I am trying to prove the following statement:
Set of all infinite subsets of natural numbers is equipotent with the power set of natutal numbers.
My thought is
Let the set of all infinite subsets of natural numbers be $S$.
We need to show that there is a bijection from $S$ to the power set of na...
Mmm.... anyone have any idea what $\omega^{CK}_{\omega_1+1}$ is? Where $\omega^{CK}_1$ is the Church-Kleen ordinal
@SimplyBeautifulArt I know almost nothing about computability, so it§s unlikely I'll be able to help you. But at least this is what I found on Wikipedia.
From the WP article Admissible ordinal: "One sometimes writes $\omega_\alpha^{\mathrm{CK}}$ for the $\alpha$-th ordinal which is either admissible or a limit of admissibles; an ordinal which is both is called recursively inaccessible."
This is given as a reference there: Friedman, Sy D. (2010), "Constructibility and class forcing", Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, pp. 557–604.
But the only relevant part I see there is this sentence: "An ordinal is recursively inaccessible if it is admissible and also the limit of admissibles."
After a brief look I do not see the notation $\omega_\alpha^{\mathrm{CK}}$ introduced there.
But I suppose there are some users on this site who know about this stuff. Although I'm not sure how often they come to chat. Quick search returns some mentions of this ordinal in chat, for example:
@Matt: While all countable ordinals are countable (duh) there is no reason to expect that they are all effectively enumerable. (To make an analogy with more modern notions, the countable ordinals above $\omega_1^{\text{CK}}$ (the Church-Kleene ordinal) have no recursive representation as a well-ordering on $\omega$, and so is you want to show that there is a recursive enumeration of a set, you cannot go beyond this point.) [cont...]
BTW your question "what $\omega^{CK}_{\omega_1+1}$ is?" was meant in the sense: What does this notation mean? or in the sense What is this ordinal equal to?
I mean that you answer to: "Is it A or B?" with: "Yes. It is A or B." (Which does not give additional information - which was probably your intended joke.)
The question I am trying to prove is as follows:
Let $\{B_i\}_{i \in I}$ and $\{C_i\}_{i \in I}$ be families of mutually disjoint sets. If $B_i \approx C_i$ for each $i \in I$, prove that $\prod_{i \in I}B_i \approx \prod_{i \in I}C_i$.
My attempt:
Since $B_1 \approx C_1$, and $B_2 \approx C_2...
