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What are "sets that don't contain itself" in Russell's paradox?

Since I think that for a set having to contain itself is intuitive and if a set doesn't contain itself, then the cardinality ought to be less than.

However, I don't understand how sets of different sizes could even be considered to be equal? Thus, isn't a set that doesn't contain itself an impossibility?

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  • $\begingroup$ In ordinary mathematics (algebra, analysis, number theory, geometry), set self-membership is not an issue. Nevertheless, many set theorists believe it must be explicitly banned from consideration at the foundational level. IMHO, such an explicit ban may not be necessary. As I see it, the problem with RP wasn't set self-membership, but an axiom of unrestricted comprehension in an early attempt to axiomatize set theory by G. Frege. $\endgroup$
    – Dan Christensen
    Nov 27, 2017 at 15:44

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The issue here is the interpretation of the verb "contain". "Contain" here means contain as an element, not as a subset. By this interpretation, almost no natural sets one would think of contain themselves, and thus it's far from an impossibility. For instance, the empty set does not contain itself. It doesn't contain anything. Also the set $\{3\}$ does not contain itself. Its only element is $3.$ $\{3\}$ is not an element of it.

It's harder to think of a set that does contain itself. A non-rigorous example is "the set of all things that are not vanilla ice cream." Since "the set of all things that are not vanilla ice cream", whatever it is, is certainly not vanilla ice cream, this set is an element of itself.

And yet, per Russell, even though it seems a perfectly normal property for a set to not contain itself, the set of all sets that do not contain themselves is a paradoxical construct, since this set would contain itself if and only if it did not contain itself.

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Note: In the Russell paradox containing does not mean that a set is a subset of itself. The set contains itself as an element (and not as a subset.) Having this is mind let's step forward.

The size of a finite set depends on the number of elements in that set. If the elements are sets then the number of elements in the contained sets do not count when we calculate the number of elements in the containing set. So, let's see the following example:

enter image description here

Here the number of elements of the outer set is $4$ (two sets and two "element-elements") and not $6$, the number of elements we can see from above.

To consider infinite sets the problem is a little more complicated but is essentially similar.

As far as a set containing itself as an element:

enter image description here

Here the number of elements in $A$ is $3$ even if $A$ is contained in $A$ as an element.

The problem is not that with $A$... You can sense the problem if you try to depict $A$ like I did and stopped at the $3^d$ level.

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  • $\begingroup$ Yes, these are great diagrams to spell out the difference between 'element of' and 'subset'. At risk of causing more confusion than education for OP, I will emphasize that your last example, and more basically, 'a set consisting of only itself' are not inherently paradoxical, even though they are outlawed in ZF by foundation, $\endgroup$
    – spaceisdarkgreen
    Nov 24, 2017 at 9:47

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