# Axiom of Separation Problem

• Aug 13th 2011, 02:51 AM
Markeur
Axiom of Separation Problem
Let $\phi (v)$ be a formula of the languages of set theory. Let X be a set. Then the following is a set: $\{a \in X | \phi [v] \}$, ie all the elements of X with the property $\phi$ form a set.

Write down a formula $\phi (v)$ 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?
• Aug 13th 2011, 07:41 AM
MoeBlee
Re: Axiom of Separation Problem
Quote:

Originally Posted by Markeur
Write down a formula $\phi (v)$ 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]