Timeline for What if Current Foundations of Mathematics are Inconsistent? [closed]
Current License: CC BY-SA 3.0
51 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Sep 10, 2013 at 13:01 | review | Reopen votes | |||
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| Sep 9, 2013 at 12:58 | review | Reopen votes | |||
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| Nov 29, 2012 at 6:33 | comment | added | Gerry Myerson | Interesting that this question was closed as "no longer relevant". Does that mean it has been proved that the current foundations of mathematics are consistent? [just kidding - I know the closure options are very limited] | |
| Dec 25, 2011 at 21:24 | history | closed |
Felipe Voloch Andrés E. Caicedo user6976 user9072 Simon Thomas |
no longer relevant | |
| Dec 25, 2011 at 1:24 | answer | added | tab | timeline score: 9 | |
| Dec 24, 2011 at 22:26 | answer | added | none | timeline score: 5 | |
| Sep 13, 2011 at 1:58 | comment | added | Thomas Riepe | Dear Pierre-Yves, thanks for correcting that! I wonder how that happened. | |
| Sep 12, 2011 at 14:41 | comment | added | Pierre-Yves Gaillard | Dear @Thomas: Is this golem.ph.utexas.edu/category/2011/09/voevodsky_on_fom.html the link you wanted to give? | |
| Sep 2, 2011 at 3:35 | answer | added | Taylor Dupuy | timeline score: 24 | |
| Sep 1, 2011 at 14:44 | answer | added | Mathieu Vidal | timeline score: 7 | |
| Aug 28, 2011 at 9:22 | history | edited | Pierre-Yves Gaillard | CC BY-SA 3.0 |
removed a sentence
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| May 19, 2011 at 7:21 | history | edited | Thomas Riepe | CC BY-SA 3.0 |
added 154 characters in body
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| May 18, 2011 at 22:51 | answer | added | Monroe Eskew | timeline score: 6 | |
| Feb 20, 2011 at 10:33 | comment | added | David Roberts♦ | I have added a vote to close, for being no longer relevant. I feel this question is attracting more answers that are not 'adding value', as they say. Discussion at meta if necessary. | |
| Feb 20, 2011 at 8:52 | answer | added | Anixx | timeline score: -20 | |
| Feb 19, 2011 at 22:09 | answer | added | Sergey Melikhov | timeline score: 6 | |
| Feb 8, 2011 at 0:14 | answer | added | Paul | timeline score: 6 | |
| Nov 11, 2010 at 23:13 | comment | added | Carl Mummert | @Steven Gubkin: ZFC minus infinity plus its negation can be interpreted into Peano arithmetic, in the formal logical sense of interpreted. Since the latter has been proven consistent (in multiple independent ways, in fact), the former must also be consistent. | |
| Nov 6, 2010 at 0:59 | comment | added | Steven Gubkin | ZFC without the axiom of infinity is known to be consistent? | |
| Oct 6, 2010 at 23:33 | answer | added | Daniel Tausk | timeline score: 33 | |
| Oct 6, 2010 at 17:37 | comment | added | Carl Mummert | Given that ZFC without the axiom of infinity is known to be consistent, it would make as much sense to say that the axiom of infinity would be the first to go, since that will certainly solve the problem :) | |
| Oct 6, 2010 at 15:57 | history | edited | Pierre-Yves Gaillard | CC BY-SA 2.5 |
EDIT clearly indicated.
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| Oct 6, 2010 at 4:08 | vote | accept | Pierre-Yves Gaillard | ||
| Oct 4, 2010 at 23:25 | comment | added | Timothy Chow | @Kevin: Replacement will be chopped only if doing so fixes whatever inconsistency happens to shows up. It doesn't make much sense to speculate on what the fix will be when it ain't broke yet. | |
| Oct 4, 2010 at 22:10 | comment | added | Peter LeFanu Lumsdaine | Actually, my error: Todd Trimble points out to me off-list that you can indeed construct the free monoid on a set $X$ without replacement, with a bit more care (as eg a subset of the set of partial functions from $\mathbb{N}$ to $X$). What I had in mind is that without replacement, one can’t always make constructions of the form $\bigsqcup_n F^n(X)$, (or $\bigprod$, $\bigcup$, $\varinjlim$ etc.) where $F$ is an arbitrary function on sets. I'm fairly sure I've seen at some point a simple, mathematically well-known example which isn't salvageable in the way “free monoid” was; but now I forget… | |
| Oct 4, 2010 at 17:03 | comment | added | Peter LeFanu Lumsdaine | @Kevin Buzzard: as the (non-joking) background in Todd's link points out, while Bourbaki don't include Replacement in the same form that ZF has it, they do include an essentially equivalent axiom: that the union of any set-indexed family of sets forms a set. ZF with replacement dropped (aka Zermelo Set Theory) is not powerful enough for much mathematics at all: for instance, iirc, it can't construct the free monoid on an arbitrary set, nor hence eg polynomial algebras in noncommuting variables. | |
| Oct 4, 2010 at 15:07 | answer | added | Timothy Chow | timeline score: 96 | |
