Comment on The chromatic numbers of the Erdos-Hajnal graphs by saf

saf — Fri, 18 May 2012 13:58:12 +0000

A PDF file with lecture notes for today’s talk on this subject, are available here.

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Comment on Waiting for the Polymath revolution — thoughts from a bystander by Mike Pawliuk

Mike Pawliuk — Thu, 17 May 2012 21:38:00 +0000

“It could be that the groups of mathematicians that influence the
conversations and developments within the mathematical community, e.g.,
tenured faculty at the top math departments, consist entirely of the
“wrong” people, unable to be in the right place at the right time.
Simply because it’s not why they got the job — they are where they are
because they are excellent at doing research in the current model, the steampunk approach of “papers in journals”.”

+1. It is hard enough to get grants if you publish lots of papers in respectable set theory journals, why would you want to publish in outlets that the grant-giving agencies don’t recognize?

I also get the feeling that this sort of project if lead by non-expert experts would just fail. Part of it seems like Gowers is trying to right a conventional paper by doing something harder than usual. I have difficulty enough writing papers when everything goes my way, let alone with one hand tied behind my back. It seems like someone like Gowers would be immensely useful for this type of project.

Comment on Secret Santa 3: The Paradox. by Micheal Pawliuk

Micheal Pawliuk — Thu, 17 May 2012 13:24:20 +0000

I forgot that I allowed 0 to be chosen. My argument needs to be adjusted in the obvious way.

Comment on Secret Santa 3: The Paradox. by Micheal Pawliuk

Micheal Pawliuk — Thu, 17 May 2012 13:21:06 +0000

So you certainly have one half of the paradox down. Yeah, so how can I conclude that 1220 is either the higher or the lower number?

Also, remember that in this game we are perfect logicians. (Which isn’t a good assumption in real life.)

IF we both know that 1,2 and 3 can’t be chosen, then the writer can’t write down ’4′ (because either the picker will get ’3′ and know that he has the smallest number, or he will get ’5′, say nothing, then the writer will know he has the smaller number). I guess your issue here is that ahead of time both players need to agree that 1,2,3 are not writable. Then they would have to agree that 4 is not writable, and same with 5 or 6 or whatever.

What might be not satisfying here is that the induction kind of hides the actual algorithm for the players to decide who has the smaller number in Jacob’s game. The induction argument seems to produce a type of “there exists an algorithm” without actually producing this algorithm.

The algorithm in this case is pretty horrendous, but is something computers could do. Maybe the game goes like this:
Player 1 writes. (Note he cannot write 1.)
Player 2 picks. (If he has ’1′ he announces “the game is broken”. Else, he says “all good for now”.)
Neither player has ’1′
Player 1 – If he has ’2′ he announces “the game is broken”. Else, he says “all good for now”.
Player 2 – If he has ’2′ he announces “the game is broken”. Else, he says “all good for now”.
Neither player has 1 or 2
Player 1 – If he has ’3′ he announces “the game is broken”. Else, he says “all good for now”.
Player 2 – If he has ’3′ he announces “the game is broken”. Else, he says “all good for now”.
Neither player has 1,2 or 3
etc.

This back and forth will end after O(the writer’s number) many steps.

I think this should make it clearer.

Comment on Secret Santa 3: The Paradox. by Chris

Chris — Thu, 17 May 2012 06:09:40 +0000

I’m unconvinced by your induction here, as I don’t see how the k+1 step generalizes to larger numbers. If you write down 1220 and I pick either 1219 or 1221, how does one of us deduce that the game has been broken?

Comment on The omega one of infinite chess, New York, 2012 by This Week in Logic at CUNY | Set Theory Talks

This Week in Logic at CUNY | Set Theory Talks — Tue, 15 May 2012 03:49:59 +0000

[...] Set Theory Seminar Friday, May 18, 2012 10:00 am GC 6417 Professor Joel David Hamkins (The City University of New York) The omega one of infinite chess [...]

Comment on yay, I’m an editor at ScienceSeeker.org by Peter Krautzberger

Peter Krautzberger — Sun, 13 May 2012 01:19:00 +0000

Vielen Dank, Patrick. Grüße nach Bonn!

Comment on yay, I’m an editor at ScienceSeeker.org by Patrick Siklossy

Patrick Siklossy — Sat, 12 May 2012 20:54:00 +0000

OK, not scientific – but equally important: HAPPY BIRTHDAY! lieber Peter.
Liebe Grüsse aus Bonn, ich wünsche Dir einen herrlichen Tag und viele zufriedene Tage im kommenden Jahr.
Mach et jot. Patrick

Comment on The chromatic numbers of the Erdos-Hajnal graphs by Assaf Rinot: On incompactness for chromatic number of graphs | Set Theory Talks

Assaf Rinot: On incompactness for chromatic number of graphs | Set Theory Talks — Sat, 12 May 2012 20:14:16 +0000

[...] We shall discuss Shelah’s paper #1006. [...]

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Comment on The countable models of ZFC, up to isomorphism, are linearly ordered by the submodel relation; indeed, every countable model of ZFC, including every transitive model, is isomorphic to a submodel of its own $L$, New York, 2012 by Joel David Hamkins

Joel David Hamkins — Wed, 09 May 2012 10:58:16 +0000

I’m glad you like it. I’m working furiously on the article, which should be ready fairly soon, and I’ll make a post when it is. The theorem shows that all forcing extensions of a countable model of set theory are isomorphic to a submodel of the original model. This submodel, of course, will not be an amenable class in the original model, and the isomorphism is visible only from an external perspective to the model. Nevertheless, it is not currently clear to what extent we may hope to prove or refute the hypothesis that there is a class embedding of V into L that is elementary for quantifier-free assertions.

Comment on The countable models of ZFC, up to isomorphism, are linearly ordered by the submodel relation; indeed, every countable model of ZFC, including every transitive model, is isomorphic to a submodel of its own $L$, New York, 2012 by Thomas Benjamin

Thomas Benjamin — Wed, 09 May 2012 06:21:20 +0000

Very, Very Interesting! How does this apply to the countable models which are forcing extensions? How does this relate (as countable models are deemed ‘toy models’) to the notion of the set-theoretic multiverse?

Comment on Models of $\rm{ZFC}^-$ that are not definable in their set forcing extensions by Victoria Gitman

Victoria Gitman — Mon, 07 May 2012 13:55:44 +0000

Hi Everett. Thanks for your excellent questions! First, the amount of choice needed for Solovay’s theorem: the principle ${\rm DC}_\lambda$ holds in $L(V_{\lambda+1})$, and so this is not sufficient to split the cof $\omega$ ordinals below $\lambda^+$ into disjoint stationary sets. Unfortunately, I do not know too much else about the situation. It seems natural to try to reduce the $I_0$ -cardinal assumption by looking for other ${\rm ZF}$-models where Solovay’s theorem fails. During my talk on friday, Joel Hamkins actually brought up the question whether definability of ground models holds for ${\rm ZF}$ models. No answers yet! I will be posting a slightly less error ridden draft of the notes today, so check in later for it!

Comment on Models of $\rm{ZFC}^-$ that are not definable in their set forcing extensions by Everett Piper

Everett Piper — Sun, 06 May 2012 01:18:51 +0000

Hi Victoria. I read your paper on definability of the universe in a generic extension of ZFC^-. I have a question related to the argument involving the I_0 cardinal. The argument there relies on being able to partition the ordinals smaller $\lambda^+$ of countable cofinality into two disjoint stationary sets. It also seems like the argument generalizes to posets which collapse other cardinals as well. My question is: how much choice suffices to prove Solovay’s theorem that stationary subsets of regular cards can be partitioned into stationary sets? It seems like enough of the power of choice is available in models of ZFC^- to carry out this argument. Do you know of other natural models of ZF+”weak choice” (“weak choice” could be uniformization or DC or something else) where Solovay’s theorem fails and you can carry out the non-definability of the ground universe argument?

Comment on Review: Is classical set theory compatible with quantum experiments? by saf

saf — Sat, 05 May 2012 04:36:09 +0000

I agree!

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Comment on Models of $\rm{ZFC}^-$ that are not definable in their set forcing extensions by This Week in Logic at CUNY | Set Theory Talks

This Week in Logic at CUNY | Set Theory Talks — Tue, 01 May 2012 02:40:03 +0000

[...] Set Theory Seminar Friday, May 4, 2012 10:00 am GC 6417 Professor Victoria Gitman (NYC College of Technology (CUNY)) Models of ZFC minus powerset that are not definable in their set forcing extensions [...]

Comment on The countable models of ZFC, up to isomorphism, are linearly ordered by the submodel relation; indeed, every countable model of ZFC, including every transitive model, is isomorphic to a submodel of its own $L$, New York, 2012 by Joel David Hamkins

Joel David Hamkins — Tue, 01 May 2012 02:12:23 +0000

Thanks for the vote of confidence, Vika. It means a lot to me.

Comment on The countable models of ZFC, up to isomorphism, are linearly ordered by the submodel relation; indeed, every countable model of ZFC, including every transitive model, is isomorphic to a submodel of its own $L$, New York, 2012 by Victoria Gitman

Victoria Gitman — Mon, 30 Apr 2012 23:36:38 +0000

Wow! This is an amazing theorem! I also never thought I would see surreal numbers used in a set theoretic paper, it makes me want to learn all about them. I can’t wait to hear the details. Hopefully, I can make arrangements to come to the city.

Comment on The mate-in-n problem of infinite chess is decidable by The omega one of infinite chess, New York, 2012 | Joel David Hamkins

The omega one of infinite chess, New York, 2012 | Joel David Hamkins — Sat, 28 Apr 2012 23:59:06 +0000

[...] article | slides This entry was posted in Talks and tagged chess, computability, definibility by Joel David Hamkins. Bookmark the permalink. [...]

