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Aug
9
revised For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice
rewording
Aug
9
revised For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice
correction
Aug
9
comment For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice
Yes, @Asaf. The argument is essentially the same as yours, but it avoids the use of ordinals as the OP requested.
Aug
9
answered For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice
Jul
26
awarded  Enthusiast
Jul
21
awarded  Yearling
Jul
19
comment Uncountable dense subset whose complement is also uncountable and dense
@tomcuchta: Yes, because a Bernstein set must meet every closed interval that are not singletons.
Jul
19
comment Uncountable dense subset whose complement is also uncountable and dense
It's even simpler to write $(-\infty,0] \setminus \mathbb{Q}$...
Jul
19
comment Uncountable dense subset whose complement is also uncountable and dense
The only drawback is that you need some Axiom of Choice to get a basis for $\mathbb{R}$ over $\mathbb{Q}$...
Jul
19
comment Uncountable dense subset whose complement is also uncountable and dense
It doesn't get much simpler than that! (Note that intersecting $(-\infty,0]$ with $\mathbb{R}$ is a bit redundant.)
Jul
19
answered Uncountable dense subset whose complement is also uncountable and dense
Jul
19
comment Silver indiscernibles and definable injections
Apostolos has the right idea. Every Silver indiscernible is in fact inaccessible in $L$. There will be a constructible injection $\lambda \to \omega\times\alpha^{<\omega}$ if and only if $\lambda < \max((|\alpha|^+)^L,\omega_1^L)$. If $\alpha < i_\alpha$, then $i_\alpha$ is necessarily much larger than $\max((|\alpha|^+)^L,\omega_1^L)$.
Jul
19
comment Silver indiscernibles and constructibility
No problem! It was a pleasure!
Jul
18
revised Silver indiscernibles and constructibility
clarification; changed terminology to match the question
Jul
18
answered Silver indiscernibles and constructibility
Jul
18
revised How is Kleene's T predicate defined?
added wikipedia links
Jul
18
answered How is Kleene's T predicate defined?
Jul
17
comment Estimation of factoring time of a $n$-digit number (current state of art) on a desktop
@Gerry: I used the formula from Wikipedia, which says that $c = (64/9)^{1/3} + o(1)$. I did ignore the $o(1)$, but all I wanted is to illustrate the fast growth of this function.
Jul
17
comment Do there exist interesting binary relations satisfying reflexivity and symmetry, but not transitivity?
These are well-known graphs with some very interesting open problems. But your post doesn't even mention why anyone should care about these graphs!
Jul
17
comment Estimation of factoring time of a $n$-digit number (current state of art) on a desktop
@TaoLee: The number field sieve is only one step along the path of better and better factoring algorithms. It is conceivable that a clever new algoritm could factor RSA-2048 in a matter of days.