The maximum principle in forcing and the axiom of choice

In this paper we prove that the maximum principle in forcing is equivalent to the axiom of choice. We also look at some specific partial orders in the basic Cohen model. Lately we have been thinking about forcing over models of set theory which do not satisfy the axiom of choice (see Miller [8, 9]). One of the first uses of the axiom of choice in forcing is: Maximum Principle p ∃x θ(x) iff there exists a name τ p θ(τ). Recall some definitions. For a partial order P = (P,E) and p, q ∈ P we say that p and q are compatible iff there exists an r ∈ P with rEp and rE q. Otherwise p and q are incompatible. A subset A ⊆ P is an antichain iff any two distinct elements of A are incompatible. It is maximal iff every p ∈ P is compatible with some q ∈ A. The standard definition of p ∃x θ(x) is given by: p ∃x θ(x) iff ∀q E p ∃r E q ∃τ r θ(τ) here p, q, r range over P and τ is a P-name. The usual proof of the maximum principle is to choose a maximal antichain A beneath p of such r and then choose names (τr : r ∈ A) such that r θ(τr) for each r ∈ A. Finally name τ is constructed from (τr : r ∈ A) in an argument which does not use the axiom of choice. For details the reader is referred to Kunen [7] page 226, who calls it the Maximal Principle. Shelah [11] and Bartosyznski-Judah [1] refer to the maximum principle as the “Existential Completeness Lemma”. Takeuti-Zaring [13] use “Maximum Principle” to title their Chapter 16.