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Results tagged with set-theory
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user 461
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
17
votes
Cardinality of the permutations of an infinite set
As already mentioned, we have $$2^k\le k^k\le(2^k)^k=2^{kk}=2^k,$$ and thus $2^k=k^k$. The inequality $k!\le k^k$ is obvious. To check $2^k\le k!$, note that $2^k$ subsets of $X$ are the set of fixed …
- 4,416
10
votes
1
answer
407
views
Does every set have a rigid self-map?
The question was asked on Mathematics Stackexchange
but has remained unanswered so far.
A self-map is a map $f:X\to X$ from a set $X$ to itself. There is an obvious notion of morphism, and thus of iso …
- 4,416
1
vote
How would one even begin to try to prove that a simple number-theoretic statement is undecid...
EDIT
Here is my problem. To prove that statement S is undecidable is to
(1) prove that one cannot prove S.
I think I understand the meaning of the second "prove". (It depends of course on the cont …
- 4,416
3
votes
Subset of the plane that intersects every line exactly twice
Here is a minor variation of the proof.
We show that there is a subset $Z$ of $\mathbb R^2$ which intersects each line exactly twice.
We define the ordinals in such a way that each ordinal $a$ is th …