Adam Epstein
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Registered User
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Math Under Toad
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Mar 25 |
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Consistency of the concept of the collection of all collection That sounds nice, I'm looking forward to it. |
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Mar 24 |
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Consistency of the concept of the collection of all collection When you specialize Cantor's proof (that no function from a set to its power set is surjective) to the identity - from the set of all sets to its power set, that is itself - the witness to nonsurjectivity is the Russell entity {$x: x\notin x$} which, not being a set, contradicts nothing. |
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Mar 24 |
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Consistency of the concept of the collection of all collection deleted 3 characters in body; added 5 characters in body |
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Mar 24 |
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Can one escape from the “mirror-image” of Russell’s Paradox? In ZF-Foundation you can have as many Quine atoms (sets which are their ownn unique element) as you like. For example, see Chapter IIIA of Felgner's "Models of ZF Set Theory". |
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Mar 24 |
answered | Consistency of the concept of the collection of all collection |
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Mar 24 |
answered | Self-containing structures |
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Mar 24 |
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Self-containing structures +1 for the Gromov-Hausdorff example. |
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Mar 23 |
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finite dimensional real division algebras Thanks, I'll have a look! |
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Mar 20 |
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Partial linearization near a hyperbolic fixed point--Classical scattering Even in a finite diimensional setting there would be the issue of possible resonances among the eigenvalues, and little is known about those numbers |
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Mar 20 |
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Partial linearization near a hyperbolic fixed point--Classical scattering I'd look at Section 6. For example, Theorem 6.3 gives a hyperbolic splitting of the tangent space. Subsequent results discuss stable and unstable manifolds. If you want an actual reduction to normal form, this does not seem to be considered, and on reflection I am not aware that such a result has been formally claimed in a complex analytic setting. |
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Mar 20 |
answered | Partial linearization near a hyperbolic fixed point--Classical scattering |
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Mar 19 |
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On the category of virtual species For clarification, perhaps someone could say something about this matter is (or is not) related to considerations in Blass's "Seven Trees in One" and Schanuel's "Negative sets have Euler Charactieristic and dimension". |
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Mar 14 |
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Bijective-equivalent collections of proper classes in set theory Dear Gerard Lang, No worries - it was clear that you didn't mean to separately count the result of precomposing by some transposition :) |
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Mar 14 |
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Bijective-equivalent collections of proper classes in set theory There surely won't be a uique bijection between POrd and a proper subclass, but maybe you meant that there is a preferred one in which they are listed in order. |
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Mar 13 |
answered | Topology, the board game |
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Mar 6 |
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Where in ordinary math do we need unbounded separation and replacement? added 238 characters in body |
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Mar 5 |
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Where in ordinary math do we need unbounded separation and replacement? added 93 characters in body |
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Mar 5 |
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Where in ordinary math do we need unbounded separation and replacement? added 683 characters in body |
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Mar 5 |
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Where in ordinary math do we need unbounded separation and replacement? added 127 characters in body |
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Mar 5 |
answered | Where in ordinary math do we need unbounded separation and replacement? |
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Mar 4 |
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Is there a constructive proof of Cantor–Bernstein–Schroeder theorem ? The inclusion $g$ seems the more obvious to me. For $f$, I imagine you would start with a Turing machine which loops on input 0, and then for each $n$ attach a subroutine which effectively lets it sit idle that many steps before looping? |
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Feb 26 |
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The point of view of semicats in functional analysis I gather from the second example that semicats needn't have identity morphisms. Perhaps you could append the relevant definition? |
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Feb 24 |
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When does $A^A=2^A$ without the axiom of choice? Which for some reason is excluded from the initial statement :) |
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Feb 23 |
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Undecidability and holomorphic functions (Reference request) @Qfwfq I noticed the reference to Krantz and Di Biase only just now. Nothing plausble came up for Di Biase on MathSciNet, and Krantz has uncountably many publications. Do you have more precise coordinates? |
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Feb 23 |
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Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ? E.g. "The Kunen inconsistency result, the assertion that there is no nontrivial elementary embedding j:V→V, becomes trivial when one treats all classes as definable. One can easily rule out all such definable j, if one only cares to consider the case in which j is first-order definable with parameters, and one needs neither the axiom of choice nor any infinite combinatorics to do it." from Joel David Hamkins's answer to mathoverflow.net/questions/71765/…. On the other hand, first-order definability is surely crucial for Reflection Principles. |
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Feb 23 |
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Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ? My sympathies, Gerard. I myself once lost 120 points overnight - upvotes to my most popular question mathoverflow.net/questions/120598/…due to my own overzealous editing which inadvertently rendered it community wiki. I'll admit to having been more annoyed than I might have expected :( And yes, as Asaf points out, you can always simulate atoms by Quine atoms |