Comment on My favorite proof that $\mathbb R$ is uncountable by Zach Teitler

Zach Teitler — Fri, 27 Apr 2012 16:03:22 +0000

Matt, a decreasing sequence of intervals in the real numbers always has a nonempty intersection (a point in the intersection of all the intervals), but a decreasing sequence of intervals in the rationals may have an empty intersection.

Consider, say, the intervals [a_n, b_n] where a_1 = 1, b_1 = 2, b_{n+1} = (a_n+b_n)/2, and a_{n+1} = 2/b_{n+1}. Then the endpoints are all rational (if that matters) and satisfy a_n < a_{n+1} < b_{n+1} < b_n. There's no rational number in the intersection of these intervals; the intersection contains only the irrational square root of 2.

Comment on The order-type of clubs in a square sequence by Shelah’s approachability ideal, part 1 | Assaf Rinot

Shelah’s approachability ideal, part 1 | Assaf Rinot — Wed, 25 Apr 2012 04:27:55 +0000

[...] then $I[lambda^+]=mathcal P(lambda^+)$. (the definition of $square_lambda$ may be found in here.) Proof hints. 1) Let $mathcal D_alpha:=[alpha]^{<omega}$ for all $alpha<lambda$. 2) [...]

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Comment on An inconsistent form of club guessing by Shelah’s approachability ideal, part 1 | Assaf Rinot

Shelah’s approachability ideal, part 1 | Assaf Rinot — Wed, 25 Apr 2012 04:09:43 +0000

[...] Thus, we shall assume that $aleph_0<kappa<kappa^{++}<lambda$. In an earlier post, we proved that $E^{aleph_3}_{aleph_1}$ carries a club guessing sequence, and since here $kappa$ [...]

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Comment on yay, I’m an editor at ScienceSeeker.org by Igorcarron

Igorcarron — Tue, 24 Apr 2012 14:02:00 +0000

Congrats (as the Chinese proverb says, be careful what you wish for ).

One of the feature that was a little off putting when I initially looked at the earlier generation called v 2.0 was this need to connect posts to a peer review article. I did not bother too much making that connection between their aggragtor and Nuit Blanche then.

On Nuit Blanche, or any blogs relying on conferences/arxiv style prepublications, when it is peer reviewed, it is already too late and so I could not even mention any of my posts then because none of them would appear before “a certain time”. I also worried that my blog entries would not be taken seriously because there would never be a “peer reviewed article” stamp on these entries. I wonder if the over-representation of life science articles there is not directly related to this need of relying on peer-reviewed papers and I wonder how blogs that rely on arxiv/conferences/unreviewed presentation would fit into the more “explanational” nature of blog entries written on already published works.

Comment on yay, I’m an editor at ScienceSeeker.org by Dave Munger

Dave Munger — Tue, 24 Apr 2012 14:01:00 +0000

Andreas, we are looking to add some editors in the physical sciences. If you know anyone who you think would be good, send them our way — I’m @davemunger on twitter.

Comment on yay, I’m an editor at ScienceSeeker.org by Peter Krautzberger

Peter Krautzberger — Mon, 23 Apr 2012 14:15:00 +0000

Thank you both, Andreas and François.

Yes, the biological sciences dominate in blogging just like everywhere else in academia (last time I checked, extra-mural funding in the EU went to 40% to the life sciences with another 40% to engineering).

But there’s a lot to learn from them, too. The way top bloggers help make the inner workings of their scientific community transparent is impressive — from being hired to being on hiring committees, from grant writing to grant panels from graduate student experiences to mentoring.

Mathematical researcher bloggers are way behind when it comes to using blogs to help our community, especially the younger generations (math teachers on the other hand are quite impressive already).

Comment on My favorite proof that $\mathbb R$ is uncountable by Joel David Hamkins

Joel David Hamkins — Mon, 23 Apr 2012 00:17:08 +0000

Sam, although sometimes people make a big fuss about this proof being different from the diagonalization proof using digits, I don’t see them as truly different. After all, fixing an initial segment of the digits of a real amounts to fixing a certain interval, the interval of reals whose digits begin that way. And so choosing the digits to avoid a certain real is just the same thing as choosing that interval to avoid that real. So the arguments seem fundamentally alike to me, just a different way of saying the same thing. The important idea of diagonalization, underlying both arguments, is that one may accomplish a countably infinite list of tasks, by accomplishing the $n^{th}$ task on the $n^{th}$ step of a convergent procedure.

But you do have good company, such as: http://mathoverflow.net/questions/23953/earliest-diagonal-proof-of-the-uncountability-of-the-reals

Comment on yay, I’m an editor at ScienceSeeker.org by Andreas Blass

Andreas Blass — Sun, 22 Apr 2012 22:16:00 +0000

Congratulations!

Looking at the list of editors and their areas, I get the impression that science consists of the biological sciences plus some peripheral stuff. So please do a great job representing our periphery.

Comment on yay, I’m an editor at ScienceSeeker.org by François G. Dorais

François G. Dorais — Sun, 22 Apr 2012 15:08:00 +0000

Congratulations, Peter!

Comment on My favorite proof that $\mathbb R$ is uncountable by Samuel Coskey

Samuel Coskey — Fri, 20 Apr 2012 19:55:58 +0000

In the argument we construct a decreasing sequence of intervals $[a_n,b_n]$. We then argue that there must be a point $x$ in the intersection of all these intervals—but we never argued that $x$ is rational! So this wouldn’t give any contradiction if you confined yourself to just the rational numbers.

Comment on My favorite proof that $\mathbb R$ is uncountable by Matt

Matt — Fri, 20 Apr 2012 19:49:43 +0000

I understand why the proof works, but what is to stop us from using the same argument on the rational numbers? Because they are dense, couldn’t we always choose an interval that excludes the most recent element in the sequence? Where is the contradiction?

Comment on Review: Is classical set theory compatible with quantum experiments? by Ido Ben-Dayan

Ido Ben-Dayan — Thu, 19 Apr 2012 21:39:53 +0000

Perhaps we should really look this up. Though the number of new things I’m thinking of is really becoming huge. Any chance you can learn epsilon analysis. At least I’m pretty sure that both me and Misha will be interested if we can say something about the path integral measure.

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Comment on Review: Is classical set theory compatible with quantum experiments? by saf

saf — Thu, 19 Apr 2012 18:36:51 +0000

Thanks! The patience here came from my impression that the speaker is trying to do an honest job, and that the source of confusions is simply the lack of firm background in set theory.
I contacted Arnon Avron, asking for an electronic copy of Guy Gildor’s thesis. While he does not have one, he pointed out that Zuzana Hanikova and Petr Hájek have some joint works on the subject, so my tip for the speaker would be: talk to these people!

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Comment on Review: Is classical set theory compatible with quantum experiments? by saf

saf — Thu, 19 Apr 2012 18:26:54 +0000

Thanks for your comment, Ido!
The more I come to think of it, the more it seems like Set Theory in Fuzzy Logic is indeed an example of framework that is worth testing with respect to such QM experiments (assuming this wasn’t done before). Just like non-standard analysis is sometime the right angle to see and prove theorems, Boolean-Valued models of set theory may be found to be convenient here.

As for the conrente/random number: $20k$. Don’t worry! $20k$ is as small as $10^{20k}$ from the partitioning point of view.

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Comment on Review: Is classical set theory compatible with quantum experiments? by Ido Ben-Dayan

Ido Ben-Dayan — Thu, 19 Apr 2012 17:45:21 +0000

Assaf sent me an email asking me what purpose math is supposed to serve in the context of physics. So here’s my very personal opinion.
Math is a fascinating science in its own. It’s not an experimental one, but rather based only on logic. Math gets a lot of intuition from physics, but it obviously does not restricts itself to what we “observe” in nature. Actually, a large part of physics today is analyzing toy models which obviously are not met in nature, in the hope that after understanding them, we may be able to build a model which does describe some natural phenomena. Personally what draws me to math is the pathologies and wackiness. Investigating PDES/ODES are interesting only when it is attached to some physical problem. We attempt to describe nature in terms of equations. That’s the physics part, but from there on the mathematical rules of solving the equations apply. It is true that we gloss over some mathematical rigour. For example people used Dirac delta function long before it was finally “mathematically” defined. Similarly people use the path integral for decades, even though its measure is still ill defined. However, I can partially defend both uses, by saying that the measure is defined on a lattice, so it’s not completely out of the blue. As far as I know in all occasions mathematicians eventually managed to construct a rigorous notion. The reason we gloss over, is because the main interest is predicting an observable phenomena, or explaining a phenomena which makes sense in terms of physical principles. Therefore since the path integral has successfully predicted endless number of experiments, it is a useful physical tool and the mathematicians are the ones who should work on it now. What physicists might expect from math is: 1) Use mathematical concepts to describe physics. Whenever anew mathematical concept was introduced into physics, it was revolutionary, from Newton through Einstein, and in the last 50+ years groups and topology. 2) Physics is limited by the mathematical knowledge, so if there’s new math being done, which could be applied to physics that would again be huge. I’m an old fashioned guy, but the amount of data being accumulated now in astrophysics and particle physics is becoming difficult to manage with current knowledge. So any mathematical concept that will enable different understanding of the data, not just a more efficient way of processing it, will again be a revolution. Last but not least, to the best of my knowledge the 20K is just a random number, he could’ve said any number larger than 20. Don’t take that part too literally.

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Comment on Review: Is classical set theory compatible with quantum experiments? by Ari B.

Ari B. — Thu, 19 Apr 2012 08:01:24 +0000

I haven’t looked at the slides so I’m just commenting based on the quote in your description. I wonder whether it’s ever possible to “conclude that the results of [any physical] experiment are incompatible with classical set theory”. I would say that, at worst, the results could be incompatible with the way someone thought classical set theory should be applied to the experiment.

