Timeline for What is Realistic Mathematics?
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
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| Sep 20, 2020 at 22:27 | comment | added | Timothy Chow | While thinking about this today, I persuaded myself that if one could define a notion of a "tame" function space and use DC to prove Hahn-Banach, Krein-Milman, etc. for tame function spaces, then analysts might consider switching over. Unfortunately, it seems that even $\ell^\infty$ is not tame. I think it will be a hard sell to get analysts to accept $(\ell^\infty)^* = \ell^1$ since they already "know" that it's false. If I'm right, it's sort of a shame because otherwise $(\ell^\infty)^* = \ell^1$ could be viewed as a feature rather than a bug! | |
| Dec 30, 2017 at 22:24 | comment | added | Andreas Blass | @ThomasBenjamin For some reason I didn't see your latest comments until now. I would expect that, in the result you quoted from Simpson's book, adding "unitary" to the hypothesis won't affect the reverse mathematical strength. Indeed, I would expect that existence of prime ideals in all (nontrivial) countable Boolean algebras is equivalent (over RCA${}_0$) to WKL${}_0$ (and that this should be easy to prove). | |
| Dec 10, 2017 at 10:25 | comment | added | Thomas Benjamin | @AndreasBlass: The phrase, "...is some weakening of $WKL_{0}$ equivalent to the Boolean prime ideal theorem" should read "...is some weakening of $WKL_{0}$ equivalent to the Boolean prime ideal theorem for countable Boolean algebras" | |
| Dec 10, 2017 at 10:03 | comment | added | Thomas Benjamin | @AndreasBlass: In "The Godel Hierarchy and Reverse Mathematics". Simpson writes: "$WKL_{0}$ is equivalent over $RCA_{0}$ to...Every countable commutative ring has a prime ideal". Since the subclass of countable commutative unitary rings is obviously contained in the class of countable commutative rings, is some weakening of $WKL_{0}$ equivalent to the Boolean prime ideal theorem which in turn is equivalent to the Tychonoff theorem for countable, second-countable compact Hausdorf spaces? Is that what you were referring to in your earlier comment to me? | |
| Dec 8, 2017 at 16:39 | comment | added | Andreas Blass | @ThomasBenjamin As far as I know, the Boolean prime ideal theorem is equivalent to the existence of prime ideals in commutative unitary rings. It is strictly weaker than the axiom of choice, which is equivalent to the existence of maximal ideals in commutative unitary rings. | |
| Dec 8, 2017 at 8:39 | comment | added | Thomas Benjamin | (cont.) David Corwin's Mathoverflow question, "Result that follows from $ZFC$ and not $ZF$ but are strictly weaker than choice", Francois Dorais' arXiv preprint [arXiv:1110.6555v1 [math.LO] 29 Oct 2011], "Reverse Mathematics of Compact Countable Second-Countable Spaces" [Section 8], and Stephen Simpson's November 24, 2009 draft of "The Godel Hierarchy and Reverse Mathematics [where $ACA_{0}$ is stated to be equivalent to "Every countable commutative ring has a maximal ideal" over $RCA_{0}$])? | |
| Dec 8, 2017 at 8:23 | comment | added | Thomas Benjamin | @AndreasBlass: Wouldn't the exact reverse mathematics strength of the Tychonoff theorem for (say) countable, second countable compact Hausdorff spaces be $ACA_{0}$ since the Tychonoff theorem, when restricted to Hausdorff spaces, is equivalent to the Boolean prime ideal theorem, which in turn is equivalent to the existence of maximal ideals in commutative unitary rings (these commutative unitary rings, of course, would be restricted to the countable case in order for the existence of maximal ideals for the countable case to be equivalent to $ACA_{0}$--see Stefan Geschke's answer to | |
| Dec 1, 2017 at 23:45 | comment | added | Andreas Blass | @ThomasBenjamin I have no good idea about an answer to your latest, question, partly because of my ignorance of most of the current study of physical reality, and partly because of the vagueness of the terms in the question (as described, for example, in the answer from Gowers). | |
| Dec 1, 2017 at 21:49 | comment | added | Thomas Benjamin | @AndreasBlass: How much of the Tychonoff theorem is actually necessary and sufficient for the current study of physical reality, anyway? | |
| Dec 1, 2017 at 15:56 | comment | added | Andreas Blass | @ThomasBenjamin Although your last comment was addressed to another Andreas (and the previous comment might have been also), let me comment that second-order arithmetic can't even formulate the general Tychonoff theorem, even for Hausdorff spaces. The special case of compact metric spaces can be formulated, and its exact reverse-mathematics strength is undoubtedly in Simpson's book. (I'd expect it to be WKL${}_0$.) | |
| Dec 1, 2017 at 14:51 | comment | added | Thomas Benjamin | @AndreasThom: Why was your last question not, "Is there any mathematical application to the study of physical reality which is not captured by some subsystem of second-order arithmetic?", since (if I understand correctly) all mathematics sufficient for the current study of physical reality can be proved in such subsystems? | |
| Dec 1, 2017 at 12:10 | comment | added | Thomas Benjamin | Question: Which of the subsystems of second-order arithmetic usually studied in Reverse Mathematics proves the Tychonoff (Product?) Theorem (if any)? I myself would hold (whatever that's worth) that a viable candidate for "Realistic Mathematics" would be whatever susbsystem of second-order arithmetic is necessary and sufficient to prove all the theorems needed to 'do' physics at its current state of development (I realize that 'do' possibly hopelessly vague). | |
| Nov 20, 2010 at 19:51 | vote | accept | Andreas Thom | ||
| Sep 3, 2010 at 6:30 | vote | accept | Andreas Thom | ||
| Nov 20, 2010 at 19:51 | |||||
| Aug 8, 2010 at 14:37 | comment | added | Mariano Suárez-Álvarez | "One of the endearing things about mathematicians is the extent to which they will go to avoid doing any real work." :P | |
| Aug 7, 2010 at 21:33 | comment | added | Andreas Blass | Yes, Solovay's theorem really needs only (the consistency of) an inaccessible cardinal. As for Hahn-Banach, it is, as Gerald Edgar says, strictly weaker than AC, but it's nevertheless false in the Solovay model. I think what I said about Tychonoff's theorem may apply here as well: The particular uses of Hahn-Banach that are needed in "realistic mathematics" might be available even though the general theorem isn't. And people might not want to keep track of which uses are available and which aren't. | |
| Aug 7, 2010 at 20:41 | comment | added | Gerald Edgar | Hahn-Banach is strictly weaker than AC. | |
| Aug 7, 2010 at 20:32 | comment | added | Johannes Hahn | Is it really "ZFC+there is an inaccesible cardinal" in Solovays theorem? I always thought the large cardinal axiom that has to be used is a measurable cardinal... Also: Functional analysis would be pretty ugly without the Hahn-Banach-theorem (I think that it is equivalent to the AC, but I'm not sure) and with ugly Functional analysis, the theory of PDEs would become (even more) ugly. This could be another reason why analysts don't prefer ZF+DC+"P(IR)=Lebesgue sets" over ZFC. | |
| Aug 7, 2010 at 20:22 | history | answered | Andreas Blass | CC BY-SA 2.5 |