When proving that every field has an algebraic closure, you have to be careful. In this article https://proofwiki.org/wiki/Field_has_Algebraic_Closure, and as I have been told on this site, if we have a field F. The "collection of all algebraic extensions of F" is not a set.
Is there a simple way to explain why this is not a set, and we can not apply zorns lemma on it? Or do you need a lot of reading in deep set-theory and logic to understand this? I have seen the russel paradox, but that is basically how much I know about this.
What also is very confusing is that in real analysis we have that "the space of continuous functions on [0,1] is a vector space". So there is a set of continuous functions? This doesn't sound any more mysterious than "all algebraic extensions of a given field F", however one of them gives rise to a set, and one doesn't?