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Problem 1.45 (Boolean-valued ordinals) (J.Bell, boolean-valued models and indipendence proof, 3rd edition)

(i) Show that, for any formula φ(x), $ [[ ∃α .φ(α)]] = \bigvee_α [[φ( \hat{α})]]$ and $[[∀α.φ(α)]] = \bigwedge_α [[φ(\hat{α})]]$. Thus, quantifications over ordinals in $V^{(B)}$ can be replaced by suprema and infima in B over standard ordinals.

(ii) Show that the following conditions on $u ∈ V^{(B)}$ are equivalent:

(a) [[Ord(u)]] = 1;

(b) there is a set A of ordinals and a partition of unity {$a_ξ$: ξ ∈ A} in B such that $[[ u = \Sigma _{ξ∈A} a_ξ · \hat{ξ} ]]= 1 $.

someone can help me?

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I think that this was asked before on the site. – Asaf Karagila Dec 8 '12 at 21:03
At least something related:… – Asaf Karagila Dec 8 '12 at 21:19

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