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Comment on Models of $\rm{ZFC}^-$ that are not definable in their set forcing extensions by Joel David Hamkins

Joel David Hamkins — Thu, 19 Apr 2012 07:27:52 +0000

Very nice, Vika, I shall look forward to the talk!

Comment on One day in Colorado or Strongly summable ultrafilters are rapid by Peter Krautzberger

Peter Krautzberger — Wed, 18 Apr 2012 15:05:00 +0000

Thanks, Andreas. I agree that this is more elegant for people who know P-points well enough. I admit I simply didn’t think of it at the time of writing.

Of course, doing it this way would mean discussing the topological definition of addition and also the fact that the $(n+q)_{nin omega}$ actually are different ultrafilters – and this would have complicated things. Oh well, maybe next time

Comment on One day in Colorado or Strongly summable ultrafilters are rapid by Peter Krautzberger

Peter Krautzberger — Wed, 18 Apr 2012 14:57:00 +0000

Thanks for supporting my memory somewhat on this

Comment on Rapid idempotent ultrafilters by Peter Krautzberger

Peter Krautzberger — Wed, 18 Apr 2012 14:56:00 +0000

Thank you for your comments, Andreas! Why I give you and Neil credit is because the proof is from your paper. It’s just that your paper doesn’t actually mention that the proof therein shows rapidity, instead refers to Matet.

On the other hand, I couldn’t reconstruct Matet’s proof from his paper. I remember that I once understood it and that back then I thought both proofs are the same but looking at Matet’s paper again for this post I found it hard to get back into his notation of filters on partitions.

Comment on Is the dream solution of the continuum hypothesis attainable? by Joel David Hamkins

Joel David Hamkins — Tue, 17 Apr 2012 21:07:07 +0000

See Andres Caicedo’s blog posts critically discussing my article.

Comment on Rapid idempotent ultrafilters by Andreas Blass

Andreas Blass — Sun, 15 Apr 2012 22:05:00 +0000

Ignore (or better delete) the second half of my previous comment. I was thinking too much about your theorem and not about Matet’s.

Comment on Rapid idempotent ultrafilters by Andreas BLass

Andreas BLass — Sun, 15 Apr 2012 22:03:00 +0000

I think you’re giving me (and Neil Hindman) more credit than we deserve. In the theorem that you attribute to us and to Pierre Matet, the part about rapidity of max is, if I remember correctly, due solely to Matet. I believe the only occurrence of “rapid” in that joint paper by Neil and me is in the paragraph acknowledging Pierre’s work.

Also, at the end of the proof of that theorem, you get that the function f can take any value k at most k times on the y’s. But it’s the FS set generated by the y’s that is in the ultrafilter, and on that set it seems that f can take the value k about $2^k$ times. That does no real harm; this weaker conclusion still implies rapidity because you can compose f with an exponential function.

Comment on One day in Colorado or Strongly summable ultrafilters are rapid by Andreas BLass

Andreas BLass — Sun, 15 Apr 2012 21:36:00 +0000

I like your direct argument that sums can’t be P-points. As you surely know, but other readers might not, I generally view this fact as a consequence of the fact that a sum, p+q, is the limit, with respect to p, of the translates n+q of q (by finite numbers n). A P-point, on the other hand, is never a limit point of a countable set of other ultrafilters. (Indeed, this is the definition of “weak P-point,” and it follows immediately from the characterization of P-points in terms of countable intersections of neighborhoods in beta(N).)

Comment on One day in Colorado or Strongly summable ultrafilters are rapid by Andreas Blass

Andreas Blass — Sun, 15 Apr 2012 21:29:00 +0000

Your conjecture, that you caused me to pay insufficient attention to the next BLAST talk, is correct, at least to the extent that I don’t remember what talk that was.

Comment on state of foundational research by Carl Mummert

Carl Mummert — Sun, 15 Apr 2012 11:46:00 +0000

I did give a short answer there, but there is a lot more that I don’t have a clear way to express.

I think a particular difficulty with logic is that it is a young enough field that the seminal papers are all easily available and written in a way that they can (mostly) be read today. Only someone studying history would read Galois to learn about abstract algebra, but plenty of people try to read Goedel’s original paper to learn about the incompleteness theorems. The problem with that is that those old papers are beautiful but lack 75 years of improved perspective.

What I don’t know of is a good book or paper that describes the current perspective on “foundations of mathematics”. There are email posts by Harvey Friedman, but they might be too polemical for an undergraduate who doesn’t have the perspective to appreciate exactly what argument is being made. On the other hand there are textbooks like Enderton’s book, which I like very much but which scrupulously avoids the topic of foundations. Does anyone else know of something that could be used to introduce an undergraduate to the modern idea(s) of foundations?

Comment on state of foundational research by Samuel Coskey

Samuel Coskey — Sat, 14 Apr 2012 16:23:00 +0000

So whose responsibility is it to make sure pure math receives funding?

Comment on state of foundational research by Peter Krautzberger

Peter Krautzberger — Sat, 14 Apr 2012 03:59:00 +0000

I’m aware of that. But it still leaves me with the impression that there’s a lack of involvement in the necessary politics. To put it differently, this wouldn’t have happened to the AMS with its active campaigning and an office in DC.

Comment on state of foundational research by David Roberts

David Roberts — Sat, 14 Apr 2012 02:42:00 +0000

“(This was and is an embarrassment and failure of the leaders of the UK
math community, really. How could the major grant agency make such major
policy changes without anyone noticing?)”

It was more of a unilateral move on the behalf of the grant agency without consulting the field. The decision was presented as a fait accompli and then leaders in the UK math community, including people like Atiyah and his ilk, made a fuss to no avail.

Comment on Quiz on public peer review by Victoria Gitman

Victoria Gitman — Thu, 12 Apr 2012 20:53:19 +0000

Well, I always suspected that the BR crowd were troublemakers, but now it turns out they are aspiring “irresponsible” researchers as well! On a more serious note, the presence of such a question does seem to indicate a certain level of insecurity in the establishment.

Comment on Pointwise definable models of set theory by Must there be numbers we cannot describe or define? Pointwise definability and the Math Tea argument, Bristol, April 2012 | Joel David Hamkins

Thu, 12 Apr 2012 19:29:24 +0000

[...] article This entry was posted in Talks and tagged Bristol, definibility, forcing, HOD by Joel David [...]

Comment on Computable roots of computable functions by Carl Mummert

Carl Mummert — Tue, 10 Apr 2012 11:09:01 +0000

The computable IVT works fine. For simplicity assume the interval is $[0,1]$ and that $f(0) < 0 < f(1)$ and that the function has no rational roots (because these are all computable). Then $f(1/2)$ is either strictly positive or strictly negative, and with enough computation we can tell which of these alternatives is the case. Thus we have can replace either $0$ or $1$ with $1/2$. We iterate this process, halving the interval again and again, which gives a quickly converging Cauchy sequence for some root.

There is a small subtlety that a different algorithm is needed if the function has a rational root, and there is no uniform way to tell whether a function has no rational roots. The uniform version of the IVT is not provable in $\mathsf{RCA}_0$ (it is equivalent to $\mathsf{WKL}_0$) while the non-uniformized version is provable in $\mathsf{RCA}_0$ as above.

Comment on Computable roots of computable functions by Carl Mummert

Carl Mummert — Tue, 10 Apr 2012 11:00:03 +0000

Thanks for the detailed reading! I edited the post and made the last two changes you suggested.

The function from Theorem 6 is constructed as follows. Enumerate the effective open set as a sequence $(I_n)$ of open intervals. On interval $I_n$, put a nonnegative bump function that is nonzero exactly on $I_n$ and has height $2^{-n}$. Then let $f$ be the pointwise sum of all these bump functions. This function $f$ is nonzero exactly on the open set, and is nonnegative everywhere. Also $\mathsf{RCA}_0$ is able to verify that this construction works.

Comment on Computable roots of computable functions by Quinn Culver

Quinn Culver — Tue, 10 Apr 2012 03:18:44 +0000

I just found this, which answers my question: http://mathoverflow.net/questions/9000/intermediate-value-theorem-on-computable-reals.

Comment on Computable roots of computable functions by Quinn Culver

Quinn Culver — Tue, 10 Apr 2012 02:32:52 +0000

Great post. Some comments/questions:

1. Notice that if $f$ actually crosses the $x$-axis at $\alpha$, then Theorem 1 is pretty easy to prove directly: fix a rational interval $[p,q]$ such that $\alpha$ is $f$’s only root in $[p,q]$. Then use the intermediate value theorem to hone in on $\alpha$. I guess if $f$ is nonnegative (or nonpositive), then it’s not so easy.

2. Algorithmic randomness gives a simple proof of Theorem 3 (and for the same reason a simple proof of the existence of a $\Pi^{0}_{1}$ class in Cantor space with no computable member): Let $K_n=[0,1]-U_n$, where $\{U_n\}_{n \in \omega}$ is the universal Martin-L\:{o}f test. Then each $K_n$ is effectively closed and has positive measure, so is uncountable. But each member of $K_n$ is Martin-L\:{o}f random, hence incomputable.

3. I’m curious to know what the function (which would have a given effectively closed set as its root set) in Theorem 6 looks like. In particular, would it be a violation of the “computable intermediate value theorem”? That is, would there be rationals $p$ and $q$ such that $f(p)<0<f(q)$, but $f$ has no computable root between $p$ and $q$? This wouldn't be the case if the function were always non-negative. Someone recently told me that this "computable intermediate value theorem" is indeed a theorem, but I was suspicious.

4. A (minor) typo:
"Theorem 4. There is an computable continuous…"

5. I think saying "computable continuous" is redundant since computable implies continuous. (Note that this occurs twice.)

Comment on Quiz on public peer review by Samuel Coskey

Samuel Coskey — Mon, 09 Apr 2012 22:24:46 +0000

“True or false: [insert idea we don't like]”

-translation-

Conform or be punished.

