Axioms 1 and 2 jointly entail that any positive property is consistent. For a property is
inconsistent just in case it entails its own negation. Per Axiom 1, if a property is positive,
its negation is not positive. But per Axiom 2, if a property is positive, it entails only
positive properties. So no positive property entails its own negation.
If every positive property is consistent, and being divine is positive, being divine is
consistent. It is necessarily so per A. 4. We can confirm this another way: being divine is
having all and only positive properties essentially. But if positive properties entail only
positive properties (A. 2), and no negation of any positive property is positive (A. 1), no
positive property entails the negation of any positive property. But then the set of all
positive properties is consistent; none of its members entails the negation of any of its
members.^10 Suppose now that if being divine is consistent, it is instanced in some
possible world. Then given what we've argued so far, there is in some possible world a
necessarily existent necessarily divine being: that is, it is possibly necessary that “a divine
being exists” is true. Given this and the Brouwer axiom, it follows that a divine being
exists.
Gödel's argument faces two basic questions. One is whether there is a con
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tentful, theologically appropriate gloss of “positive” on which the axioms are true. The
other is whether there is a sort of possibility such that (a) a concept's being syntactically
consistent entails that it is possible in that sense that it be instanced, and (b) the Brouwer
axiom is true for that sort of possibility and necessity.
The answer to the first question is yes. Talk of God as a perfect being is certainly
appropriate theologically, and perfect being theology has been the main tool to give
content to the concept of God philosophically almost as long as there has been
philosophical theology. And on Anderson's gloss, the axioms come out true.
Anderson's gloss validates Axiom 1. Suppose that a property F is positive. Then by
Anderson's gloss, if A lacks F, A is imperfect. If A has not-F, A lacks F. So if A has not-
F, A is imperfect, and so not-F is not compatible with perfection, and so not positive.
Anderson's gloss validates Axiom 2. On Anderson's gloss, if a property is not positive,
either it is not necessary for or it is not compatible with perfection. If having a property F
entails having some property that is not compatible with perfection, having F is not
compatible with perfection—and so any property that entails something for this reason
nonpositive is itself nonpositive. If a property entails a property not necessary for
perfection, it entails a property a divine being can lack. Any property a divine being can
lack is not part of its essence. A divine being's essence includes or entails whatever
properties it has necessarily (D. 2); so any property a divine being can lack is contingent.
But only properties had contingently entail the having of contingent properties. So any
property that entails a property not necessary for perfection is itself contingent and not
part of a divine being's essence. But a divine being's essence includes all positive
properties (D. 1). So any property entailing a property that is not positive in this second
way is itself not positive. Axiom 3 seems patent, for given D. 1, being divine amounts to
a conjunction of all positive properties, and it's hard to see how such a conjunction could
fail to be positive. As to Axiom 4, on Anderson's gloss, a property's being positive
consists in two facts about property-entailment. It's plausible that properties entail what