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I am currently learning Set Theory from Paul Halmos' Naive Set Theory. In the section about the axiom of specification, while dealing with Russell's Paradox, the author defined the set $B$ as:

$B=\{x \epsilon A: S (x)\}$

with $S(x)$ as

not$(x\epsilon x)$

How is $\epsilon$ used here? If it symbolises the relation 'belongs to' then how can an element belong to itself? It also proceeds, in the proof, to use such statements as:

...if $B \epsilon A$, then either $B \epsilon B$ also or else $B \epsilon' B$...

How can this set $B$ 'belong' to $A?$ $B$ can be is a subset of $A$, and it had been previously made clear that '$\epsilon$' and '$\subset$' are very different relations between elements and sets, or along sets.

Any help is appreciated. Thank You!

Asaf Karagila
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USSeR
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  • Membership is not even defined in most axiomatic systems of set theory. We have a collection of axioms and rules of inference and we see what we can infer. For example Russell's paradox can be given as : It is paradoxical to assert the existence of a widget that dapples all those & only those widgets that don't dapple themselves. – DanielWainfleet Jan 07 '21 at 23:49
  • I am confused about the usage of the notation $\epsilon$ to symbolise a set 'belongs to' another set, or an element 'belongs to' itself(an element). – USSeR Jan 07 '21 at 23:57
  • In the formal language used in ZF & ZFC there is no word "set" nor "element". There are axioms that use the undefined symbols $=$ and $\in$. Intuitively, in ZFC a set is anything whose existence is implied by the axioms. – DanielWainfleet Jan 08 '21 at 00:24
  • Try the short introductory book Set Theory, by Suppes. – DanielWainfleet Jan 08 '21 at 00:27
  • @DanielWainfleet thank you for the suggestion! Will try that out... – USSeR Jan 08 '21 at 10:27

1 Answers1

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In most formulations of set theory, all elements of sets are themselves sets. This is called the hereditary property. Without axioms to prevent it, it is totally possible for sets to contain themselves. In ZFC this is prevented by the axiom of well-foundedness. In particular, when unrestricted comprehension is allowed over all sets, it is possible to define a set which is the collection of all sets that do not contain themselves.

subrosar
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  • But isn't the singleton subset {2} (as a subset of, say, {1,2,3,4}) not equal to the element 2 of thar set? – USSeR Jan 07 '21 at 23:16
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    @USSeR You are right, but that does not rule out the possibility that there will be some set that contains itself as an element. – subrosar Jan 07 '21 at 23:18
  • but how can the set B 'belong' to the set A? It was defined to be a subset of A... – USSeR Jan 07 '21 at 23:23
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    @USSeR We can certainly have $X\subseteq Y$ and $X\in Y$ hold simultaneously: for example, let $X=\emptyset$ and $Y={\emptyset}$. (That is, while $\in$ and $\subseteq$ are different, neither precludes the other - they're just not the same thing, like "even number" and "square number.") – Noah Schweber Jan 07 '21 at 23:24
  • Now I see some of it, but is this correspondence allowed? Even if 'belonging' and 'being a subset of' had been shown to be satisfying different, rather contrasting properties? – USSeR Jan 07 '21 at 23:33
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    @USSeR Is what correspondence allowed? Sometimes $\in$ and $\subseteq$ coincide, other times they don't. Moreover, some set theories rule out self-containing sets (like $\mathsf{ZFC}$) while others permit or indeed require them (like $\mathsf{NFU}$). You have to pay careful attention to the details of the system you're working with. – Noah Schweber Jan 07 '21 at 23:39
  • @NoahSchweber as I said, I am just begun with set theory, with Halmos' book...are there any prerequisites to be met before start with this book? – USSeR Jan 07 '21 at 23:43
  • @USSeR Not that I'm aware of. But you have to be careful to not make unjustified (if natural-feeling) assumptions about how sets behave. You don't need to know what $\mathsf{ZFC}$ or $\mathsf{NFU}$ are at the moment, I mentioned them just for reference, the point is that whipping up a "working" set of rules for set theory is not easy and there are various strategies for doing so. Russell's paradox presents an issue that any such strategy must address: that "full comprehension" is just not something we can ever have. – Noah Schweber Jan 07 '21 at 23:47
  • So would you suggest any supplementary notes or texts that would go along? Since I am self-studying this topic, so any advice is appreciated... – USSeR Jan 08 '21 at 00:00