Comment on Hindman’s Theorem, partial semigroups and some of my most lacking intuitions (part 5) by Rapid idempotent ultrafilters | Peter Krautzberger

Rapid idempotent ultrafilters | Peter Krautzberger — Sun, 08 Apr 2012 17:59:15 +0000

[...] After I got around to writing my argument up properly after the conference, Jana asked me whether there are could be other rapid idempotent ultrafilters. In particular, could there be so-called minimal idempotents which are rapid? This, again, sounded rather drastic to me. Minimal idempotents have extremely rich algebraic properties, in particular, any set in them is central and thus all versions of the Central Sets Theorem hold for such sets (as opposed to FS-sets where no FS-set with the growth condition satisfies even the simplest Central Sets Theorem). [...]

Comment on One day in Colorado or Strongly summable ultrafilters are rapid by Rapid idempotent ultrafilters | Peter Krautzberger

Rapid idempotent ultrafilters | Peter Krautzberger — Sun, 08 Apr 2012 17:58:59 +0000

[...] Krautzberger Skip to content HomePapersResources ← One day in Colorado or Strongly summable ultrafilters are rapid by Peter Krautzberger | 2012/04/08 · 1:58 pm ↓ Jump to [...]

Comment on Infinitary computability publications by A course in infinitary computability, Fall 2012, CUNY Graduate Center, CSC 85020 | Joel David Hamkins

Sun, 08 Apr 2012 10:00:48 +0000

[...] Infinitary computability [...]

Comment on Automorphism tower publications by What happens when one iteratively computes the automorphism group of a group? Temple University, Philadelphia 2012 | Joel David Hamkins

Sat, 07 Apr 2012 09:05:52 +0000

[...] Automorphism towers [...]

Comment on Quiz on public peer review by Peter

Peter — Fri, 06 Apr 2012 15:27:25 +0000

I exaggerated a little. I also believe in peer review in principle. But the inflation of papers makes “when done well” a more and more remote possibility.

Comment on Quiz on public peer review by Carl Mummert

Carl Mummert — Fri, 06 Apr 2012 15:07:10 +0000

I’m a moderate about peer review – it certainly has some benefits when done well, and problems when done poorly. But the idea that peer review will magically evolve on its own to eliminate problems seems to be particularly farfetched.

Comment on Quiz on public peer review by Peter

Peter — Fri, 06 Apr 2012 15:02:30 +0000

This would be hilarious if it wasn’t real.

The answer is “it won’t work” and then some general bs about peer review evolving (how exactly?) to become awesomesauce!!eleventy!! stuff — how, uh, refined.

At least the reverse implication of the predicted evolution is honest: right now, peer review is biased, full of conflicts of interest, neither fair nor rigorous. Now that’s something we could agree on…

Comment on MA and its effect on Tree Partitions by Another Combinatorial Result

Another Combinatorial Result — Wed, 04 Apr 2012 18:13:29 +0000

[...] (NonSpecial Tree) $notrightarrow$ (NonSpecial Tree, $omega+2)^2$”, which I explained here. This entry was posted in Full Article, Presentation and tagged MA, Partition Relation, Ramsey [...]

Comment on Idempotent Ultrafilters, an introduction (Michigan Logic Seminar Nov 09, 2011) by One day in Colorado or Strongly summable ultrafilters are rapid | Peter Krautzberger

Tue, 03 Apr 2012 01:21:46 +0000

[...] then). One afternoon, I was sitting somewhere on campus with Jana Flašková and she asked me if idempotent ultrafilters could be [...]

Comment on Julia sets and the Mandelbrot set by Victoria Gitman

Victoria Gitman — Thu, 29 Mar 2012 23:36:27 +0000

Oh yes! I spent some time reading Rhapsody in Numbers, it is awesome !

Comment on Julia sets and the Mandelbrot set by Peter Krautzberger

Peter Krautzberger — Thu, 29 Mar 2012 22:36:11 +0000

Cool slides! By the way, my favorite source for fractals is the blog Rhapsody in Numbers.

Sam, have you checked out s5? Pandoc supports it, especially from markdown.

Comment on Julia sets and the Mandelbrot set by Samuel Coskey

Samuel Coskey — Thu, 29 Mar 2012 20:20:35 +0000

Your slides look really great. Good job!

I completely agree that Beamer is not sufficient or even desirable for a complex presentation with many pictures and animations. And on the other hand, the web provides a really nice home for such things.

One day, I would like to see a free and easy-to-use tool to create html5 presentations. I know there are several attempts out there, but I doubt they are at a point where they can give you the kind of flexibility you would want.

Comment on Prelude to a small experiment by Peter Krautzberger

Peter Krautzberger — Thu, 29 Mar 2012 16:52:00 +0000

Yes, discoverability is an issue (then again, googling or duckduckgoing for idempotent ultrafilters puts some of my stuff on the front page) .

I think we’ll never get enough motivation for a good solution for this without creating and sharing the content that we want to be discovered. So I worry about that first while keeping an eye on the discoverability. Personally, I favor decentralized social networks via tools like wordpress to host the content as well as specialized search engines // aggregators to discover content and the people behind it.

Comment on Prelude to a small experiment by Peter Krautzberger

Peter Krautzberger — Thu, 29 Mar 2012 16:49:00 +0000

Looking forward to your post on this!

Comment on Prelude to a small experiment by Carl Mummert

Carl Mummert — Wed, 28 Mar 2012 11:32:00 +0000

The “microresult” idea is very similar to the motivation I have for experimenting with online publishing. I agree with many of Felix Breuer’s comments in his blog post, that it is becoming increasingly impossible to keep up with the volume of mathematics (this has been a problem for several decades, of course). But at the same time I think that we don’t publish *enough*. There are many microresults that we discover, possibly write in a notebook, and then leave because there’s not a good way to fit them into a peer-reviewed paper. These often get passed around as folklore in larger departments, discussed over tea. After our discussion last night, I’ve started to think about a post for my blog to try to collect my ideas on this.

Comment on Prelude to a small experiment by fbreuer

fbreuer — Tue, 27 Mar 2012 17:22:00 +0000

Thanks for the mention! I am looking forward to your double-post!

I am all for micro-contributions! With their “open” nature they explore the territory “beyond theorems”, even if they are micro-theorems themselves.

The main challenge I see is that of discoverability. How will we discover relevant micro-contributions if we are already flooded with macro-contributions? Dror Bar-Natan’s pensieve is an interesting example in this regard. On
the one hand it is absolutely brilliant in its radicality! (Just the
thought of putting all my xournal notes online is scary.) On the other
hand, who is going to dig through all these notes to sift out the great
ideas?

I am sure, though, that in the case of your double-post, discoverability will not be much of an issue as your blog gets plenty of readers. (I can tell by the amount of traffic you have sent my way.

Comment on Indestructibility for Ramsey cardinals by Simon Thomas; Victoria Gitman | Set Theory Talks

Simon Thomas; Victoria Gitman | Set Theory Talks — Tue, 27 Mar 2012 00:17:59 +0000

[...] Logic Seminar Monday April 2nd, 5:00-6:20 pm Victoria Gitman, CUNY Indestructibility for Ramsey cardinals This entry was posted in Seminars and tagged Simon Thomas, Victoria Gitman. Bookmark the [...]

Comment on Bumper sticker by Joel David Hamkins

Joel David Hamkins — Mon, 26 Mar 2012 06:32:00 +0000

Honk! Honk!

Comment on Indestructibility for Ramsey cardinals by Victoria Gitman

Victoria Gitman — Thu, 22 Mar 2012 11:58:10 +0000

Thanks Erin! See you soon .

Comment on Indestructibility for Ramsey cardinals by Erin Kathryn Carmody

Erin Kathryn Carmody — Thu, 22 Mar 2012 02:49:36 +0000

Victoria, I am very much looking forward to your talk.

Comment on Possibly true. Necessarily funny. by Sam

Sam — Mon, 19 Mar 2012 18:51:00 +0000

My partner “got me” with this article! It’s perfect because Kripke is really quite a personality.

Comment on Why markdown, not $\LaTeX$? by Peter Krautzberger

Peter Krautzberger — Fri, 16 Mar 2012 13:43:00 +0000

iiuc, you should be able to add arbitrary content via css, in particular counters and anchors.

I can’t help but say that this is asking for more than what LaTeX does (without the help of packages) — it’s a sore point in many discussions that people want MathJax to do work that LaTeX doesn’t…As I said, MathJax most likely won’t ever add this because it’s not a math issue, it’s a general authoring problem that should be solved elsewhere. (Well, actually, thanks to the sponsor AMS desiring it, MathJax 2.0 offers ref and label so Davide might be convinced to add this and people can always write their own extensions of MathJax…)

Comment on Why markdown, not $\LaTeX$? by Carl Mummert

Carl Mummert — Fri, 16 Mar 2012 13:35:00 +0000

With CSS counters, is it possible to reference the counter somewhere else in the HTML, as the ref command would do in latex? Of course this could all be implemented in JavaScript instead of CSS or XSLT, and since MathJax is already using JavaSscript that may be a better way to go. My spring break is coming up, so I may have some time to try this out.

Comment on Why markdown, not $\LaTeX$? by Mclearc

Mclearc — Thu, 15 Mar 2012 21:06:00 +0000

the current version of pandoc (1.9.1.2) seems to integrate latex commands pretty seamlessly into its markdown and then passes them through to pdflatex or whathaveyou. Anything between “$$” will be treated as TEX math. And I regularly incorporate header/footer info as well as other little bits and bobs into my markdown. If all you’re doing is sending it to PDF it should work seamlessly from the command line. No need to even open TexShop (or whatever).

Comment on Why markdown, not $\LaTeX$? by Peter Krautzberger

Peter Krautzberger — Wed, 14 Mar 2012 13:31:00 +0000

Ah, ok!

Do you need more than CSS counters? I mean, if you want to do it properly, you theorems should have their own css and then https://developer.mozilla.org/en/CSS_Counters will get you some counters.

