Smullyan

I have just posted on my papers page a preprint of a review of

MR3379889
Smullyan, Raymond
Reflections—the magic, music and mathematics of Raymond Smullyan.
World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. x+213 pp.
ISBN: 978-981-4644-58-7; 978-981-4663-19-9

that I have submitted to Mathematical Reviews.

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Mathoverflow

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

Answer by Andrés E. Caicedo for Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$ May 10, 2018

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Does constructing non-measurable sets require the axiom of choice? October 14, 2010

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Andrés E. Caicedo

Math.StackExchange

Answer by Andrés E. Caicedo for Measure on ([0,1], P([0,1])) October 17, 2018

Surprisingly, the answer is no due to serious set-theoretic restrictions. If $X$ is an infinite set such that there is a ($\sigma$-additive) measure on $\mathcal P(X)$ that only takes the values 0 and 1, assigns 1 to $X$ and 0 to singletons, then the cardinality $\kappa=|X|$ of $X$ is much much larger than $\mathfrak c=|\mathbb R|$, the cardinality of the se […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Without the Axiom of Choice, must two nonempty sets have a surjection between them? October 9, 2018

No, you cannot show this. For instance, it is consistent to have infinite Dedekind-finite sets whose power set is still Dedekind-finite. Now, if there is a surjection from $A$ to $\omega$, then there is an injection from $\omega$ (indeed, from $\mathcal P(\omega)$) to $\mathcal P(A)$, so $\mathcal P(A)$ is Dedekind-infinite. Thus, if $\mathcal P(X)$ is infin […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Example of a concrete irrationality test October 7, 2018

First, there are some nice examples like $$ e=\sum_{n\ge0}\frac1{n!} $$ or Liouville-like numbers, mentioned in the answers by Wilem2, that can be easily proved to be irrational using the theorem, but for which typically there are simpler irrationality proofs: For $e$, we quickly get that $$0

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Set of infinite measure and subsets of finite measure. September 20, 2018

Note first that if $C$ is uncountable and for each $i\in C$ we have a real number $r_i>0$, then $\sum_i r_i=+\infty$. The point is that for some $n>0$, the family $\{i\in C: r_i>1/n\}$ is uncountable. Of course, in this case there is a countable subfamily whose sum is infinite as well. Now, let $A$ be measurable of infinite measure. If there is an u […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for What is the utility, exactly, of the Deduction Theorem? September 19, 2018

The theorem says that to prove an implication it is enough to assume the hypothesis and proceed to prove the conclusion. Proofs of that kind tend to be more natural than proofs that conclude the implication directly. Just as in regular mathematical practice: many theorems have the form "Assuming $A$, then we have $B$", and we usually prove them by […]

Andrés E. Caicedo

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