# Thread: Transitive Set Theory Problem

1. ## Transitive Set Theory Problem

Show that for any x there exists a transitive y such that $y \notin x$
x and y are sets.

Here's my idea: Take a set x and take it's cardinality. Express this number in terms of power sets of the empty set. This would give a transitive set every time.

The only problem is if the set we start off with is in terms of power sets of the empty set. In this case, map this to the cardinality of it's set +1 and express that number in terms of power sets of the empty set.

Is this a good way of proving this?

2. Here's an easy solution:

Since x is a set and ON is a proper class, there exists an ordinal $\alpha$ with $\alpha \notin x$.

3. Wow, one line solutions usually make sense.

However, we haven't been lectured about ON, proper classes or ordinals.

Can I just ask what your ON actually is? I think i'll go and read about proper classes or ordinals, it might help me understand what you mean.

4. An ordinal is a set that is transitive and well-ordered. Here are some simple examples of ordinals:

$0=\emptyset$

$1=\{ \emptyset \} = \{ 0\}$

$2=\{ 0,1 \}$
...

$\omega = \{ 0,1,2,... \}$

$\omega +1= \{ 0,1,2,... ,\omega \}$

ON is the class of all ordinals.

ON is too big to be a set, thus it is a proper class.

The ordinals should be covered in a basic set theory course. They are fundamental.