Proving Transitivity of Ordinals and V_a Sets

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In summary, the homework statement is that every ordinal is a transitive set. The Attempt at a Solution provides an example of how to show that an ordinal is transitive. The Homework Equations provide an equation for how to find the cumulative hierarchy of an ordinal. The ordinal part of the equation is definitional, but the inclusion part is not. The summary states that every ordinal is a transitive set and that the cumulative hierarchy is transitive. The attempt at a solution provides an example of how to show that an ordinal is transitive. The last part of the summary states that if an ordinal is a limit ordinal, then V_\alpha=\bigcup_{\beta<
  • #1
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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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  • #2
What is your definition of an ordinal?? I ask this because many books define an ordinal to be transitive, while other books don't...
 
  • #3
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?
 
  • #4
Take [tex]B\in \alpha[/tex]. We must prove [tex]B\subseteq \alpha[/tex]. So take [tex]\beta \in B[/tex]. By definition of B, [tex]\beta<B<\alpha[/tex]. This means that [tex]\beta\in \alpha[/tex].
 
  • #5
micromass said:
Take [tex]B\in \alpha[/tex]. We must prove [tex]B\subseteq \alpha[/tex]. So take [tex]\beta \in B[/tex]. By definition of B, [tex]\beta<B<\alpha[/tex]. This means that [tex]\beta\in \alpha[/tex].

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

each ordinal [tex]\alpha[/tex]
is the set of all smaller ordinals i.e.
[tex]\alpha = {\beta : \beta<\alpha[/tex]
 
  • #6
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?


[tex]\gamma \in \beta[/tex]

[tex]\gamma <\beta<\alpha[/tex]

[tex] \gamma \in\beta\in\alpha[/tex]
[tex]\gamma\in\alpha[/tex]
[tex]\beta\subseteq \alpha[/tex]
 
  • #7
Yes, I think that would be correct!
 
  • #8
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?
 
  • #9
Try proving it by transfinite induction.This will be the easiest way:

So you need to show the following
- [tex]V_0[/tex] is transitive (this is easy)
- [tex]V_{\alpha+1}[/tex] is transitive for all [tex]\alpha[/tex]. Use here that [tex]V_{\alpha+1}=\mathcal{P}(V_\alpha)[/tex].
- [tex]V_\alpha[/tex] 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 [tex]\alpha[/tex] is a limit ordinal, then [tex]V_\alpha=\bigcup_{\beta<\alpha}{V_\beta}[/tex]. So take [tex]A\in V_\alpha[/tex]. Then there actually exists [tex]\beta<\alpha[/tex] such that [tex]A\in V_\beta[/tex]. 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
 

1. What is the definition of transitivity for ordinals and Va sets?

Transitivity is a property of sets that states that every element of the set is also a subset of the set. For ordinals, this means that every element of the ordinal is also an ordinal. For Va sets, this means that every element of the set is also a member of the set's transitive closure.

2. How do you prove the transitivity of an ordinal?

To prove the transitivity of an ordinal, you can use the induction principle. First, show that the base case, usually 0 or the empty set, is transitive. Then, assume that the ordinal up to n is transitive and use this assumption to prove that the ordinal at n+1 is also transitive.

3. What is the significance of proving transitivity for ordinals and Va sets?

The transitivity property is important in set theory as it allows us to define operations and relations on sets in a consistent way. In particular, transitivity is necessary for the construction of arithmetic on ordinals and for the definition of the cumulative hierarchy of sets.

4. What are some common techniques used to prove the transitivity of Va sets?

One common technique is to use the Axiom of Foundation to show that every element of the set has a well-defined rank, and then use this rank to prove that the set is transitive. Another technique is to use the Axiom of Regularity to show that every element has a minimal element, and then use this minimal element to prove transitivity.

5. Can a set be transitive but not an ordinal or a Va set?

Yes, it is possible for a set to be transitive without being an ordinal or a Va set. For example, the set of all even numbers is transitive, but it is not an ordinal or a Va set. This is because it does not satisfy the other properties that define ordinals and Va sets, such as being well-ordered or being the transitive closure of a set.

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