June | 2015 | A kind of library

Update

June 29, 2015

So, we moved to Ann Arbor.

We visited the Hands-On Museum (thanks to Zach and Diana, and congratulations!),

and attended the Math Reviews Summer picnic.

I started today.

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3 Comments | Life | Tagged: Francisco, Isabel | Permalink
Posted by andrescaicedo

You are currently browsing the A kind of library blog archives for June, 2015.
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- Answer by Andrés E. Caicedo for What definitions were crucial to further understanding? December 3, 2018
  In set theory, definitely the notion of a Woodin cardinal. First, it is not an entirely straightforward notion to guess. Significant large cardinals were up to that point defined as critical points of certain elementary embeddings. This is not the case here: Woodin cardinals need not be measurable. If $\kappa$ is Woodin, then $V_\kappa$ is a model of set the […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Examples of Mathematical Papers that Contain a Kind of Research Report November 21, 2018
  The first example that came to mind was MR0270881 (42 #5764) van der Waerden, B. L. How the proof of Baudet's conjecture was found. 1971 Studies in Pure Mathematics (Presented to Richard Rado) pp. 251–260 Academic Press, London. There, van der Waerden describes some of the history as well as his proof of his well-known theorem. Another example: MR224589 […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Strong partition property + DC + existence of non-principal ultrafilter on $\omega$ August 30, 2018
  The key reference for this is MR0799042 (87d:03141). Henle, J. M.; Mathias, A. R. D.; Woodin, W. Hugh. A barren extension. In Methods in mathematical logic (Caracas, 1983), C. A. Di Prisco, editor, 195–207, Lecture Notes in Math., 1130, Springer, Berlin, 1985. There, Henle, Mathias, and Woodin start with $L(\mathbb R)$ under the assumption of determinacy (an […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Can we have an infinite sequence of decreasing cardinality all terms of which have equal sized power sets? August 12, 2018
  This is consistent, at least under a rather tame large cardinal assumption. (One can also produce examples by manipulating Dedekind finite sets, but Asaf's answer addresses this. The answer here works even in the context of $\mathsf{DC}$.) For instance, see MR3612001. Conley, Clinton T.; Miller, Benjamin D. Measure reducibility of countable Borel equiva […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Are there known examples of sets whose power set is equal in size to power set of larger sets only in absence of choice? August 11, 2018
  Let me add some comments to Asaf's answer. We are looking for situations with $|X|
  Andrés E. Caicedo
Math.StackExchange
- Answer by Andrés E. Caicedo for Can we derive a large cardinal axiom by a principle of isomorphism between cardinals and ordinals December 10, 2018
  You should really stop writing soups of symbols and just describe in words what you are doing. It saves a coding headache on your side and a decoding headache on the reader's. Anyway, the formula seems to say that if for every ordinal there is a larger cardinal with property $\psi$, then there is a bijection between the class of ordinals and that of car […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Proof that the Cardinality of Borel Sets on $\mathbb R$ is $c$ without using the ordinals . May 4, 2016
  Your idea is sound, but it requires more work. As pointed out, you have only described so far a very small subcollection of the Borel sets. Instead, show that you can associate to each Borel set a code that keeps track of the "history" of its construction starting from basic open sets, and then count the number of such codes. There is a lot of leew […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Minimal model of ZF with $0\sharp$ November 27, 2018
  Yes, the minimal such model is $L[0^\sharp]$. This model can be built by stages, just as $L$, starting with the empty set, taking unions at limit stages, and at each successor stage $\alpha+1$ taking the collection of subsets of $L_\alpha[0^\sharp]$ definable in $(L_\alpha[0^\sharp],\in,0^\sharp)$ from parameters. Here, $0^\sharp$ can be thought of as a set […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Why doesn't ZFC "know about" all of the ordinals? November 21, 2018
  There are several issues here. An obvious one is that there are only so many proofs and many more ordinals, so there are ordinals that we cannot even refer to within the theory, so of course the theory cannot prove anything about them. This is somewhat subtle, as there are models of set theory that are pointwise definable, so that any $x$ in the model is def […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Help with a sequence proof November 20, 2018
  Check first that $\liminf |a_n|=0$ if and only if some subsequence of the $a_n$ converges to $0$. Assuming that $\liminf|a_n|=0$, use the equivalence above to show that, indeed, there is a subsequence $(a_{n_k})_{k\ge0}$ that, not only converges to 0, but does it very quickly, say $|a_{n_k}|
  Andrés E. Caicedo
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