Godel's Ontological Argument is expressed symbolically as:
For those unfamiliar with modal-logic, there is an article on the general Ontological Argument here.
With respect to the theorem's axioms, WikiPedia tells us the following:
For debate:WikiPedia wrote:We first assume the following axiom:
Axiom 1: It is possible to single out positive properties from among all properties. Gdel defines a positive property rather vaguely: "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)... It may also mean pure attribution as opposed to privation (or containing privation)." (Gdel 1995)
We then assume that the following three conditions hold for all positive properties (which can be summarized by saying "the positive properties form a principal ultrafilter"):
Axiom 2: If P is positive and P entails Q, then Q is positive.
Axiom 3: If P1, P2, P3, ..., Pn are positive properties, then the property (P1 AND P2 AND P3 ... AND Pn) is positive as well.
Axiom 4: If P is a property, then either P or its negation is positive, but not both.
Finally, we assume:
Axiom 5: Necessary existence is a positive property (Pos(NE)). This mirrors the key assumption in Anselm's argument.
Now we define a new property G: if x is an object in some possible world, then G(x) is true if and only if P(x) is true in that same world for all positive properties P. G is called the "God-like" property. An object x that has the God-like property is called God.
-Is the Ontological Theorem logically valid?
-Are all the axioms of the theorem valid?
-Can the argument hold without the axioms being valid, if they are not necessarily so?