Comment on Convergence of ideas by Dana C. Ernst

Dana C. Ernst — Wed, 14 Mar 2012 02:21:00 +0000

I agree with Peter. Great post.

Comment on Why markdown, not $\LaTeX$? by Carl Mummert

Carl Mummert — Wed, 14 Mar 2012 01:45:00 +0000

Mental error – when I wrote “MathJax”, I meant “Markdown”. At the moment when I write HTML text on my blog I have to type the numbers manually myself, and as far as I can tell Markdown would have the same problem. I have been looking into using XSLT to fix this but it’s a steep learning curve.

Comment on Why markdown, not $\LaTeX$? by Peter Krautzberger

Peter Krautzberger — Tue, 13 Mar 2012 23:41:00 +0000

My perspective is a little different.

On the one hand, ref and label work in version 2.0. But theorem environments do not exist (and might never arrive) because MathJax focuses on mathematics, not text.

From my point of view, it is actually better to keep the two separated. I would prefer to have an html source that does not rely on MathJax to do something as simple as have a number next to a theorem.

Which doesn’t mean I don’t want theorem numbering. It just needs to come from somewhere else, preferably in pre-production, i.e., use tex4ht to generate html+mathml and have MathJax render it or extend the markdown syntax or choose a higher level markup language etc.

Comment on Why markdown, not $\LaTeX$? by Carl Mummert

Carl Mummert — Tue, 13 Mar 2012 19:53:00 +0000

In my opinion, the main thing that MathJax lacks is not layout. I typically just ignore the layout when I write in LaTeX, although the typesetting will still end up much better than in a web browser. The main thing that I don’t get from MathJax is automatically numbered theoremlike environments and a way to refer back to them by name in the source code of my article. This is also the main problem that Microsoft Word has, by the way. I can live with the bad typesetting in a browser, but it is a real step backward to have type “Theorem 1″ directly into my source code, as if I am using a typewriter.

Comment on Convergence of ideas by Peter Krautzberger

Peter Krautzberger — Sun, 11 Mar 2012 22:02:00 +0000

What a wonderful post. This is why I think it is so urgent that we move beyond papers — this is exemplary research activity of the future — quick dissemination, immediate peer review. There is no reason to actually press this into a paper, really, if credit would be given (not that Andreas needs it, but for the younger generation, this might is a much saner way to build a track record).

Comment on WordPress in a VM by Kokomoko 26

Kokomoko 26 — Sun, 11 Mar 2012 10:29:00 +0000

nice one!

Comment on Disqus by François G. Dorais

François G. Dorais — Fri, 09 Mar 2012 02:08:00 +0000

$LaTeX$ is now enabled in Disqus comments!

Comment on Why markdown, not $\LaTeX$? by Peter Krautzberger

Peter Krautzberger — Thu, 08 Mar 2012 14:39:01 +0000

I agree, the archival value is impressive even though I’m not as worried about stability of open formats, HTML in particular.

Comment on Facts about the Urysohn Space – Some useful, some cool by m.

m. — Wed, 07 Mar 2012 11:25:16 +0000

nice, ~~but there is a typo in the def of Katetov function~~. (Fixed. -Mike)

Comment on Why markdown, not $\LaTeX$? by John Baker

John Baker — Tue, 06 Mar 2012 21:55:00 +0000

Thanks for your thoughtful remarks. I wish I had come across them before working through a project to convert WordPress Blogs to LaTeX+PDF and EPUB and MOBI formats:

http://wp.me/pDcwA-EC
In the process I stumbled upon pandoc and markdown and even though I am a longtime LaTeX user I completely agree with your points.

One additional point in markdown’s favor is it’s inherent archival quality. As long as we can edit text files we can read it. This is not true of binary formats, ODT, DOCX and even PDF. In my lifetime I’ve already experienced “bit-loss” due to ever changing word processing files. For a format to survive over the long run (centuries) it has to be human readable, open and beyond the control of software companies and governments.

Comment on MA and its effect on Tree Partitions by Micheal Pawliuk

Micheal Pawliuk — Mon, 05 Mar 2012 17:08:06 +0000

Thanks again for the proofing. I assimilated all of them except your comment about the proof of Lemma 2.

I also clumped your comments together for ease of reading.

Comment on Generalized separation principles by Carl Mummert

Carl Mummert — Fri, 02 Mar 2012 14:49:00 +0000

There is an example in “The Atomic Model Theorem and Type Omitting” by Hirschfeldt, Shore, and Slaman. They showed that the two principles $Pi^0_1-G$ and AMT are not equivalent over $mathsf{RCA}_0$ but they are equivalent over $mathsf{RCA}_0$ plus $SIgma^0_2$ induction. This is described in their paper in the prose below Corollary 4.5.

Comment on Generalized separation principles by François G. Dorais

François G. Dorais — Fri, 02 Mar 2012 02:04:00 +0000

All the ones I’m aware of are for $Sigma^1_1$ induction. Are you aware of any for something like $Sigma^0_2$ induction?

Comment on Generalized separation principles by Carl Mummert

Carl Mummert — Thu, 01 Mar 2012 13:44:00 +0000

Another classic example of necessary induction in a reversal is that $mathsf{ACA}_0$ is equivalent to $mathsf{ACA}_0′$ in the real world and also equivalent over $mathsf{RCA}_0$ plus $Sigma^1_1$ induction, but not over $mathsf{RCA}_0$ alone. I believe there are several other examples of this in Simpson’s book, for example Theorem VII.6.9(2), although they are not emphasized as having this property.

Comment on Generalized separation principles by Carl Mummert

Carl Mummert — Thu, 01 Mar 2012 13:31:00 +0000

For those comparing this with classic reverse math literature, they would have talked about the range of a function instead of an enumerable set in the sense given here (mostly to camouflage the terminology $Sigma^0_1$). Thus the classic “Every function has a range” would translate to “Every enumerable set is decidable”. The terminology “decidable set” is also nice because it brings out the relationship with constructive systems, where “set” does not mean “decidable set”.

Comment on Special uncountable trees by saf

saf — Thu, 01 Mar 2012 01:41:29 +0000

The above is a proof of the c.c.c. property for a poset which is not Knaster, and indeed the reasoning is different than the “$\Delta$-system plus refinements” method. [btw, while the ultrafilter argument is very elegant, one can still do without it].

Note that if a poset is not Knaster, then a “$\Delta$-system plus refinements” argument simply cannot end-up with an uncountable directed set. Hence, either the refinement allows to find merely two (equivalently, countably many pairwise) compatible conditions [that's not very common], or, a different/additional type of reasoning is required – as exemplified in my post that you mentioned.

Comment on Subposets of small dimension by François G. Dorais

François G. Dorais — Wed, 29 Feb 2012 18:48:00 +0000

Yes, no problem. I’m glad your interested in thinking about this!

Any linear order has dimension $1$, of course, so an infinite chain in a poset is an infinite subposet of dimension at most $2$. The equality relation on any set $X$ is a partial order of dimension $2$ since ${=}$ is the intersection of any linear ordering of $X$ with the reverse of that linear ordering. Therefore, an infinite antichain in a poset is an infinite subposet of dimension at most $2$. The partition relation $aleph_0 to (aleph_0)^2_2$ guarantees that every infinite poset either contains an infinite chain or an infinite antichain, in either case it contains an infinite suposet of dimension at most $2$.

The Sierpinski coloring that witnesses $aleph_1 notto (aleph_1)^2_2$ comes from an uncountable $2$-dimensional poset. Recall that this coloring is defined as follows: pick $aleph_1$ real numbers $langle r_alpha rangle_{alpha lt omega_1}$ then, for each pair $alpha lt beta lt omega_1$, define $c(alpha,beta) = 1$ when $r_alpha lt r_beta$ and $c(alpha,beta) = 0$ when $r_alpha gt r_beta$. Thus $c(alpha,beta) = 1$ precisely when $alpha$ and $beta$ are comparable in the poset $(omega_1,{leq_S})$, where ${leq_S}$ is the intersection of the usual linear order ${leq}$ on $omega_1$ and the linear order on $omega_1$ defined by $alpha leq_R beta$ iff $r_alpha leq r_beta$. This is a poset of dimension $2$ with no uncountable chains nor uncountable antichains.

Comment on Subposets of small dimension by Micheal Pawliuk

Micheal Pawliuk — Wed, 29 Feb 2012 18:14:00 +0000

Like Assaf I am also most interested in the third question. I’m having a hard time grokking this dimension idea though.

Do you mind giving more detail as to why “$aleph_0 rightarrow (aleph_0)^2_2$ means that every infinite poset has an infinite subposet of dimension at most 2.” From there I should be able to start thinking about this problem.

Comment on MA and its effect on Tree Partitions by Ari Brodsky

Ari Brodsky — Wed, 29 Feb 2012 11:31:36 +0000

Corrections

Your definition of nonspecial should have a superscript 1, not 2.

You also need to change the descriptive text at the end of that sentence where you changed the number.

In the statement of the theorem, it’s confusing to have T on the right side of the arrow, as the T on the left side refers to the specific tree introduced earlier in the sentence, whereas when you put T on the right side you really mean any nonspecial tree.

Your statement of Lemma 1 is confusing, because the usual definition Aronszajn tree requires that it have cardinality $\aleph_1$. Rather than “Aronszajn tree T” you mean “tree T with no uncountable chain”.

Two comments about your Lemma 2:
1) The condition “of cardinality $2^{ℵ_0}$ is not required in the statement of the lemma.
2) In the proof, the point is that there cannot be nonspecial many nodes with special upwards cones. If there were, you get a contradiction by considering a minimal such node, as it would have a nonspecial cone above it.

The usual definition of $\hat a$ contains $<$ rather than $\leq$.

You are consistent in your use of Aronszajn throughout to mean a tree without uncountable chain, but this seems to be nonstandard.

