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?