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Feb 23 |
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Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ? In principle there are finer points somewhat suppressed in ZFC. For example, some arguments might use classes in a way that could be formalized in NBG, without any reference to structure due to definability, whereas other arguments might well require proofs by induction on quantifier complexity. Maybe some expert could weigh in about this, at least whether it is worth formulating as its own quesstion. |
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Feb 23 |
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Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ? I don't see that you are using that $x$ is well-founded, just that it is bijective to an ordinal. Am I missing something? I was also starting to wonder about the situation when Foundation+Choice is replaced by "every set is equinumerous to a well-founded set", but above you point to a less exotic failure already. |
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Feb 23 |
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Pullback measures Upvote for the first sentence. |
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Feb 23 |
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Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ? In a funny way this is different from what I was expecting. Here it seems that the use of Foundation is to guarantee that $A$ is contained in the cumulative hierarchy, rather than that $x$ is. Might there be a way to put actual meat on this observation, in some context or another, say something about Choice vs Global Choice in the absence of Foundation? |
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Feb 23 |
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Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ? Thanks, Asaf. But these are in Jech's The Axiom of Choice rather than Set Theory |
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Feb 23 |
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Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ? Upvote for a compelling statement that requires Replacement. |
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Feb 23 |
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Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ? Interesting. Would it be correct to infer that the overall strategy is to prove, presumably by transfinite induction, that it is possible to inject any $V_\alpha$? |
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Feb 20 |
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Undecidability and holomorphic functions (Reference request) Emil - I actually tried that first, and it also came out wrong, at least in the previewer. Quid -thanks. |
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Feb 20 |
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Undecidability and holomorphic functions (Reference request) So now I am going to have to read this paper. |
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Feb 20 |
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Undecidability and holomorphic functions (Reference request) I also spent well over 3 minutes trying to get an umlaut, any style umlaut, over the o in "Erdos". :) |
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Feb 20 |
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Undecidability and holomorphic functions (Reference request) You beat me to it! But to be fair, I just vaguely remembered there was such a result, and then applied creative googling to find, e.g. Problem 16 of dpmms.cam.ac.uk/study/III/2012-13/… |
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Feb 20 |
answered | Undecidability and holomorphic functions (Reference request) |
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Feb 19 |
accepted | What is a good example of a hyperspace where the base space is non-Hausdorff? |
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Feb 14 |
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What is a good example of a hyperspace where the base space is non-Hausdorff? added 446 characters in body; deleted 6 characters in body |
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Feb 14 |
answered | What is a good example of a hyperspace where the base space is non-Hausdorff? |
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Feb 4 |
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When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics? There is also Dehornoy's work on braids and distributive algebra, which while not actually requiring large cardinals was somehow revealed by them. |
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Feb 4 |
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When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics? Asaf - That's just the sort of thing I'm trolling for :) The 'small object argument' mentioned by Martin involves a long-running transfinite recursion, but I gather that the whole point is that the small object hypothesis guarantees that the recursion (which might not teminate on its) can be shut off, at least for the purpose intended (construction of factorization systems). The cardinalities involved might well be large in comparison with those ordinarily encountered in topology, but not large in the sense of set theory. So actual large cardinals...that's intriguing! |
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Feb 3 |
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When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics? Nice examples. Regarding the small object arguument, the transfinite construction has a formal parallel in Baer's proof that the categories R-mod have enough injectives and Grothendieck's abstraction to suitable abelian categories. Carrying this out requires much set theoretic infrastructure, enough for the execution of possibly unbounded transfinite recursions. This suggests that Replacement is in the air. McLarty observed that in the relevant context (Grothendieck toposes) there is an alternate route (ia the Barr cover) requiring far less set theory. How about for the small object argument? |
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Feb 3 |
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When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics? added 2 characters in body; added 2 characters in body |
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Feb 3 |
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When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics? added 14 characters in body |
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Feb 3 |
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When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics? added 2 characters in body |
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Feb 3 |
awarded | ● Nice Question |
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Feb 3 |
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When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics? deleted 23 characters in body |
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Feb 3 |
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When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics? added 10 characters in body; added 1 characters in body |
