# Since the conception of Set Theory, was Russell's Set the only problematic set found?

I've read the history section of Set Theory on wikipedia:

The next wave of excitement in set theory came around 1900, when it was discovered that Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself, and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics.

It mentions Russell's Paradox and Cantor's question as a related problem. It seems odd to me that only this set presented a problem - mostly based on the naive heuristic which states that things present problems all the time - and which I really think is not so naive. I've done some readings on set theory books (not very deep readings though) and Russell's set is always pointed as the problematic set. Are there other sets that also presented problems?

## 1 Answer

Yes. There were several paradoxes in early set theory:

1. The set of all sets, Cantor's proof that $|X|<|\mathcal P(X)|$ means that if $X$ is the set of all sets then $|\mathcal P(X)|\leq|X|$, which is a contradiction.

In $\sf ZF$ this is resolved by not having the set of all sets; in set theories like $\sf NF$ where a universal set does exist, it is solved by other ways (e.g. making sure that the surjection from the universal set onto its power set, while definable, is not a set).

2. The set of all ordinals, Burali-Forti noted that if $X$ is the set of all ordinals, then $X$ is well-ordered, therefore an element of itself, and thus contradicting the fact that it is in fact well-ordered.

You may find more information in this Wikipedia category.

• "making sure that the surjection ... is not a set". Is this a typo? – Did Dec 31 '13 at 10:30
• Why would it be a typo? (It might be a minor mistake of mine, or a misunderstanding of some sort; but it certainly read as it should read.) – Asaf Karagila Dec 31 '13 at 10:31
• Note that the rules of how to generate a set from other sets are different in $\sf NF$, and in particular not every definable subset of a set is a set itself. – Asaf Karagila Dec 31 '13 at 10:36
• Surely this is due to my lack of expertise in set theory but I was surprised that a function could be, or not be, a set. If this is what you meant, I must read more on these matters... – Did Dec 31 '13 at 10:39
• Did, in set theory... everything is a set. Including functions. Sometimes functions are not sets, but they are definable so we can talk about them in the appropriate way (i.e. class functions). But functions, much like everything else, are sets... (with the obvious exclusion of Chuck Norris. He's not a set.) – Asaf Karagila Dec 31 '13 at 10:41