 (UTC)

Second-order logic is what you get when you allow quantification over predicate letters in first-order logic (though one can also express second-order logic in set theoretical language). The official semantics for second-order logic requires that the domain of discourse for the n-ary predicate variables be all sets of n-tuples of elements of the original domain of discourse. The theory of Peano arithmetic (for example) when expressed in second-order language with this semantics has just one model (up to isomorphism). However, the actual means of formal reasoning available to us in second-order language are those of multi-sorted first-order logic, and such multi-sorted first-order theories have nonstandard models for the usual reasons. The second-order theory of a complete ordered field is categorical, in the sense that there is only one model (up to isomorphism) with the official semantics. The multi-sorted FOL theory is not categorical (again, it has nonstandard models for the usual reasons) but it is perfectly capable of expressing and proving the usual consequences of the least upper bound property (for example). When people say that second-order logic is required to express the least upper bound property (for example) they are alluding to the fact that SOL semantics are required for the usual axioms for the reals to be understood to uniquely describe the reals up to isomorphism; but this does not mean that the usual FOL approach does not actually describe the least upper bound property and prove the usual theorems of analysis; it does (and in fact the FOL approach is the only one that allows us to prove anything about the reals: second order logic has no proof theory (the "second-order logics" or "higher-order logics" about whose proof theory you will hear are multi-sorted FOL systems). WARNING: conversations on this topic can go in circles endlessly due to equivocation on the exact meaning of "second-order logic". Randall Holmes 03:36, 4 January 2006 (UTC)

Thank you

Just a 'thank you' for your "New Foundations" article (which is one of the best mathematical ones on Wikipedia) and also your website and your brilliant and free book.

Thank you for everything you are doing, both with respect to mathematics and with respect to making information free.

- Jax

My blushes! The book is only free because it is out of print and this is the only way I can keep it available. By the way, I am editing it actively at the moment, and any comments on errors and infelicities would be effective now. I am skeptical of certain kinds of intellectual property, but I am not sure I would describe myself as a campaigner for free information... Randall Holmes 03:16, 5 January 2006 (UTC)

Classes vs. sets

Hi. Two days ago I tried fixing the definition of function (mathematics) to be minimally correct. However my edits were reverted by a maths teacher(!) who insists that one should not even mention the word "set" in the whole article, because it may scare "the 90% of readers who do not undestand that language". I have spent many keystrokes arguing with him, but he probably is not convinced, and I am afraid that he may revert it again.
So I am afraid that mentioning class at the outset will increase the risk of binary relation being similarly "mushified". Would you agree with moving the offending example to the end of the head section, together with the set-->class generalization?
All the best, Jorge Stolfi 18:06, 18 January 2006 (UTC)

Hi, I have reworded the class/set section before realizing that it was new. Please see whether you agree with my edits, if not feel free to revert/fix as appropriate.
What do you think of the new head section?.
All the best, Jorge Stolfi 19:04, 18 January 2006 (UTC)

Function

I like your version better than the old version, but (forgive me) I think your sentences are too long. The big fuss, from which Jorge Stolfi has now withdrawn, is over writing about math in a way that a layperson can understand. He finds anything aimed at the layperson horrible, and rewrote the entire article from beginning to end in two days.

I think the first paragraph of any article should be written so that any intellegent reader can understand it, without prerequisites. I'm going to try again. Rick Norwood 20:23, 18 January 2006 (UTC)

I am sure we can work together, to our mutual benefit. Of course, the formal definition of a function needs to be in the article, and was before the recent edit. My problem was that Jorge moved it to the first sentence. Rick Norwood 20:35, 18 January 2006 (UTC)

In working on the "function" article, I'm reacting in part to a study reported -- somewhere (Focus???) -- that the idea of a function is where most math students get totally lost. They can understand arithmetic. They can understand simple algebra. But functions loose them. It seems to be one level of abstraction too much for most people. And, I think, they are usually taught badly, because we professors sometimes do not realize that our students are totally lost.

