As an atheist and a skeptic, I am personally driven to carefully consider any strong arguments for God's existence that may be brought forward, to continually re-evaluate my own position and verify that it remains correct. The two best-known classical arguments for God's existence are Aquinas' cosmological argument and Anselm's ontological argument.

The ontological argument has been rejected by luminaries such as Aquinas, Hume, Kant, Bertrand Russell and Hammiesink, but it has also been supported by an equally distinguished list of thinkers, including Descartes, Leibniz, Plantinga and even Kurt Gödel.

The ontological argument has been recently given a new suit of clothes by Gyula Klima. More than ever, it seems to me, this updated version confirms Bertrand Russell's observation that "it is easier to feel convinced that it must be fallacious than it is to find out precisely where the fallacy lies."

Here, for your consideration, is the modern ontological argument in its full formal dress. Please tell me precisely where the fallacy lies.

Definitions:

• A thought-object is anything that can be considered in the intellect. It need not exist in reality.
• M(x,y) is defined as “x and y are thought-objects and x can be thought to be greater than y.”
• I(x) is defined as “x is a thought-object which is only in the intellect.”
• R(x) is defined as “x is a thought-object which can be thought to exist in reality.”

Axioms:

• (1) ~∃y[M(y,g)]
  God (g) is defined as the thought-object for which no thought-object can be thought to be greater.

• (2) I(x)∧R(y)→M(y,x)
  If x is a thought-object which is only in the intellect and y is a thought-object which can be thought to exist in reality, then y can be thought to be greater than x.

• (3) R(g)
  God can be thought to exist in reality.

Derivation:

Suppose I(g). Then:

• (i) I(g)∧R(g)
  (conjunction introduction from 3 and postulate.)

• (ii) M(g,g)
  (modus ponens from 2 and i)

• (iii) ∃y[M(y,g)]
  (existential introduction from ii)

• (iv) ~∃y[M(y,g) ]
  (restatement of 1)

Since iii and iv are a contradiction, ~I(g).

Conclusion:

~I(g) and R(g) together show that God exists. If it is false that the God exists only in the intellect, this means either that God does not exist at all, even in the intellect; or that God exists not just in the intellect but also in reality. But if God can be thought to exist in reality, then God does exist in the intellect. Therefore, God exists not just in the intellect but also in reality.

EDIT: In axiom 2, for clarity I changed "y is greater than x" to "y can be thought to be greater than x." This was already stated in the definition of R and could be understood from the predicate formulation of the axiom, but was not fully spelled out in the English-language version.

EDIT 2: I have to go to bed now. I will respond to comments more tomorrow.