In the second sentence of the proof of Fact 2, “special” should be “nonspecial”.

In the statement of the Claim inside the proof of Fact 1, there is a hat missing from $A_\xi \cap \hat\alpha$.

Questions

I'm reasonably confident that the hypotheses of the theorem can be weakened. Instead of requiring the full MA and $\left|T\right| = \mathfrak c$, we can simply assume that $T$ is a tree with no uncountable chains such that $\left|T\right| = \mathfrak m$, where $\mathfrak m$ is the smallest cardinal $\kappa$ for which $MA_\kappa$ is false.
The required version of Lemma 1 is: If $T$ is a tree with no uncountable chain such that $\left|T\right| < \mathfrak m$ then $T$ must be special. (Thus nonspecial trees with no uncountable chain must have cardinality $\geq \mathfrak m$.) This is Corollary 41I of Fremlin's textbook "Consequences of Martin's Axiom".
Lemma 2 doesn't depend on the cardinality as I mentioned earlier.
I think Lemma 3 simply requires $\kappa$ to be $\leq \mathfrak p$, which is certainly true if $\kappa \leq \mathfrak m$.
Does anyone see anything wrong with this?

Comment on Special uncountable trees by Ari B.

Ari B. — Wed, 29 Feb 2012 09:32:51 +0000

“whith” should be “which”.

Comment on Special uncountable trees by Ari B.

Ari B. — Wed, 29 Feb 2012 08:21:58 +0000

Is this a counterexample to Assaf’s claim http://blog.assafrinot.com/?p=1246 that “if the notion of forcing does not have the Knaster property, then this method [using the delta-system lemma followed by further refinements] is unlikely to do the job”? Or is the point that the ultrafilters add a level of complication to the proof that would not be present if it satisfied Knaster, thereby supporting Assaf’s statement? Can we make a general claim along the lines of “To show that a non-Knaster poset is ccc using the delta-system lemma, you will need to use ultrafilters”?

Comment on Special uncountable trees by MA and its effect on Tree Partitions

MA and its effect on Tree Partitions — Tue, 28 Feb 2012 19:35:37 +0000

[...] 1970. The poset is what you might expect – finite approximation of a partition- but it is difficult to show this is [...]

Comment on Special uncountable trees by saf

saf — Mon, 27 Feb 2012 15:48:12 +0000

I think it may be easier to use the other characterization that I came up with last night. Suppose that $\langle \omega_1,\unlhd\rangle$ is an Aronszajn tree. Take a countable elementary submodel $\mathcal N$ that contains $\unlhd$ and $P$. Suppose that $A\subseteq P$ is an uncountable antichain in $\mathcal N$, and $q\in A\setminus\mathcal N$. We shall yield a contradiction by finding some $r\in A\cap\mathcal N$ which is compatible with $q$. Let $\rho:=q\cap\mathcal N$. For all $p\in A$, and $i<\omega$, denote (whenever makes sense) by $p(i)$ the $i_{th}$ element of $\text{dom}(p\setminus\rho)$. We shall be interested in lower cones of the form $\{ \alpha\in\omega_1\mid \alpha\unlhd q(i)\}$. By pigeonhole, there exists $i,j<\omega$ such that $\{ r(j)\mid r\in A\cap\mathcal N, \rho\sqsubseteq r\}$ belongs to the initial $\{ \alpha\in\omega_1\mid \alpha\unlhd q(i)\}$, and moreover, that former set is a cofinal branch in $\langle\mathcal N\cap\omega_1,\unlhd\rangle$. Now, we only need to find a formal way to get a contradiction to the fact that $\langle\omega_1,\unlhd\rangle$ is Aronszajn…

Comment on Special uncountable trees by François G. Dorais

François G. Dorais — Sun, 26 Feb 2012 20:28:39 +0000

Let’s try to finish this argument without falling back into the ultrafilter argument…

So $p$ and $q$ are not necessarily compatible since $p \cup q$ could assign the same number to two compatible nodes of $T$, one of which is an element of the domain of $p$ and level at least $\delta$.

For simplicity, let’s first consider the case where the domain of $p$ has only one node $t_0$ with level at least $\delta$. Let’s suppose further that $p(t_0) = 3$. If no good $q$ can be found in $D \cap \mathcal{N}$, then it must be the case that $q^{-1}(3)$ contains a node below $t_0$ for every extension $q$ of $p_0$ in $D \cap \mathcal{N}$. Since $p_0 \cup \{(t,3)\}$ is a legal extension of $p_0$ in $\mathcal{N}$ for every node $t$ below $t_0$ and with height less than $\delta$, we see that there are uncountably many distinct extensions of $p_0$ in $D$ and that for any two such $q,r$ the finite sets $q^{-1}(3)$ and $r^{-1}(3)$ must intersect or contain comparable nodes.

This is starting to feel just like the ultrafilter argument: my next step would be to make a $\Delta$-system out of the $q^{-1}(3)$’s and then do some variant of the index trick again… Is there any way to cleverly use $t_0$ at this point?

Comment on Generalized separation principles by François G. Dorais

François G. Dorais — Sun, 26 Feb 2012 20:05:51 +0000

Paul Shafer kindly pointed out that Steve Simpson had already asked in 2001 whether $(\exists k)\mathsf{DNR}_k$ is equivalent to the weak König lemma over $\mathsf{RCA}_0$. (See these slides.)

Comment on The Delta-System Lemma by Special uncountable trees

Special uncountable trees — Fri, 24 Feb 2012 04:50:06 +0000

[...] we assume the reader is familiar with the $Delta$-system lemma, outlined in Mike’s post here, a standard tool for showing posets are [...]

Comment on Subposets of small dimension by François G. Dorais

François G. Dorais — Thu, 23 Feb 2012 20:10:53 +0000

Thanks Assaf! The third question is the one that started all of this. I’m mostly thinking about it under PFA, but anything will do. Trees won’t work because they are 2-dimensional (lex-order in two opposite ways) but I really like the idea of using a Cohen real as a diagonalization tool. A $\Diamond$ argument also sounds likely though that probably won’t help understanding the PFA situation.

Comment on Subposets of small dimension by Assaf Rinot

Assaf Rinot — Thu, 23 Feb 2012 03:11:03 +0000

Interesting post!! I mostly liked the third question, and has a feeling that the answer should be affirmative. More specifically, adding a single Cohen real should produce a model of ZFC admitting an uncountable poset with infinite dimension. The uncountable poset would be Todorcevic’s Souslin tree which is obtained as a composition of a coherent system of injections $\langle e_\alpha:\alpha\rightarrow\omega\mid \alpha<\omega_1\rangle$ composed with the generic Cohen real. Use the fact that the forcing notion is countable to diagonalize against any finite collections of linear ordering (of any uncountable subset).

Comment on Subposets of small dimension by Peter Krautzberger

Peter Krautzberger — Tue, 21 Feb 2012 05:11:23 +0000

Great post!

Comment on If you build it, will they come? by Both Students And Professors Need Certification, and the Elsevier Boycott | QED Insight

Fri, 17 Feb 2012 01:32:25 +0000

[...] Aaronson, Nassif Ghoussoub, Cathy O’Neil (here and here), and Peter Krautzberger (here and here). The issue has even hit the mainstream, at The New York Times and The Boston [...]

Comment on A comment on Tim Gowers’s blog by Both Students And Professors Need Certification, and the Elsevier Boycott | QED Insight

Thu, 16 Feb 2012 22:22:53 +0000

[...] Scott Aaronson, Nassif Ghoussoub, Cathy O’Neil (here and here), and Peter Krautzenberger (here and here). The issue has even hit the mainstream, at The New York Times and The Boston [...]

Comment on The Delta-System Lemma by Carl Mummert

Carl Mummert — Tue, 14 Feb 2012 04:20:01 +0000

I’m interested to see the result by Arkhangel’skii you alluded to.

Comment on The Delta-System Lemma by saf

saf — Mon, 13 Feb 2012 01:36:34 +0000

Sweet!

Comment on A comment on Tim Gowers’s blog by Peter Krautzberger

Peter Krautzberger — Tue, 07 Feb 2012 18:05:22 +0000

I see your point. What might help is a culture of seeking co-authors to remedy such deficits (if I was daring, I’d suggest this as referee of such a paper but that’s most likely too late in the game).

In set theory, Saharon Shelah seems similar to the researcher you describe. Reading a Shelah paper can be very painful, but reading a Shelah-Goldstern or Shelah-Blass paper usually isn’t (of course, with 1000+(!) papers Shelah has many, many co-authors for other practical reasons).

Oh, and I wholeheartedly agree with you that we should value expository work seriously!

Comment on A comment on Tim Gowers’s blog by Izabella Laba

Izabella Laba — Tue, 07 Feb 2012 17:31:05 +0000

I’m thinking of someone in particular who, research-wise, is at the absolute top of several fields of mathematics. I do wish that his papers were easier to read. But we still read them (even if it takes an effort) and use his ideas (and there’s an abundance of them, and they’re often unexpected and have far-reaching consequences, so it’s all worth the time we invest in it). He has been incredibly influential in many areas and has developed a huge following. To me, there’s no question about either his greatness or being a valuable member of the community. Sometimes you can’t have it all.

On the other hand, I really would like to see expository work better valued. Right now, if someone writes an article making X’s unreadable but important work accessible to the community, this is dismissed as “just expository” and therefore lesser work, unless the author manages to attach some minor new result to it whereupon the paper gets upgraded immediately to the “research” category. I don’t think that this is right.

Comment on A comment on Tim Gowers’s blog by Peter Krautzberger

Peter Krautzberger — Tue, 07 Feb 2012 15:59:47 +0000

I don’t know if there’s no correlation. I’d like to think that the very best, the extraordinary mathematicians are good expositors but I have no data to support this. Ultimately, I think, it comes down to values of the community — I would call someone with a lesser “new results” record the greater mathematician if they are a great expositor. What good are results if nobody can understand them (and why we don’t value people who make them understandable)? For example, I find it hard to call Perelman anything near great; his results, yes, but the mathematician, the member of our community? Another example would be Grothendieck, thanks to this letter.

