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I need to show that the Well Ordering principle which is $(\forall X)(\exists R) [(X, R )$ - a well ordered set$]$

is true in $H(\kappa)$ where

$H(\kappa)$ is hereditary set of $\kappa$ - infinite cardinal.

I know that Axiom of Choice is true in $H(\kappa)$, but not the Power Set Axiom, and I was also advised to use the following fact

$R$ is a well founded relation on X $\to$ $H(\kappa) \models $ “R is a well founded relation on X”

but I do not know how it helps, because in the original proof of WOP well roundedness is not used

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The idea is that you don't have to prove WOP inside $H(\kappa)$ from scratch--you can just deduce it in $H(\kappa)$ from the fact that it holds in $V$. Given a set $X\in H(\kappa)$, you want to prove there exists $R\in H(\kappa)$ such that $$H(\kappa)\vDash \text{"$R$ is a well ordering of $X$"}.$$

So, how do you find such an $R$? Well, you pick a well-ordering $R$ of $X$. Now you just have to prove that $R\in H(\kappa)$, and $H(\kappa)$ believes that $R$ really is a well-ordering of $X$. You should find the fact you were advised of useful in proving this.

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  • $\begingroup$ Thanks, this is the kind of the approach I was intending to, but the hard part for me is proving that $R$ is actually a well ordering in $H(k)$. The fact helps me to prove it is well-founded, but it says nothing about the partial order, which I also need to prove $\endgroup$ – tmac_balla Apr 20 '17 at 6:33
  • $\begingroup$ How do you prove that R is total and partially ordered in $H(\kappa)?$ $\endgroup$ – tmac_balla Apr 20 '17 at 22:42

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