I am writing an essay for which I need to prove that sufficiently many graphs of a certain type exist. Is it true that for any set of sets (or set of cardinals) $S$ a countable subset $C$ exists such that

$$\forall s \in S \exists c\in C[c>s]$$

if we speak about cardinals or

$$\forall s \in S \exists c\in C[|c|>|s|]$$

when speaking about sets of sets? does this require well ordering or any other kind of explanation?


No, this is not always true. For example $S$ could be the set of cardinals less than $\aleph_{\omega_1}$. Your $C$ would be a set of alephs whose indices have $\omega_1$ as their least upper bound, but that's not possible for a countable $C$ because $\omega_1$ has cofinality $\omega_1$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.