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Transitive Sets

  • Thread starter gutnedawg
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  • #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.
 

Answers and Replies

  • #2
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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...
 
  • #3
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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
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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
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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
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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
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Yes, I think that would be correct!
 
  • #8
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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
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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
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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
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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
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how do I apply the induction hypothesis?
 
  • #13
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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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