François G. Dorais - Research - Latest Comments

Re: Disqus

jcmckeown — Wed, 27 Mar 2013 22:35:43 -0000

Here http://userscripts.org/scri... is a greasemonkey kludge that is working in my firefox right now. I don't think it will handle new comments at submission time, or pagination, but it's a start.

Re: Disqus

François G. Dorais — Sat, 16 Mar 2013 16:45:16 -0000

It's a difficult issue. My understanding is that disqus needs to provide appropriate hooks to run mathjax at the right time but they haven't done that yet and there is no clear indication that they will. I've been looking for alternatives but I didn't find anything with all the features I want. Do let me know if you find something.

Re: Disqus

jcmckeown — Sat, 16 Mar 2013 15:13:26 -0000

Yeah, I've been trying to learn GreaseMonkey instead, but I just don't understand the concurrency issues, and it doesn't seem friendly to make everyone else modify their browsers. So far, I can get the init to run in disqus' iframe, but that seems to finish before it inserts any comment text.

Re: Arithmetical consequences of the set-theoretic multiverse

François G. Dorais — Tue, 12 Mar 2013 20:43:47 -0000

Thank you, Erin! I plan to add a few more thoughts about the multiverse when I find time to think about it more. Stay tuned!

In general, an universe does not need to be a $V_\alpha$ or a transitive set in a larger compatible universe $V$. Any model of ZFC in $V$ is a compatible universe with the same standard system (from the divine point of view), if there is one such model then there is also a nonstandard model $U$ which is not wellfounded (from the point of view of $V$) and hence is not isomorphic to a transitive set in $V$.

Re: Arithmetical consequences of the set-theoretic multiverse

erin carmody — Tue, 12 Mar 2013 20:26:41 -0000

I really like your post. What an interesting perspective! I enjoyed your imagery of "trunks" of the multiverse through a finite divine lens. Regarding your compatibility theorem, if two universes are compatible, then are they both $V_{\alpha}$'s for different $\alpha$'s? To answer the final question in your post, I prefer the multiverse perspective since it helps me understand forcing, and is so enriching philosophically. I find the debate fascinating, and look forward to more discussion among mathematicians.

Re: Generalized separation principles

François G. Dorais — Sat, 23 Feb 2013 12:15:46 -0000

Edits: Fixed small typos. Reformatted references.

Re: Arithmetical consequences of the set-theoretic multiverse

François G. Dorais — Mon, 11 Feb 2013 10:12:48 -0000

Thanks Joel. Fixed.

Re: Arithmetical consequences of the set-theoretic multiverse

Joel David Hamkins — Mon, 11 Feb 2013 08:44:55 -0000

You have a typo in the statement of the finiteness mirage.

Re: Arithmetical consequences of the set-theoretic multiverse

François G. Dorais — Mon, 11 Feb 2013 01:39:23 -0000

Edit: Changed the misleading term brane to trunk following a suggestion by John Baez and David Roberts.

Re: Arithmetical consequences of the set-theoretic multiverse

François G. Dorais — Sun, 10 Feb 2013 21:02:48 -0000

Thanks, Joel. I'm really happy the arithmetic consequences of the multiverse axioms turned out to be the exactly minimum they could be. I find this much more satisfactory than the universe perspective.

Re: Arithmetical consequences of the set-theoretic multiverse

Joel David Hamkins — Sun, 10 Feb 2013 19:21:34 -0000

Great post, François, and I am glad to see this work being taken further. I really like your perspective of looking for what arithmetic consequences of the perspective there might be.

Re: Towsner’s stable forcing

Peter Krautzberger — Sat, 09 Feb 2013 21:32:52 -0000

I'm not sure, but I'm guessing the title-tag of the span is the only crucial thing since it contains structured data.

Re: Towsner’s stable forcing

François G. Dorais — Fri, 08 Feb 2013 15:25:52 -0000

The citations are MLA style, which I am not particularly fond of, and WordPress tends to strip some stuff out. What parts are necessary to make scanning work?

Re: Towsner’s stable forcing

Peter Krautzberger — Fri, 08 Feb 2013 13:46:39 -0000

Got to Mathblogging.org, hit the menu item "Generate Citation", follow instructions?

Re: Towsner’s stable forcing

François G. Dorais — Fri, 08 Feb 2013 13:32:44 -0000

How does one generate a citation?

