I've already learned that there are arbitrary larger aleph fixed point. But do all these fixed points constitute a set or a proper class?
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Proper class, since it contains, as you said, arbitrarily large cardinals. – Sassatelli Giulio Apr 13 '25 at 16:37
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Sorry for being stupid. But could you explain why arbitrarily large cardinals implies that the class of all fixed points is proper? – yiqi jin Apr 13 '25 at 17:00
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If it were a set, then its union would be a set. Its union is the class of ordinals, which isn't a set because, if it were, then it would be an ordinal itself, and therefore it would be a solution to $x\in x$. But no set satisfies $x\in x$. – Sassatelli Giulio Apr 13 '25 at 17:02
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Thanks for this simple solution and your patience！ – yiqi jin Apr 13 '25 at 17:05
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@SassatelliGiulio Are you using Ordinals as Transitive Sets well-ordered by ∈? If so, The Issue with The Class of All Ordinals, is that if it were a set- it would be an Ordinal, and so it would be an elemment if Itself, but Ordinals are irreflexive with respect to ∈. One doesnt need to make an appeal to foundation, ( that no x ∈ x) The Ordinals arent't a set, even in models with some x∈x. – Michael Carey Apr 14 '25 at 02:19
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@MichaelCarey Oh, interesting. – Sassatelli Giulio Apr 14 '25 at 04:54