I also fully agree with you that nobody should have to make every type of contribution. In fact, I hope for the exact opposite! People should contribute the best way they can. However, they can’t as long as the community does not value all types of contributions. I see a monoculture where only people who can produce the right kind of “new results” papers will get into the “right” journals and, ultimately, get the jobs. And I worry that it is not a sustainable culture.

Comment on A comment on Tim Gowers’s blog by Izabella Laba

Izabella Laba — Tue, 07 Feb 2012 05:46:45 +0000

There are truly great mathematicians who are not good expositors at all. I’m not sure that it would serve anyone’s interests to post names on a public blog, but they’re in Gowers’s league, research-wise. I’ve come to think that there might just be no correlation at all.

I’m all for recognizing a wider range of skills and contributions. I just don’t think that we should expect everyone to have every skill and make every type of contribution in the book. We have to wear a lot of hats already as it is.

Comment on Groups in $\beta \mathbb{N}$ by If you build it, will they come? | Peter Krautzberger

If you build it, will they come? | Peter Krautzberger — Tue, 07 Feb 2012 04:09:00 +0000

[...] In the summer of 2007, researchblogging.org went online. If you have ever visited a science blog, you might have come across its well known badge (I finally got around to it here). [...]

Comment on A comment on Tim Gowers’s blog by Peter Krautzberger

Peter Krautzberger — Sat, 04 Feb 2012 21:56:14 +0000

I wasn’t trying to describe why MO is attractive to some of its users. I was trying to describe why it isn’t attractive to others. Izabella Laba had a discussion with yet different reasons a while ago.

But yes, I disagree with your view that the competitive nature is key to MO’s success. It may have been in the past, but I think it can become damaging in the future.

Comment on A comment on Tim Gowers’s blog by Joel David Hamkins

Joel David Hamkins — Sat, 04 Feb 2012 13:57:08 +0000

But surely a major part of the attraction of MO is its vitality, the energetic participation by knowledgeable users. My view is that its mildly competitive nature, from a sociological design perspective, is absolutely key to its success.

Comment on A comment on Tim Gowers’s blog by Peter Krautzberger

Peter Krautzberger — Fri, 03 Feb 2012 16:26:10 +0000

Mike, unfortunately, I think your observations are on point.

The reason why mathematicians are bad speakers actually has a simple answer — we receive no training and there’s no active discussion in the community on how to present well (just check out the Journal of Number Theory’s youtube channel…). There’s also no feedback for speaker, i.e., nobody tells you what went badly in your talk or how to improve, even if you ask arond. Giving productive feedback is, of course, also a skill to be trained.

The trouble is that most people think you cannot learn this — which is plain wrong. To be a good speaker can be trained, just like anything else.

The observation regarding the job market is, I think, also correct, but for entirely different reasons. This is quite simply a (poor) decision of our community to actively favor the wrong set of skills (or rather, focus on one skill exclusively “paper writing”).

If you look around, the truly great mathematicians (Tim Gowers being a formidable example) are very good teachers and speakers. I think it is a misconception in our community that this skill is not important to further mathematical research; a misconception most likely originating from the fact that such skills could not be documented/evaluated until very, very recently.

Comment on A comment on Tim Gowers’s blog by Peter Krautzberger

Peter Krautzberger — Fri, 03 Feb 2012 16:06:01 +0000

Joel, I mean it as both criticism and praise. Praise, obviously, for giving fantastic answers.

But it is also criticism. I see mathoverflow activity as something that adds to the research reputation of, in particular, untenured faculty. In fact, I sincerely hope that activities such as mathoverflow will soon find their rightful place in the research section of people’s CV. (And I’m baffled that not even Anton is getting the recognition he deserves — mathoverflow has probably created more impact than an entire year of Inventiones papers.)

The problem I see is that nobody else has a chance of building a reputation by *answering* questions because of power users such as yourself.

This is not a specific logic/set theory problem, since I’ve seen colleagues in other fields make the same observation. It’s probably easier to spot in a small field such as set theory.

I think it would help if there was a culture of less hurry on MO. Why not leave a question unanswered for a day or two even if you have an answer? The question is hardly life-and-death… But younger people (such as graduate students, postdocs) might get a chance to take a first step into the community.

I’m not saying that there is a solution — there might very well not be. In fact, when I say that MO deserves to be in the research section of a CV, I don’t mean that it should be the only thing there. We need many, many more platforms to allow researchers to contribute in other forms than old-fashioned papers.

I should add, this isn’t about me. There are other reasons why I don’t participate on MO anymore (though I like reading it).

Comment on An incompleteness theorem for βn models by Carl Mummert

Carl Mummert — Fri, 03 Feb 2012 14:33:36 +0000

I’m glad you pointed out the set theoretic side. I am usually only thinking about second-order arithmetic, but many of these results work equally well there and in set theory (which would be a nice subject for another post).

The result of your second paragraph is essentially Friedman’s result, although that result is in the context of second-order arithmetic, using countable coded $omega$-models. It’s Theorem VIII.5.6 in Simpson’s SOSOA. There is an interesting twist: the base theory has to include ACA₀ in order for the result to go through. Every countable $omega$-model of WKL₀ does contain a countable coded (omega)-model satisfying WKL₀, notwithstanding the incompleteness theorem. The difference is that ACA₀ is able to form the satisfaction predicate for a countable coded $omega$ model.

By the way, I have enabled a comment editing plugin on this blog – please try it out. There is a long delay when it starts, in my browser at least, and I noticed that it removes backslashes from comments when they are saved, so I will need to look into that.

Comment on A comment on Tim Gowers’s blog by Joel David Hamkins

Joel David Hamkins — Thu, 02 Feb 2012 22:11:47 +0000

Peter, I’m unsure whether to take your remark in the second comment as criticism or as joking praise… I would think that vigorous participation on MO by a logic user such as myself or Andreas Blass (or Emil, Carl, Francois, etc. etc.) would be seen as fundamentally encouraging of a vigorous logic activity there.

Comment on A comment on Tim Gowers’s blog by Micheal Pawliuk

Micheal Pawliuk — Thu, 02 Feb 2012 00:09:00 +0000

I get the impression from your posts Peter that all mathematicians are terrible presenters and writers.

The problem I suppose is that good presentation skills are (unfortunately) often orthogonal to good research skills. I have even heard that some people with particularly strong teaching backgrounds actually have a harder time getting an academic job. (sigh)

Comment on A comment on Tim Gowers’s blog by Peter Krautzberger

Peter Krautzberger — Mon, 30 Jan 2012 00:52:00 +0000

Thank you, Carol. I’m sure there’s a tail of awful contributions. That’s why I’m speaking of an attention economy instead of a production economy. We need to share what we think about each others work. On the other hand, I think if we had activities other than writing papers which we could still put in the research section of our CV (such as open reviewing, helping other people make progress in their work), then the pressure to write as many papers as possible could be reduced, leading to fewer awful papers. But nobody can do that if we do not experiment more in this respect.

By the way, I’ve sometimes seen the argument “but then the cranks take over” which I find hilarious since I’ve yet to meet a mathematician who cannot spot a crank from a mile away.

Comment on Testing… by Dana Ernst

Dana Ernst — Tue, 24 Jan 2012 23:52:08 +0000

Yada yada. Blah blah blah.

Comment on Why write papers? by Peter Krautzberger

Peter Krautzberger — Thu, 29 Dec 2011 21:55:44 +0000

No, there wasn’t anything missing. It was just poor writing on my part.

I meant “Reinventing discovery”, Michael Nielsen’s book about how the web will change the way research is done. I just finished part 1, hoping to post a review at some point. It’s an interesting read so far, partially well known to me (e.g. polymath), partially surprising, partially confusing.

What I meant with “Shelah’s model” is the idea of massive collaboration, spreading and helping as much as possible. With “Wiles’s model” I meant the idea of hiding preliminary results for years so that nobody can scoop. We need a different attitude towards making influences transparent (and the tools to give credit) so that we can find ways to modularize research much further.

Comment on Why write papers? by Micheal

Micheal — Tue, 20 Dec 2011 23:11:35 +0000

Great post! This reminds me a lot of how Stevo Todorcevic writes his papers. As few details as possible while still allowing the (dedicated) reader to follow along.

Comment on Why write papers? by sam

sam — Mon, 19 Dec 2011 01:22:24 +0000

Andreas,
That last remark makes me think: we should be writing papers to ourselves ten years ago!

Comment on Why write papers? by Andreas Blass

Andreas Blass — Sun, 18 Dec 2011 01:16:21 +0000

Although the intended readers of my papers are other mathematicians, not myself, I agree with the advice to put in enough information so that the paper will make sense to me 10 years from now. It’s embarrassing to write “clearly,” “obviously,” etc., and later be unable to remember why these things were clear, obvious, etc. I hope that, by making things clear to my future self, who won’t remember my present network of ideas, I also make things clear to my present readers who (I expect) have not yet explored that network.

Comment on Why write papers? by François

François — Sat, 17 Dec 2011 15:44:13 +0000

Peter, it seems there are some missing parts in your second paragraph. I’m assuming you don’t mean Michael’s textbook on quantum computing. What are these models you’re talking about? I don’t see any sense in which Shelah, Wiles, Grothendiek, and Perelman would constitute realistic models for run-of-the-mill mathematicians.

