This is from an article talking about Russell's Paradox and why it was so significant. Could someone please explain what it means?

The reason this conclusion was so groundbreaking was because it showed that "one cannot both hold that there is meaningful totality of all sets and also allow an unfettered (unrestricted) comprehension principle to construct sets that must then belong to that totality."

The conclusion was Russell's Paradox.

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    $\begingroup$ It is flowery language that essentially says "We can't have a universal set of all sets without things breaking" $\endgroup$
    – JMoravitz
    May 3 '21 at 17:44
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    $\begingroup$ Basically, the following two statements are not consistent, although both are desirable:$$\text{There is a set of all sets}$$$$\text{For any logical formula, there exists a set consisting of exactly those sets satisfying that formula}$$ $\endgroup$ May 3 '21 at 17:45
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    $\begingroup$ @JMoravitz Not really. It says we can't have unrestricted comprehension without things breaking. If we restrict comprehension to stratified formulas (as in NF), then we can have a set of all sets. If we restrict comprehension to only allow comprehension over sets (as in ZF), then we can't have a set of all sets. Both ways presumably avoid Russell's paradox. But I think even in the latter case the proper class of all sets still constitutes a "meaningful totality of all sets". $\endgroup$ May 3 '21 at 18:05
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    $\begingroup$ @DonThousand You don't need the first of those though, since it follows immediately from the second (take $x=x$). $\endgroup$ May 3 '21 at 18:38
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    $\begingroup$ @MauroALLEGRANZA Yes, sorry about that... Got confused; all around the same time anyway, foundational crises and all that. $\endgroup$ May 4 '21 at 10:44

Say we have some "meaningful totality of all sets", called $V.$ One thing we might like is to be able to define the set of all sets satisfying a given property $P$. In other words, we might like an axiom of the form

Axiom (Unrestricted comprehension.) For any property $P$ defined on $V,$ there is a set $X_P\in V$ such that for all sets $x\in V,$ $x\in X_P$ if and only if $P(x)$ holds.

This is proven inconsistent by Russell's paradox, by taking $P(x):=x\notin x.$

Note that, contrary to the comments, this has nothing to do with whether $V\in V$ or not, i.e. whether there is a set of all sets. (It does imply there is a set of all sets, by applying comprehension to $x=x$, as Noah points out in the comments, but that’s not the source of the paradox.)

However, the most popular way of eliminating this paradox is to instead use the following restricted comprehension axiom:

Axiom (Separation) For any property $P$ defined on $V$ and any set $A\in V,$ there is a set $X_{P,A}$ such that for all $x\in V,$ $x\in X_{P,A}$ if and only if $x\in A$ and $P(x)$ holds.

This does not eliminate the paradox if $V\in V,$ since we could just apply the separation axiom to $P(x):=x\notin x$ and $A=V$ to get a contradiction. However, if $V\notin V,$ we are fine, because going through the same motions with an arbitrary set $A$ doesn't generates an inconsitency and simply tells us that $X_{P,A}\notin A,$ which is totally possible if $A$ does not contain all sets. So if we go this route, we must also accept that there is no set of all sets. This approach is the basis of the standard set theory ZF(C).

There was another famous attempt: the stratified comprehension principle of Quine's New foundations, that avoids the paradox while allowing a universal set. But this proved less popular for foundational purposes.

  • $\begingroup$ Thank you so much! $\endgroup$
    – Javs
    May 4 '21 at 7:09

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