12 | March | 2009 | A kind of library

580 -Cardinal arithmetic (11)

March 12, 2009

4. Strongly compact cardinals and ${{sf SCH}}$

Definition 1 A cardinal ${kappa}$ is strongly compact iff it is uncountable, and any ${kappa}$ -complete filter (over any set ${I}$ ) can be extended to a ${kappa}$ -complete ultrafilter over ${I.}$

The notion of strong compactness has its origin in infinitary logic, and was formulated by Tarski as a natural generalization of the compactness of first order logic. Many distinct characterizations have been found.

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2 Comments | 580: Topics in set theory | Tagged: Alfred Tarski, compactness, covering property, fine measure, Jussi Ketonen, Matteo Viale, regular ultrafilter, Robert Solovay, sch, strongly compact cardinal, ultrapower, weakly normal | Permalink
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- Answer by Andrés E. Caicedo for Who introduced direct limits? April 19, 2020
  As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwa […]
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- Answer by Andrés E. Caicedo for Atlas-like websites on specific areas of mathematics March 8, 2020
  A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Are infinite planar graphs still 4-colorable? December 16, 2013
  As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]
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- Answer by Andrés E. Caicedo for Riemann series theorem and uncountable number of sums which sum to every value November 1, 2019
  Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0
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- Answer by Andrés E. Caicedo for "Inner Regularity" of probability measure on totally ordered sets October 7, 2019
  Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]
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Math.StackExchange
- Answer by Andrés E. Caicedo for Do any proofs/properties rely on the distinction between some uncountable size and a larger uncountable size, in order for the proof/property to hold? May 12, 2020
  Sure. A large class of examples comes from the partition calculus. A simple result of the kind I have in mind is the following: Any infinite graph contains either a copy of the complete graph on countably many vertices or of the independent graph on countably many vertices. However, if we want to find an uncountable complete or independent graph, it is not e […]
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- Answer by Andrés E. Caicedo for Why is ZFC the foundational theory of mathematics? May 5, 2020
  I think that, from a modern point of view, there is a misunderstanding in the position that you suggest in your question. Really, "set theory" should be understood as an umbrella term that covers a whole hierarchy of ZFC-related theories. Perhaps one of the most significant advances in foundations is the identification of the consistency strength h […]
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- Answer by Andrés E. Caicedo for Invoking Continuum Hypothesis to prove the cardinality of a set May 2, 2020
  I'll only discuss the first question. As pointed out by Asaf, the argument is not correct, but something interesting can be said anyway. There are a couple of issues. A key problem is with the idea of an "explicitly constructed" set. Indeed, for instance, there are explicitly constructed sets of reals that are uncountable and of size continuum […]
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- Answer by Andrés E. Caicedo for One-to-one mapping from Vitali set to $\mathbb{R}$ April 26, 2020
  The question seems to be: Assume that there is a Vitali set $V$. Is there an explicit bijection between $V$ and $\mathbb R$? The answer is yes, by an application of the Cantor-Schröder-Bernstein theorem: there is an explicit injection from $\mathbb R$ into $\mathbb R/\mathbb Q$ (provably in ZF, this requires some thought, or see the answers to this question) […]
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- Answer by Andrés E. Caicedo for On the meaning of the Axiom Of Regularity in terms of " repeatedly taking the union" . April 20, 2020
  If a set $X$ is well-founded (essentially, if it contains no infinite $\in$-descending chains), then indeed $\emptyset$ belongs to its transitive closure, that is, either $X=\emptyset$ or $\emptyset\in\bigcup X$ or $\emptyset\in\bigcup\bigcup X$ or... However, this does not mean that there is some $n$ such that the result of iterating the union operation $n$ […]
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