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 Question about Jech's proof of V = L implies GCH June 3, 2019

You assume $\omega_\alpha\subseteq M$ and $X\in M$ so that $X$ belongs to the transitive collapse of $M$ (because if $\pi$ is the collapsing map, $\pi(X)=\pi[X]=X$. You assume $|M|=\aleph_\alpha$ so that the transitive collapse of $M$ has size $\aleph_\alpha$. Since you also have that this transitive collapse is of the form $L_\beta$ for some $\beta$, it fol […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Smallest $\beta$ such that it is provable that $2^{\aleph_\beta} > 2^{\aleph_0}$ January 16, 2019

Perhaps the following may clarify the comments: for any ordinal $\delta$, there is a Boolean-valued extension of the universe of sets where $2^{\aleph_0}>\aleph_\delta$ holds. If you rather talk of models than Boolean-valued extensions, what this says is that we can force while preserving all ordinals, and in fact all initial ordinals, and make the contin […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Arguments against large cardinals October 29, 2010

I do not know of any active set theorists who think large cardinals are inconsistent. At least, within the realm of cardinals we have seriously studied. [Reinhardt suggested an ultimate axiom of the form "there is a non-trivial elementary embedding $j:V\to V$". Though some serious set theorists found it of possible interest immediately following it […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Nonessential use of large cardinals December 10, 2010

There is a fantastic (and not too well-known) result of Shelah stating that $L({\mathcal P}(\lambda))$ is a model of choice whenever $\lambda$ is a singular strong limit of uncountable cofinality. This is a consequence of a more general theorem that can be found in 4.6/6.7 of "Set Theory without choice: not everything on cofinality is possible", Ar […]

Andrés E. Caicedo

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

Math.StackExchange

Answer by Andrés E. Caicedo for What is the consistency strength of generalized failure of the continuum hypothesis. April 14, 2019

The precise consistency strength of the global failure of the generalized continuum hypothesis is somewhat technical to state. As far as I know, it has not been published, but I think we have a decent understanding of what the correct statement should be. The most relevant paper towards this result is MR2224051 (2007d:03082). Gitik, Moti Merimovich, Carmi. P […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Is "integrability" equivalent to "having antiderivative"? March 9, 2014

There are integrable functions that are not derivatives: Any function that is continuous except at a single point, where it has a jump discontinuity, is an example. (Derivatives have the intermediate value property.) More interestingly, we can ask whether the existence of an antiderivative ensures integrability. The answer depends on what integral you are co […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Meaning of $0^{¶}$ in set theory May 2, 2019

$0^¶$ is the sharp for an inner model with a strong cardinal in the same sense that $0^\dagger$ is the sharp for an inner model with a measurable cardinal. In terms of mice, this is the first mouse containing two overlapping extenders. The effect of this is that, by iterating its top measure throughout the ordinals, you extend the bottom extender in a variet […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for ¿Is, for $1\le n May 1, 2019

Clearly $\omega^\omega\le(\omega+n)^\omega$. Also, $(\omega+n)^\omega\le (\omega^2)^\omega=\omega^{(2\cdot \omega)}=\omega^\omega$, and the equality follows. If you do not feel comfortable with the move from $(\omega^2)^\omega$ to $\omega^{(2\cdot \omega)}$, simply note the left-hand side is $\omega\cdot\omega\cdot\omega\cdot\dots$, where there are $2\cdot\o […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Does $2^{O(n)} $ mean $O(2^n)$? If not, what does $2^{O(n)} $ mean? April 24, 2019

A function $f:\mathbb N\to\mathbb R$ is $2^{O(n)}$ if and only if there is a constant $C$ such that for all $n$ large enough we have $f(n)\le 2^{Cn}$. We can think of the $O$ notation as decribing a family of functions. So, $2^{O(n)}$ would be the family of functions satisfying the requirements just indicated. In contrast, a function $f$ is $O(2^n)$ if and o […]

Andrés E. Caicedo

	Anthea M. on No durians
	580 -Some choiceless… on 580 -Some choiceless results…
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	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…

A kind of library

Smullyan

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A kind of library

Smullyan

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