| Oct 4, 2010 at 13:27 | comment | added | Todd Trimble | @Andras: Joking aside, there is a serious point behind Taylor's post: that the replacement axiom is indeed enormously powerful, even if not strictly speaking necessary for most of core mathematics (which can be developed within a universe of sets whose rank is less than $\omega+\omega$, where replacement fails). In fact, Taylor's book Practical Foundations of Mathematics culminates in a discussion of how to formulate the replacement axiom in the context of dependent type theory. | |
| Oct 4, 2010 at 12:45 | comment | added | András Salamon | @Todd: not fair, you are only allowed to resurrect those jokes once a year! | |
| Oct 4, 2010 at 0:23 | comment | added | Todd Trimble | Um, yes, I know all this, Andras?! Was joking around myself?! | |
| Oct 4, 2010 at 0:09 | comment | added | András Salamon | @Todd: Lawvere points out a few messages in that this was a (slightly irresponsible) April Fool. The final comment of the thread is appropriate: "I suppose that it is possible that somebody, browsing at random, *might* find it in the CATEGORIES archives, in years to come, and not read ahead to find out what the world had had to say about this discovery back in the mad, exciting days of the late C20." Note that this does not detract from the larger issue that ZF may well be inconsistent. | |
| Oct 3, 2010 at 23:10 | answer | added | Felipe Voloch | timeline score: 72 | |
| Oct 3, 2010 at 21:30 | comment | added | Todd Trimble | Kevin Buzzard: yup. See mta.ca/~cat-dist/catlist/1999/zf-010499. | |
| Oct 3, 2010 at 20:54 | comment | added | Kevin Buzzard | Bourbaki dropped the axiom(-scheme) of replacement in their development of mathemetics, so they don't, I think, have enough mathematics to build the ordinals. However their work seems to indicate that they had enough to do an awful lot of mathematics (probably all of the mathematics I've ever done and will do won't need replacement). My guess is that if ZFC is inconsistent then replacement will be the first axiom for the chop. | |
| Oct 3, 2010 at 19:53 | history | edited | Kaveh |
retag
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| Oct 3, 2010 at 19:43 | answer | added | Todd Trimble | timeline score: 12 | |
| Oct 3, 2010 at 19:09 | answer | added | Pietro Majer | timeline score: 22 | |
| Oct 3, 2010 at 18:31 | comment | added | The Mathemagician | I think he needs to be a little more specific about what he means by this question.WHICH foundations? ZFC set theory? Categorical foundations a la Lawvere? What? | |
| Oct 3, 2010 at 16:16 | answer | added | Rob Simmons | timeline score: 24 | |
| Oct 3, 2010 at 13:50 | answer | added | Jim Conant | timeline score: 28 | |
| Oct 3, 2010 at 13:42 | answer | added | Dick Palais | timeline score: 83 | |
| Oct 3, 2010 at 13:36 | comment | added | muad | I don't understand his objection to Gentzen's proof at 29:00. Why would someone be skeptical about well foundedness of $\epsilon_0$? | |
| Oct 3, 2010 at 12:37 | comment | added | Pierre-Yves Gaillard | @Gerry Myerson - Sorry about the confusion. The question is "What if the current foundations of Mathematics are inconsistent?" A subsidiary question is "Has this kind of opinion been expressed before?"... At least that's how I see it. You're welcome to answer any of the questions, to ask others, to edit the question... - An implicit question is "What do you think of Vladimir Voevodsky's talk?" | |
| Oct 3, 2010 at 12:33 | comment | added | Michael Bächtold | +1 just for pointing out this talk. | |
| Oct 3, 2010 at 12:15 | comment | added | Gerry Myerson | I'm confused. Is the question, "What if the current foundations of Mathematics are inconsistent?" Or is the question, "Has this kind of opinion been expressed before?" | |
| Oct 3, 2010 at 12:11 | comment | added | José Figueroa-O'Farrill | Does anyone know what Voevodsky means by the "current troubles between mathematics and physics" in his talk?! | |
| Oct 3, 2010 at 11:45 | comment | added | Todd Trimble | What opinion? That ZFC or even Peano Arithmetic is in fact inconsistent? The trouble is that this sort of thing is not too likely to be put in print by a reputable mathematician, even if he expresses his (perhaps occasional) doubts in private. But yeah, I can think of a few reputable mathematicians who sometimes express sentiments ("opinion" is maybe too strong a word) along these lines. Backing that up is another story altogether. | |
| Oct 3, 2010 at 11:20 | comment | added | Piero D'Ancona | Then the unreasonable effectiveness of mathematics would become slightly more unreasonable | |
| Oct 3, 2010 at 10:24 | history | edited | Pierre-Yves Gaillard | CC BY-SA 2.5 |
fixed grammar
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| Oct 3, 2010 at 10:17 | history | asked | Pierre-Yves Gaillard | CC BY-SA 2.5 |