Re: Towsner’s stable forcing

Peter Krautzberger — Fri, 08 Feb 2013 12:48:01 -0000

Excellent piece of research blogging. You could generate a citation via mathblogging.org and add it.

Re: Towsner’s stable forcing

François G. Dorais — Fri, 08 Feb 2013 12:32:20 -0000

Edit: I redacted a claim that the second result follows from the first using arguments similar to those Jared Corduan and I used in On the indecomposability of $\omega^n$. That unfortunate claim was at best incomplete and possibly false. Since the claim was only tangential to the topic, early redaction was the best course of action.

Re: Disqus

François G. Dorais — Sat, 02 Feb 2013 18:02:21 -0000

The $\LaTeX$ fix appears to be broken after a recent disqus upgrade.

Re: Back to Cantor?

Mark Gomer — Sun, 27 Jan 2013 16:32:50 -0000

Type theory gives you the best of both worlds, and unlike ZFC or ECTS it doesn't presuppose first-order logic!

http://golem.ph.utexas.edu/...

Re: Back to Cantor?

Mike Shulman — Thu, 10 Jan 2013 00:01:35 -0000

I agree that the three concepts are really different for a structuralist -- they are different for me, even though I'm not exactly a structuralist any more. However, that doesn't prevent a structuralist from systematically using implicit coercions (which mathematicians traditionally call "abuses of notation") to avoid introducing separate notations for the three concepts. This sort of thing is common among materialists too: a group is a different concept from its underlying set, but we have no problem using the same name for both.

Re: Disqus

François G. Dorais — Tue, 08 Jan 2013 14:32:34 -0000

It's a trick found by Christian Perfect and implemented here by Sam Coskey. Here is the source code:

  function typeset() {
    MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
  }
  function disqus_config() {
    this.callbacks.onNewComment = [typeset];
    this.callbacks.onInit = [typeset];
    this.callbacks.onPaginate = [typeset];
  }

Re: Back to Cantor?

François G. Dorais — Tue, 08 Jan 2013 12:36:44 -0000

Thanks for the comments, Andreas. I'm not sure exactly when ZFC was "crowned" but it has to be after key papers by von Neumann and Skolem in the late 1920s when the current formulation took form and before 1940 when Gödel published the details of his construction of L. (Technically, Gödel used NBG but that is morally the same as ZFC.)

Re: Back to Cantor?

Andreas Blass — Tue, 08 Jan 2013 11:42:59 -0000

A minor comment about "ZFC is now approaching 100 years of reign as the foundation of mathematics": Is there a clear indication of when the reign actually began? Gödel's paper proving the incompleteness theorems refers, in its title, to Principia Mathematica, suggesting that this was still the dominant foundation at that time. Just a few years later, Gödel used NBG as the framework for his consistency proof of AC and GCH, but that choice may have been heavily influenced by the nature of the proof. Since NBG is conservative over ZF, one might regard this as an indication that ZF was beginning its reign (or at least preparing for a coup d'état), but I'm not sure conservativity is a strong enough connection to support such a claim. So when did ZF "really" start to reign?

A slightly related point: In early work of Erdös on the partition calculus, there are notational distinctions between a cardinal number, the initial ordinal of that cardinality, and the set of ordinals below that initial ordinal. For today's set theorist, this looks like a horrible, unnecessary proliferation of notation, wasting the reader's mental energy on keeping track of the notation, when the energy is really needed for the mathematical content. But from a structural point of view, these notational distinctions make perfect sense; the three concepts really are different. The materialist has just taken advantage of the nice well-founded structure of the ZFC world to identify all three concepts, and we've become so accustomed to the identification that its absence strikes us as very unpleasant.

Re: Disqus

jcmckeown — Mon, 07 Jan 2013 10:53:33 -0000

Yes, evidently... would it be too much trouble to highlight the salient points in making that happen? For whatever reason my brain doesn't seem able to compile the docs.mathjax.org, and so...

Re: Back to Cantor?

Mike Shulman — Mon, 07 Jan 2013 02:24:49 -0000

Thanks; now I understand better. I don't think we're really disagreeing about anything, except maybe words (e.g. I feel like possibly you mean something a bit different than I do by "well-foundedness"). What you say confirms the general feelings I've gotten from reading set theory. It's an interesting challenge to try to marry the ideas of modern set theory with the structural perspective coming from category theory and "algebraic/topological" mathematics, and I'm glad there are other people thinking about it and approaching it from other perspectives.