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I am reading Halmos's Naive Set Theory. I am really enjoying though I never read such a book. But there are some things that I am unable to grasp. Notably in the second chapter after stating the axiom of specification he presents a condition that (x does not belong to x). After that he uses this condition by say that {x belongs to A: x does not belongs x}. What I don't understand is that how can we talk about an element belonging to itself ? and what does it mean ?

Sandeep Jassal
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Maybe it is simpler to approach it in terms of predicates (expressing properties of objects). See original Russell's formulation The Principles of Mathematics (1903)]:

Among predicates, most of the ordinary instances cannot be predicated of themselves, though, by introducing negative predicates, it will be found that there are just as many instances of predicates which are predicable of themselves: predicability, as is evident, is predicable, i.e. it is a predicate of itself. But the most common instances are negative: thus non-humanity is non-human, and so on [i.e. the predicate humanity applies to human being and not to predicates, and thus it does not apply to itself; on the contrary, the predicate non-humanity is not a human being, and thus applies to itself]. The predicates which are not predicable of themselves are, therefore, only a selection from among predicates, and it is natural to suppose that they form a class having a defining predicate.

Here we can see the basic ingredient: the Comprehension principle stating that every predicate (expressed by a formula $\varphi$ of the language of sets) identifies a unique set $S$, i.e. the set (or class) defined by the predicate "to be a predicate that cannot be predicated of itself".

Regarding "not belonging to itself", we can try with the following example.

We know the natural numbers: $0,1,2,\ldots$ and we have the set $\mathbb N$ of all naturals, such that $0,1,2 \in \mathbb N$.

We have that $\mathbb N \notin \mathbb N$: the set of all naturals is not itself a natural number.

Assume now that we can define the set of non-naturals:

$\overline {\mathbb N} = \{ x \mid x \text { is not a natural number } \}$;

we have that this set is not itself a natural, and thus:

$\overline {\mathbb N} \in \overline {\mathbb N}$.

Mauro ALLEGRANZA
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  • I thought of approaching the question by looking at '$\in$' as a relation but it leads to a circular dependency as the definition of a relation makes use of '$\in$' itself. Do you know if there is some independent definition of the "belongs-to" symbol? – Aelx Oct 26 '21 at 07:18
  • In Halomos's words 'belonging is primitive (undefined) principal' – Sandeep Jassal Oct 26 '21 at 07:22
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    @СССР - In set theory $\in$ is part of the language: it is a binary predicate and it is the "basic property" of sets: sets have elements and are in turn elements of other sets. In set theory relations are specific type of sets: a binary relation is a set of pairs. – Mauro ALLEGRANZA Oct 26 '21 at 07:33
  • Doesn't it then answer the question of what $x \in x$ means? If "belongs-to" symbol is defined as a basic property of sets, then $x \in x$ implies that $x$ is assumed to be a set (not element), otherwise $x \in x$ would not be a valid notation at all. – Aelx Oct 26 '21 at 07:39
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    @СССР - $\in$ is a symbol of the language and thus $x \in x$ is a formally correct formula, like $x=x$. Only with Regularity axiom we can prove that "circular chains" of sets like $(x \in y) \land (y \in x)$ are ruled out. – Mauro ALLEGRANZA Oct 26 '21 at 07:51