(Speaking of students totally lost, I taught an Introduction to Modern Algebra II class today in which I discovered students who had passed Intro to MA I and did not know what the words "identity" and "inverse" mean.) Rick Norwood 23:56, 18 January 2006 (UTC)

NBG Set Theory

Thanks for adding the axioms of NBG set theory to that page. Information about NBG is surprisingly hard to find on the net, and I've been hoping someone would fill out that page for some time. Thanks a lot! -- Zarvok | Talk 15:04, 19 January 2006 (UTC)

I've added a set of my parents' axioms for NBG to the talk page. I'm fairly sure my mother's Set Theory for the Mathematician mentions that the comprehension schema can be replaced by a finite number of axioms, as does Equivalents of the Axiom of Choice, II, but I'm not sure it lists them. Arthur Rubin | (talk) 23:46, 24 January 2006 (UTC)

function (mathematics) and function (set theory)

Just for the record (to avoid any confusion) I got frustrated and went over to edit my proposed function (set theory) (to be paired with the not-yet-existant function (analysis) just before you (very sensibly) protected the article. I was not circumventing your protection but stepping out of the edit war... Randall Holmes 21:46, 19 January 2006 (UTC)

Yes, that was clear to me, because you had already floated the idea before. Nevertheless, I find it a very bad idea, I don't think the subject can be split like that, and I am sad that you felt it necessary to do this. -- Jitse Niesen (talk) 21:56, 19 January 2006 (UTC)

Theory of relations

• JA: Thanks, I appreciate it. Jon Awbrey 04:34, 28 January 2006 (UTC)

Relational Methods in Computer Science, Tarski & Givant, etc.

• JA: Yes, I spent the 90's working on a kind of capstone diss in a Systems Engineering department, and one of my advisors was very much into the "relational programming" paradigm, co-authored a book on it and all. There's a loose group that has yearly conferences and a mailing list:

• RelMiCS-l -- Relational Methods in Computer Science

• JA: Jon Awbrey 04:50, 28 January 2006 (UTC)

Cantor and naïve set theory

I will definitely defer to you on the topic, since the ratio of your experience with the world of set theory to mine has an infinite limit, but this does create a contradiction with the actual naive set theory article, which explicitly credits it to Cantor. The article on Russell's paradox was crediting it to Frege. While Russell may have specifically ruptured Frege's overarching theory, it still seems inaccurate to attribute all of naive set theory to Frege, when Cantor (it appears to me) is generally credited as the father of set theory.

At any rate, if I'm in error about this, I appreciate your correcting me. But, if so, would you mind editing the naive set theory article as well, to remove this subtle discrepancy?

Mineralogy 22:05, 29 January 2006 (UTC)

second order ZF

Hi Randall-

I was on Trovatore's talk page for some other questions, and your conversation with him about whether Morse-Kelley set theory was a second order language or not caught my eye. I noticed you mentioning second order ZF, and I am curious about what that is. Through your comments, I gather it's a second order logic with sets and classes, or something like that? I looked for it under Zermelo-Fraenkel set theory, but of course there's no mention. And about languages versus logics, second order logical language redirects to second order logic. The article doesn't talk about any distinction. hmm... -lethe ^talk 09:45, 30 January 2006 (UTC)

Imps of math

Oh, I haven't read it for content yet. Just doing what I normally do on first scan of a paper draft. Just noticed the line breaks because one of them was keeping a link from highlighting, and it took me a while to figure out what was going on, er, not going on. The chatter on the [[Naive set theory]] page, now that's emusing. Jon Awbrey 01:36, 31 January 2006 (UTC)

Quantity in mathematics

Randall, would you mind having a look at the section quantity in mathematics within Quantity^talk if your are able and inclined to do so? It could really use someone with your sort of background in my view. For that matter, any comments on the article would be appreciated. Cheers, Holon 03:38, 26 February 2006 (UTC)

limitation of size, Cantor

Hi Randall,

I wonder if you could help out with my historical questions at limitation of size and Georg Cantor. (The limitation of size notion I'm primarily interested in is Cantor's, as in Hallett's book Cantorian set theory and limitation of size; unfortunately I don't have the book.) The issues with Cantor can be found by searching for "fact check needed" in the Georg Cantor page and looking at Talk:Georg Cantor#Fact check needed. Thanks, --Trovatore 04:30, 26 February 2006 (UTC)

either "vita" or "curriculum vitae", not "curriculum vita"

You used that last phrase on your home page.

"Vita" is Latin for "life". "Curriculum vitae" is Latin for "course of life". "Curriculum vita" would be "course life", with "of" omitted. Michael Hardy 23:42, 31 March 2006 (UTC)

Categories in userspace

Hi! I noticed that in your sandbox User:Randall Holmes/Sandbox/relation (mathematics) you have the categories still activated, so it's showing up in Category:Logic and Category:Set theory. Could I suggest that you deactivate them (by putting a colon before 'Category' in the link) until such time as the article is in the mainspace rather than the userspace? (As per WP:CG, "If you copy an article to your user namespace (for example, as a temporary draft or in response to an edit war) you should decategorize it".) Cheers, Ziggurat 22:59, 10 April 2006 (UTC)

Given that you don't seem to be around I'll remove them myself. All the best, Ziggurat 03:39, 9 May 2006 (UTC)

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