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In his book on Set Theory, Jech says:

(1) Axiom schema of comprehension: If $P$ is a property, then there exists a set $Y=\{x\,\,:\,\, P(x)\}$.

(2) [The he mentions] Russell's Paradox: Consider the set $S$ whose elements are all those sets that are not members of themselves; then $S\in S$ or $S\notin S$ always leads to a contradiction.

(3) [Then he states] Schema of Separation: If $P$ is a property then for every set $X$ there exists a set $Y=\{ x\in X \,\,:\,\, P(x)\}$.

And now he mentions following:

Once we give up the full Comprehension Schema, Russell's paradox is no longer a threat; moreover it provides this useful information: The set of all sets does not exist. (Otherwise, apply the Separation Schema to the property $x\notin x$.) In other words, it is the concept of the set of all sets that is paradoxical, not the idea of comprehension itself.

I am confused by the last statement. He already derived Russell's paradox from Schema of Comprehension, and how he is saying that the idea of comprehension is not paradoxical?

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    $\begingroup$ Separation is a weaker form of the first comprehension axiom that does not lead to the same paradox. $\endgroup$
    – Karl
    Dec 24, 2021 at 17:38

1 Answer 1

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Jech makes two claims here:

  1. The idea behind comprehension is not paradoxical.

  2. The idea of a universal set is.

The key word doing all the heavy lifting in idea 1 is "idea:" for Jech, the full comprehension scheme and the full separation scheme are each implementations of the comprehension idea, and the fact that the former (which is more obvious) is inconsistent doesn't mean that the underlying idea itself is inconsistent. Note that there's an implicit claim here that separation does not represent a new idea but rather a more careful formulation of the same idea behind comprehension. I tentatively agree with Jech here.

Claim 2, to be honest, I have little sympathy for. Certainly there are consistent set theories which admit a universal set (e.g. $\mathsf{NFU}$ - while the consistency of $\mathsf{NF}$ relative to $\mathsf{ZF}$ is (sorta-kinda-)open, it's easy to show that $\mathsf{NFU}$ is consistent relative to even much less than $\mathsf{ZF}$).

One possible response to my objection to claim 2 is that theories like $\mathsf{NFU}$ while technically having a universal set aren't faithful to the idea behind a universal set - that part of sethood is separation. If we grant this, then it is indeed the case that "the idea of a universal set" is contradictory; however, this goes against the putative bundling of separation and comprehension two paragraphs prior.

Ultimately, I disagree with Jech here: I think there are two genuinely different ideas, each of which is consistent on its own but which are mutually inconsistent.

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  • $\begingroup$ Thanks for the clarifications and further comments. Could you mention what is NFU and NF? $\endgroup$ Dec 25, 2021 at 3:15
  • $\begingroup$ @MathsRahul $\mathsf{NF}$ is Quine's New Foundations; $\mathsf{NFU}$ is a variant without the axiom of extensionality (the "U" stands for "urelement"). $\mathsf{NF}$ at first glance looks no more dubious than $\mathsf{NFU}$, but it turns out that it's a much stranger beast; not only is it not known whether $\mathsf{NF}$ is consistent even granting the consistency of $\mathsf{ZFC}$ + large cardinals (I'm taking the I-think-standard position that Holmes' claimed consistency proof has not yet been accepted) but $\mathsf{NF}$ is known to disprove choice. $\endgroup$ Dec 25, 2021 at 4:10
  • $\begingroup$ The disproof of choice is rather technical; see the discussion here. By contrast, $\mathsf{NFU}$ has lots of well-understood models and is compatible with choice. (As an amusing aside, note that Jech himself has done some work on $\mathsf{NF}$.) The Stanford Encyclopedia article on alternative set theories has a lot of good information on $\mathsf{NF(U)}$. $\endgroup$ Dec 25, 2021 at 4:12

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