0
$\begingroup$

I am trying to learn set theory from a book, Elements of Set Theory by Herbert Enderton. His book has some practice exercises and I was hoping you could help me determine if I understand them and, if not, where I went wrong.

  1. Define the rank of a set c to be the least "a" such that c ⊆ Va (where Va is the general notation he uses for each iteration in a hierarchy of sets). Compute the rank of {{∅}}. Compute the rank of {∅,{∅},{∅,{∅}}}

Answer: V0 = ∅. V1 = V0 ⋃ P(V0) = {∅,{∅}}. V2 = V1 ⋃ P(V1) = {∅,{∅},{{∅}},{∅,{∅}}}. Therefore, {{∅}} is rank 1 and {∅,{∅},{∅,{∅}}} is rank 2.

  1. We (the author) have stated that Va+1 = A ⋃ P(Va) (where A = ∅). Prove this for at least a < 3.

Answer: I'm not sure how to prove this except using the kind of expansion I tried to do above??

Please let me know if I have misunderstood, the text is fairly concise and well-written but not simple to grasp when trying to learn it on my own. Thank you!

$\endgroup$

1 Answer 1

Reset to default
0
$\begingroup$

You should prove by induction that for all natural numbers $\alpha$, $V_\alpha \subseteq P(V_\alpha)$.

For $\alpha = 0$, this is trivial since $V_\alpha = \emptyset$.

For $\alpha = \beta + 1$, we see that $V_\alpha = V_\beta \cup P(V_\beta) = P(V_\beta)$. Since $V_\beta \subseteq V_\alpha$, we have $V_\alpha = P(V_\beta) \subseteq P(V_\alpha)$.

Note that you can extend this to all ordinals $\alpha$ by doing ordinal induction. We need only add the following clause:

For $\alpha$ a limit ordinal, we see that $V_\alpha = \bigcup\limits_{\beta < \alpha} V_\beta \subseteq \bigcup\limits_{\beta < \alpha} P(V_\beta) \subseteq P(\bigcup\limits_{\beta < \alpha} V_\beta) = P(V_\alpha)$.

$\endgroup$

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.