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In studying more on the mathematics in the past of Frege, Russell, and Zermelo, and I was wanting to learn more about impredicative/predicative definitions to solve some inquiries I had.

1. How does banning impredicative definitions avoid Russell's Paradox?

From what I read, The Vicious Circle Principle played a role where "No entity can be defined in terms of a totality to which this entity belongs". From this, I can see that this does indeed ban the definition of Impredicativity. Is there more to this that I'm missing?

2. Does ZFC allow impredicative definitions? If it does, how does it avoid Russell's Paradox?

Zermelo and Fraenkel, developed the ZFC and they did allow impredicative definitions as they did not allow the existence of universal sets and only referred to Pure Sets/proper classes and prevents its model from containing elements of sets that are not themselves sets. Were there other factors that ZFC had to avoid Russell's paradox?

Thanks for reading & helping!

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    For 1., see [Russell's Paradox](https://plato.stanford.edu/entries/russell-paradox/#ERP) : Russell's solution is the theory of types: this theory implements VCP and thus the definition of "Russell's set" is forbidden. – 2017-02-02
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    For 2., see [Zermelo's Axiomatization of Set Theory](https://plato.stanford.edu/entries/zermelo-set-theory/) : the Axiom of Separation avoid Russell's Paradox. – 2017-02-02

1 Answers 1

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(1) You are right, with caveats. The caveat is that "impredicative" is an intuition that Russell tried to pin the blame for the paradox on -- and then he spent reams of words and many years trying to define what exactly "impredicative" means, such that banning it would both avoid the paradoxes and still allow ordinary mathematics. The results were not exactly successful -- at least they didn't catch on.

(2a) Yes, ZFC allows impredicative definitions. For example let's define

A natural number $n$ is called hooplish if every subset $A$ of $\mathbb N$ with the property that every prime power is a sum of at most $n$ elements of $A$ must contain an arithmetic sequence of length $n$.

(The details of this don't matter -- in fact, I have no idea which numbers are or are not hooplish, or whether the concept is trivial or not). What matters is that "$n$ is hooplish" can certainly by defined by a formula in the language of set theory, and therefore ZFC's Axiom of Separation allows us to define $$ H = \{ n\in\mathbb N \mid n\text{ is hooplish} \} $$ According to this definition, in order to figure out whether some number is in $H$, we need to quantify over all subsets of $\mathbb N$, including $H$ itself. That is by every reasonable standard impredicative! But ZFC has no problem with it; it promises us that there is a set with which property.

And nobody has, so far, been able to leverage that guarantee into an proof of a contradiction.

The philosophical underpinning of this is the view that the Axiom of Separation does not generate the subsets of $\mathbb N$ -- in the intended interpretation they are all there from the beginning, and the axiom just explains that we can pick one of them in such-and-such way.

(2b) ZFC avoids Russell's paradox by not having an axiom that guarantee that $\{x\mid x\notin x\}$ describes a set. ZFC doesn't say that the problem with the definition is that it is "impredicative", but simply that it doesn't fit into any of the precisely enumerated kinds of definitions that ZFC does allow.

Russell thought that banning impredicative definitions would be one way to avoid the paradoxes while preserving ordinary mathematics. Just because he said so, however, doesn't mean that he was right -- opinions seem to be divided whether in his quest to preserve ordinary mathematics he didn't, effectively, open a back door to at least some kind of impredicative definitions.

And in any case, I don't think Russell claimed such a ban would be the only way to reach the goal (though he evidently was of the opinion it would be the best way, if only the details could be gotten right). ZFC simply follows a different strategy, one that seems to be pretty successful so far.