- #1

- 263

- 4

## Main Question or Discussion Point

In page 6 of Naive Set Theory by Halmos, he introduces the definition of the axiom of specification, then sets up one example based on the axiom, in which he changes ##S(x)## to ##x \not \in x## to illustrate something. I understand that this mean ##x## doesn't belong in ##x##.

Afterwards comes this excerpt,

Or is it by inferring from the definition like this:

##(*)## ##B \in B## if and only if ##(B \in A \text{ and } B \not \in B)##

But then clearly there's something wrong since it cannot be both true that ##(B \in B## and ##B \not \in B)##? I mean yes if ##B \in A## then according to the definition ##B \not \in B## but then if we accept ##B \in A## such that ##B \not \in B##, clearly according to the definition ##B \in B##.

Could you guys explain this part further? I also think that I haven't fully appreciated this example. What does the writer want to show to us? That a set cannot have everything and there must be at least something that is not in the set, hence the Axiom of Specification?

Thank You

Afterwards comes this excerpt,

How does one infer that if ##B \in A## then either ##B \in B## or ##B \not \in B##. Is it like when I am shopping to the market ##A## and there's a basket ##B## there. If a product is in the market it is possible to have a product either in the basket or not in the basket i.e in the shelves.It follows that, whatever the set A may be, if ##B = \{x \in A \mid x \not \in x \}## then,

for all ##y##,

##(*)## ##y \in B## if and only if ##(y \in A \text{ and } y \not \in y)##

Can it be that ##B \in A##? We proceed to prove that the answer is no. Indeed, if ##B \in A## then either ##B \in B## also (unlikely, but not obviously impossible), or else ##B \not \in B##.

Or is it by inferring from the definition like this:

##(*)## ##B \in B## if and only if ##(B \in A \text{ and } B \not \in B)##

But then clearly there's something wrong since it cannot be both true that ##(B \in B## and ##B \not \in B)##? I mean yes if ##B \in A## then according to the definition ##B \not \in B## but then if we accept ##B \in A## such that ##B \not \in B##, clearly according to the definition ##B \in B##.

Could you guys explain this part further? I also think that I haven't fully appreciated this example. What does the writer want to show to us? That a set cannot have everything and there must be at least something that is not in the set, hence the Axiom of Specification?

Thank You