Proving Transitivity of Ordinals and V_a Sets

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Homework Help Overview

The discussion revolves around proving that every ordinal is a transitive set and that each level V_a of the cumulative hierarchy is also a transitive set. Participants are exploring definitions and properties related to ordinals and transitive sets.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants are attempting to define ordinals and discuss their properties, particularly focusing on the transitive nature of ordinals and V_a sets. Questions about definitions and the implications of these definitions are raised, along with discussions on how to demonstrate inclusion and transitivity.

Discussion Status

Some participants are providing guidance on how to approach the proof, including suggestions for using transfinite induction. There is an ongoing exploration of the definitions and properties of ordinals and V_a sets, with various interpretations being discussed.

Contextual Notes

Participants mention issues with definitions varying across sources, and there are references to specific steps in proofs that are not fully articulated, indicating a need for further clarification on certain points.

gutnedawg
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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.
 
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What is your definition of an ordinal?? I ask this because many books define an ordinal to be transitive, while other books don't...
 
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?
 
Take B\in \alpha. We must prove B\subseteq \alpha. So take \beta \in B. By definition of B, \beta&lt;B&lt;\alpha. This means that \beta\in \alpha.
 
micromass said:
Take B\in \alpha. We must prove B\subseteq \alpha. So take \beta \in B. By definition of B, \beta&lt;B&lt;\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&lt;\alpha
 
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 &lt;\beta&lt;\alpha

\gamma \in\beta\in\alpha
\gamma\in\alpha
\beta\subseteq \alpha
 
Yes, I think that would be correct!
 
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?
 
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...
 
  • #10
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
 
  • #11
If \alpha is a limit ordinal, then V_\alpha=\bigcup_{\beta&lt;\alpha}{V_\beta}. So take A\in V_\alpha. Then there actually exists \beta&lt;\alpha such that A\in V_\beta. Now apply the induction hypothesis...
 
  • #12
how do I apply the induction hypothesis?
 
  • #13
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
 

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