Comment on Why write papers? by François

François — Sat, 17 Dec 2011 15:36:21 +0000

I’m glad Joel confirmed this. The more I think about this, the more I realize how great advice this is. My next paper will be tacitly addressed to my future self. That’s much less awkward than writing for an imaginary reader who may never materialize…

Comment on Why write papers? by Peter Krautzberger

Peter Krautzberger — Fri, 16 Dec 2011 21:43:16 +0000

I like this statement. For me it also makes a very good point why we should separate scientific progress from publication. In fact, it reminds me of Claire Mathieu’s post reacting to the publishing debate spawned by Gowers: publish other people’s work. It’s the old “communicated by” idea taken seriously, we should never publish our own results — or, a little softer, we should never publish alone.

Reading Michael Nielsen’s book I’m starting to think that Shelah’s model is actually the future (though not as papers). We cannot afford the Wiles, the Grothendieck or the Perelman model of research.

Comment on Czech Winter School by sam

sam — Fri, 16 Dec 2011 16:43:19 +0000

Cheers! I’ll post it on the set theory talks blog.

Comment on Why write papers? by Joel David Hamkins

Joel David Hamkins — Fri, 16 Dec 2011 14:30:25 +0000

I like to read my own papers from 10 years ago…sometimes I learn something useful.

Comment on Why write papers? by sam

sam — Thu, 15 Dec 2011 20:49:15 +0000

Haha, that’s a pretty self-centric point of view. We write papers so that ten years later we may remember them?

Comment on Czech Winter School by KT

KT — Tue, 13 Dec 2011 17:09:30 +0000

Yes I was there last year – it was a really terrific atmosphere. In order not to be thought a wuss, I chose the extreme branch of the “experts’ hike” for the excursion.We hiked some of the end in pitch darkness and returned so late as to almost miss dinner, but it was a great (and exhausting) experience.

Would be great if you could make it, but it is quite the ways from Michigan. :(

Comment on Czech Winter School by Peter Krautzberger

Peter Krautzberger — Mon, 12 Dec 2011 19:52:37 +0000

I love the Czech winterschool! I don’t know if I can make it myself though :(

Be sure to go on a hike every day during the lunch break! It’s such a lovely area and the organizers know it inside out.

Comment on XKCD on the Axiom of Choice… by Peter Krautzberger

Peter Krautzberger — Sun, 27 Nov 2011 19:56:58 +0000

Ha! I had missed the alternate text. Awesome!

Comment on Santa Exists! by François

François — Mon, 21 Nov 2011 13:56:10 +0000

This is explained later on in the post. The forward implication is $S \in S \lthen (S \in S \lthen \text{Santa exists})$ which gives $S \in S \lthen \text{Santa exists}$ by contraction.

Of course, this is an invalid proof in linear logic, which does not admit the contraction rule, but then I would have to be more precise about what implication means…

Comment on Santa Exists! by Buschi Sergio

Buschi Sergio — Mon, 21 Nov 2011 13:40:53 +0000

This is simply (seem to me) a wrong proof (no a different version of Russel paradox).

When you write: “The forward implication of (∗) gives that…” forget to mention $if\ S\in S$

Comment on Uncountable perfect graphs by François

François — Thu, 17 Nov 2011 02:48:46 +0000

David Eppstein just wrote a post where he generalizes chordal graphs to the infinite case in a manner much different than I do. It is interesting to compare the two…

Comment on Oracle forcing Part I by KT

KT — Mon, 31 Oct 2011 10:56:09 +0000

James: flattery will get you far, but sending the promised correspondence would get you even farther. ;)

Peter: thanks for noticing the missing words (which also happened as some of the markdown turned html simply disappeared) – I will put those in anon.

Part II is forthcoming as public – there are 1 or 2 proofs missing that I thought I had somewhere….

Comment on On computing complex square roots by François

François — Sun, 30 Oct 2011 18:21:44 +0000

Thanks Peter! This is an unusual post because the results are kind of interesting but not enough to publish, so Booles’ Rings is just the right place!

(The result of Theorem A did get a mention in my paper Classical consequences of continuous choice principles from intuitionistic analysis.)

Comment on On computing complex square roots by Peter Krautzberger

Peter Krautzberger — Sun, 30 Oct 2011 15:03:09 +0000

When I caught up on last week’s Booles’ Rings activity, I read this and wanted to write something — but couldn’t quite formulate, why I like this post. I didn’t want to just leave a spammish “Great post!” comment and just now realized what’s so great about your post: it’s a perfect example for the range of content I hoped for when Sam and I started to think about this project.

Thank you, François!

Comment on Oracle forcing Part I by krautzberger

krautzberger — Sat, 29 Oct 2011 22:16:09 +0000

Those markdown troubles are really odd. It sounds more like a bug somewhere to me and I’d love to hunt it down — I never had the experience you describe :(

I also wondered if the first definition is missing some text (like “oracle”). Also, the first sentence after M-cc is defined seems to be missing some text.

Comment on Oracle forcing Part I by James Cummings

James Cummings — Fri, 28 Oct 2011 23:22:00 +0000

Aren’t you in fact the young queen of set theory? This is much the most interesting
blog post I have read this week, can I have the sooper sekrit password to
the sequel or is that reserved for younger set theorists than me?

Comment on Oracle forcing Part I by KT

KT — Thu, 27 Oct 2011 13:14:06 +0000

Ok, I am going to stop using markdown now in wordpress. All and even sometime quotes are changed *in the html editor* (!) to their respective ASCII codes as soon as I press update. This didn’t happen last summer! The visual editor is disabled, etc.. I cannot use blockquotes as these are changed before the markdown processor can get to it. Back to good old html for now…

Comment on A game-theoretic proof of Fraïssé’s Theorem by François

François — Mon, 24 Oct 2011 11:59:55 +0000

Thanks Andreas! You are completely correct, I had proved the forward implication and then its contrapositive. I have deleted the redundant proof.

I wonder if there is a game-theoretic proof of the backward implication…

Comment on A game-theoretic proof of Fraïssé’s Theorem by Andreas Blass

Andreas Blass — Sun, 23 Oct 2011 22:00:48 +0000

It seems to me that you’ve essentially proved one direction of the theorem twice and ignored the other. You’ve shown that if Duplicator has a winning strategy then A and B are n-quantifier equivalent, and that if A and B are not n-quantifier equivalent then Spoiler has a winning strategy. But the theorem also claims that if Duplicator has no winning strategy (equivalently since the game is finite: if Spoiler has a winning strategy) then A and B are not n-quantifier equivalent. As far as I can see, your argument doesn’t establish this. (If I remember correctly, this is the part of the proof where function symbols make life unpleasant. One has to count applications of function symbols as if they were quantifiers, to ensure that there are only finitely many inequivalent formulas of any fixed quantifier depth.)

Comment on Santa Exists! by François

François — Sun, 23 Oct 2011 02:13:20 +0000

Right! The same argument works replacing “Santa exists” by $\bot$…

I was thinking of a different argument (which I think is how the paradox is usually described): Since $R \in R \lthen (R \notin R \land R \in R)$ and $R \notin R \lthen (R \notin R \land R \in R)$, from $R \in R \lor R \notin R$ we conclude $R \notin R \land R \in R$.

I guess the real difference is that Curry’s version doesn’t require negation (or $\bot$).

Comment on The enumeration of permutations sortable by pop stacks in parallel by krautzberger

krautzberger — Sat, 01 Oct 2011 16:58:54 +0000

Sorry about that. I didn’t check your reaction carefully enough.

Comment on The enumeration of permutations sortable by pop stacks in parallel by Vince Vatter

Vince Vatter — Sat, 01 Oct 2011 05:59:07 +0000

I don’t see what the problem is. I deleted the link, so this spam wouldn’t hurt anyone. If you don’t reverse your decision, please explain.

Comment on The enumeration of permutations sortable by pop stacks in parallel by krautzberger

krautzberger — Sat, 01 Oct 2011 05:09:20 +0000

Vince, I have taken the liberty of unapproving the spam comment. Comment spam is about search engine optimization, so approving comments helps the spammers to increase their page rank.

Comment on The enumeration of permutations sortable by pop stacks in parallel by Vince Vatter

Vince Vatter — Fri, 30 Sep 2011 18:51:00 +0000

Glad to be of service, spammer.

Comment on The enumeration of permutations sortable by pop stacks in parallel by Sac Hermes Birkin

Sac Hermes Birkin — Fri, 30 Sep 2011 18:44:23 +0000

We’re grateful which discovered this blog, just the correct information and facts which i wanted!

Comment on Hello subjects and fans! by Joel David Hamkins

Joel David Hamkins — Fri, 23 Sep 2011 02:15:41 +0000

But Katie, I would expect that everyone will want to visit the Queen of young set theory!

Comment on Of pancakes, mice and men by krautzberger

krautzberger — Thu, 11 Aug 2011 22:24:57 +0000

I see — that might have helped

Comment on How to make slides from handwritten notes using potrace by krautzberger

krautzberger — Thu, 11 Aug 2011 22:19:41 +0000

Yeah, Sam and I’ll have to look into that… TeX and Zip should definitely be included.

Comment on Of pancakes, mice and men by Vince Vatter

Vince Vatter — Thu, 11 Aug 2011 13:43:22 +0000

Thanks Peter. Plus is always happy with unsolicited submissions, but also, Colva is good friends with their editor.

Comment on Of pancakes, mice and men by krautzberger

krautzberger — Thu, 11 Aug 2011 13:32:08 +0000

That’s very cool. How did you get in touch with them?

Comment on Maximal independent sets in graphs with at most $r$ cycles by Maximal and maximum independent sets in graphs with at most $r$ cycles | Vince Vatter

Wed, 10 Aug 2011 21:48:09 +0000

[...] navigation ← Previous Next [...]

Comment on Hello subjects and fans! by KT

KT — Thu, 04 Aug 2011 11:17:45 +0000

Haha no way, this might be the only non-me comment I ever get. ;)

Comment on Hello subjects and fans! by krautzberger

krautzberger — Sun, 31 Jul 2011 17:08:22 +0000

Love the subtitle already. Feel free to delete this :)