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Andrés Caicedo
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Mathoverflow- Answer by Andres Caicedo for If ZFC has a transitive model, does it have one of arbitrary size? May 5, 2013In a comment to François's answer, I point out that the least $\alpha$ such that $L_\alpha$ is a model of $\mathsf{ZFC}$ is countable. In what follows, "model" means "transitive model of $\mathsf{ZFC}$." If $M$ is transitive, then $L^M=L_\beta\models\mathsf{ZFC}$ for $\beta=\mathsf{ORD}\cap M$, so the least height of a model is count […]Andres Caicedo
- Answer by Andres Caicedo for How additive is Lebesgue measure in ZF+AD ? December 7, 2010Ricky: I think I see how to answer the problem under a stronger assumption. Rather than $\mathsf{AD}$, work in $$ {\sf AD}^+ + V=L({\mathcal P}({\mathbb R})). $$ This is a bit unsatisfying, since it is very possible the question can be answered assuming only $\mathsf{AD}$. In any case, $\mathsf{AD}^+$ is potentially harmless, since it may be that $\mathsf{AD […]Andres Caicedo
- Answer by Andres Caicedo for Basis theorem (due to Solovay?) May 1, 2013The result is indeed Solovay's Basis Theorem. It is a consequence of Moschovakis's Coding Lemma, and sometimes it is referred to as (a version of) the reflection theorem (for example, in section 8 of Steel's Handbook article). Perhaps the optimal reference (it is self-contained, and easily accessible) is Section 2.4 of Peter Koellner, and W. H […]Andres Caicedo
- Answer by Andres Caicedo for Counterintuitive consequences of the Axiom of Determinacy? April 28, 2013Let's see: $\mathsf{AD}$ implies that all sets of reals are Lebesgue measurable, have the Baire property, and the perfect set property (so, a version of the continuum hypothesis holds). It is conjectured that it also implies that all sets of reals are Ramsey. All these statements fail (badly) in the presence of choice. (There is a technical strengthenin […]Andres Caicedo
- Answer by Andres Caicedo for Models of Determinacy March 14, 2013Certainly, $L(\mathbb R)$ is not a mouse (rather, a weasel) over a countable set, and the only way I see of thinking of it as a mouse and still capturing all the reals is making it a mouse over $\mathbb R$, in which case the answer is trivial. There is interesting work on $\mathbb R$-premice, but I do not think this is what you had in mind here. More relevan […]Andres Caicedo
- Answer by Andres Caicedo for If ZFC has a transitive model, does it have one of arbitrary size? May 5, 2013
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- Answer by Andres Caicedo for When does equality holds in $A\subseteq P(\cup A)$ May 17, 2013If equality holds, then $A=\mathcal P(X)$ for some set $X$, namely $X=\bigcup A$. Conversely, note that if $A=\mathcal P(X)$, then $\bigcup A=X$. That is: $A=\mathcal P(\bigcup A)$ iff $A$ is the power set of some set $X$, in which case $X=\bigcup A$.Andres Caicedo
- Answer by Andres Caicedo for The relationship of ${\frak m+m=m}$ to AC May 16, 2013Suppose that $A$ is infinite and Dedekind finite. Then $\mathfrak m=|A\cup\mathbb N|$ satisfies that $|A|Andres Caicedo
- Answer by Andres Caicedo for Growth-rate vs totality May 11, 2013There is a hierarchy of fast-growing functions $f_\alpha:\mathbb N\to\mathbb N$ indexed by ordinals $\alphaAndres Caicedo
- Answer by Andres Caicedo for Model theory question with finiteness May 13, 2013[Edit: I address first the original version of the question, where the requirement that $\Psi$ is transitive was not stated. The last paragraph addresses this case.] As stated, this is false, and it requires too many wrinkles to fix appropriately: (Assume that) $\mathsf{ZF}$ is consistent, and that $\Psi$ is a model of $\mathsf{ZF}+\lnot\mathrm{Con}(\mathsf{ […]Andres Caicedo
- Answer by Andres Caicedo for What are some good open problems about countable ordinals? May 3, 2013There is still much to explore in the area of partition calculus. The notation $$\alpha\to(\beta,\gamma)^2$$ means that if one colors the $2$-element subsets of $\alpha$ with two colors, red and blue, then there is an $H\subseteq\alpha$, all of whose $2$-sized subsets have the same color and, either the color is red and $H$ has order type $\beta$, or the col […]Andres Caicedo
- Answer by Andres Caicedo for When does equality holds in $A\subseteq P(\cup A)$ May 17, 2013
