Axiom of Separation Problem

Let $\displaystyle \phi (v)$ be a formula of the languages of set theory. Let X be a set. Then the following is a set: $\displaystyle \{a \in X | \phi [v] \}$, ie all the elements of X with the property $\displaystyle \phi$ form a set.

Write down a formula $\displaystyle \phi (v)$ saying that the set v has exactly one element.

I'm actually stuck here for very long. Is there really an abstract formula that represents all the concrete examples that the set has exactly one element?

Re: Axiom of Separation Problem

Quote:

Originally Posted by

**Markeur** Write down a formula $\displaystyle \phi (v)$ saying that the set v has exactly one element.

[I'll use 'E' for 'there exists', 'e' for 'is an element of' and 'A' for 'for all']

Ex(xev & Ay(yev -> y=x))

By the way, you probably meant not

{aeX | P[v]}

but rather

{veX | P[v]}

where the variable 'X' does not occur free in P[v]

or

{aeX \ P[a]}

where the variable 'X' does not occur free in P[v]