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Halmos states in Section 1

"The most basic property of belonging is its relation to equality, which can be formulated as follows.

Axiom. of extension. Two sets are equal if and only if they have the same elements.

With greater pretentiousness and less clarity: a set is determined by its extension.

It is valuable to understand that the axiom of extension is not just a logically necessary property of equality but a non-trivial statement about belonging. One way to come to understand the point is to consider a partially analogous situation in which the analogue of the axiom of extension does not hold. Suppose, for instance, that we consider human beings instead of sets, and that, if x and A are human beings, we write $x \, \epsilon \, A$ whenever x is an ancestor of A. (The ancestors of a human being are his parents, his parents' parents, their parents, etc., etc.) The analogue of the axiom of extension would say here that if two human beings are equal, then they have the same ancestors (this is the "only if" part, and it is true), and also that if two human beings have the same ancestors, then they are equal (this is the "if" part, and it is false)."

Questions

I do follow the logic of humans, ancestors but do not get how the relation between belonging and equality is shown either by the axiom or the analogy ?

In his analogy, there is the second part also that if two human beings have the same ancestors, then they are equal which is false. Why does the axiom as stated does not mention the second part? I have read elsewhere it has something to do with logic theory in Jech.

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  • $\begingroup$ Can someone answer question 2 atleast. I got @Bram28's answer with the new concept called set-identity $\endgroup$
    – New Student
    Sep 17, 2019 at 7:39

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Questions

I do follow the logic of humans, ancestors but do not get how the relation between belonging and equality is shown either by the axiom or the analogy ?

In his analogy, there is the second part also that if two human beings have the same ancestors, then they are equal which is false.

I think the two questions are intertwined, so I'll answer them as one.

Yes, you're right, two people with the same ancestors are not necessarily the same: a sister and a brother, for example.

But, that was just the point of the example that is given here. That is, Halmos shows how, if sets and membership thereof would be a relation like this, then you can have the exact same 'membership', without having identity.

An example I like to use is this: Suppose I take the set of all people liking peanut butter sandwiches, and I compare that to the set of all people born in September, and suppose that it turns out that the people liking peanut butter sandwiches are exactly the same people that are born in September. This, of course, would be a complete coincidental fluke. Indeed, we could say that these two sets have the same members ... but we also have the intuition that these two sets are really different sets. So, maybe set-identity is something over and above what the members of the sets are.

As such, one could make a difference between the extension of a set (its contents), and its intention (what it 'is') ... roughly the difference between what is in the 'bag' .. and the 'bag' itself. So, you could have two different bags .. but with the same contents.

In fact, in the time that I wrote this and you read this, probably someone was born somewhere. So, there is one more person born in September than before. So, we would like to say that the set 'gained one more member' ... but that is impossible if a set is defined by its elements!

In sum: it is by no means a logical necessity that set-identity should be defined by its members. Or, as Halos states:

... the axiom of extension is not just a logically necessary property of equality but a non-trivial statement about belonging.

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  • $\begingroup$ A newborn affected belonging of the September set and hence its equality with the peanut set became unequal. Seems to me belonging and equality affect each other ( 2 sides of the same coin so to speak ) but that did not answer should axiom of extension have a part 2 like the human/ancestor analogy ? If yes, what should it be ? Is it axiom restated with subsets ? FYI I get the concept of identity but Halmos did not mention it :-( $\endgroup$
    – New Student
    Sep 17, 2019 at 1:20
  • $\begingroup$ @NewStudent sorry if I wasn't clear ... the example of the newborn had nothing to do with the peanut vs September-born set ... but rather was supposed to support the intuition that a set, just by itself, can gain elements. $\endgroup$
    – Bram28
    Sep 17, 2019 at 1:47
  • $\begingroup$ It can gain elements by itself but that (potentially) can affect the relationship between belonging and equality right? 1. Is that the point of Halmos saying the axiom of extension is not just a logically necessary property of equality but a non-trivial statement about belonging. 2. When Halmos says relation, is the mathematical term relation he means between belonging and equality or just the english way of speaking since relation in terms of math is not introduced in Section 1 or mentioned as a prerequiste $\endgroup$
    – New Student
    Sep 17, 2019 at 2:37
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    $\begingroup$ @NewStudent Note that if we are ok with the idea that a set can gain elements by itself, then we have a problem with the idea that a set is defined by its elements, since the set before the addition of new elements would have to be a different set from the set after the addition of elements. But, are you a different person just because you got a haircut? No, you are still you, even as you change your properties. Likewise, we could say a set can change its members while staying 'that set. $\endgroup$
    – Bram28
    Sep 17, 2019 at 2:56

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