Proving Transitivity of Ordinals and V_a Sets

[gutnedawg]

Homework Statement

show every ordinal is a transitive set

show that every level V_a of the cumulative hierarchy is a transitive set

Homework Equations

The Attempt at a Solution

I understand that these are transitive sets, I'm just not sure how to show this. I feel like the ordinal part is just definitional.

 

[micromass]

What is your definition of an ordinal?? I ask this because many books define an ordinal to be transitive, while other books don't...

 

[gutnedawg]

my definition is each ordinal a is the set of all smaller ordinals, i.e. a={B: B<a}

this mean B ε a

I'm not sure how to get the inclusion, I mean I know that B is included in a but is this obvious or should I show this?

 

[micromass]

Take $$B\in \alpha$$. We must prove $$B\subseteq \alpha$$. So take $$\beta \in B$$. By definition of B, $$\beta<B<\alpha$$. This means that $$\beta\in \alpha$$.

 

[gutnedawg]

Take $$B\in \alpha$$. We must prove $$B\subseteq \alpha$$. So take $$\beta \in B$$. By definition of B, $$\beta<B<\alpha$$. This means that $$\beta\in \alpha$$.

I was in a hurry so I meant to type out beta instead of B

each ordinal $$\alpha$$
is the set of all smaller ordinals i.e.
$$\alpha = {\beta : \beta<\alpha$$

 

[gutnedawg]

for some reason Latex is giving me grief

for a gamma in beta we have gamma<beta<alpha and thus gamma in beta in alpha

then gamma is in alpha meaning that beta is contained in alpha

is this a sound demonstration?


$$\gamma \in \beta$$

$$\gamma <\beta<\alpha$$

$$\gamma \in\beta\in\alpha$$
$$\gamma\in\alpha$$
$$\beta\subseteq \alpha$$

 

[micromass]

Yes, I think that would be correct!

 

[gutnedawg]

alrighty, now for V_alpha

V_alpha={a : rk(a)<alpha}

let rk(V_beta)= beta for all beta<alpha then V_beta is in V_alpha for every beta<alpha by the definition of V_alpha

do I just do the same thing as I did above pick a gamma and solve?

 

[micromass]

Try proving it by transfinite induction.This will be the easiest way:

So you need to show the following
- $$V_0$$ is transitive (this is easy)
- $$V_{\alpha+1}$$ is transitive for all $$\alpha$$. Use here that $$V_{\alpha+1}=\mathcal{P}(V_\alpha)$$.
- $$V_\alpha$$ is transitive for all limit ordinals. This shouldn't be to difficult...

 

[gutnedawg]

alright well TI was one of my questions that I had so I'm glad you typed this out

-V_0 = the empty set which is transitive since y in V_0 is the empty set and the empty set is contained in the empty set

-V_a+1 I'm not sure, I know I have the power set in my notes I just can't find them right now...

-Not sure how to show this last step

 

[micromass]

If $$\alpha$$ is a limit ordinal, then $$V_\alpha=\bigcup_{\beta<\alpha}{V_\beta}$$. So take $$A\in V_\alpha$$. Then there actually exists $$\beta<\alpha$$ such that $$A\in V_\beta$$. Now apply the induction hypothesis...

 

[gutnedawg]

how do I apply the induction hypothesis?

 

[gutnedawg]

are you saying that since A in V_b and A in V_a then for all V_b in V_a
-> V_b is contained in V_a