Godel's workarounds to the ontological argument

This is part of my series on debugging the ontological argument.

In the previous post, I discussed Plantinga's modal ontological argument, which is about the simplest modal ontological argument you can get.  A simply-stated argument is quite valuable, but I believe that as a matter of technical feat, Plantinga's argument was surpassed by Gödel's Ontological argument (GOA).

This series does not focus on history, but I will dryly summarize a few established facts.  Gödel formulated the argument in 1941, but only told people about it in 1970.  Gödel did not actually believe that the argument proved God, and was afraid that publishing it would send that message.  He was more interested in it from the perspective of logical investigation (which is my interest as well).  It was published posthumously in 1987, which as far as I know predates Plantinga's argument.

Long story short, I've gotten several people who said stuff like (actual quote), "The first premise you must accept is that Godel was almost certainly much, much, smarter than you."  And besides being the worst kind of argument from authority it's also historically wrong about Gödel's motivations.

But it is true, after all, that Gödel was very smart, and I find the GOA to be quite clever in ways that are not immediately obvious.  This post will focus on several things that the GOA got right.

1. Possibility is proven, not assumed

In the conceptual ontological argument, we had difficulty with the premise that God was "conceivable" in the sense required by the proof.  In the modal ontological argument, we had difficulty with the premise that God was "possible" in the sense required by the proof.  In the GOA, we don't need to assume that God is possible.  Instead, we prove it from more basic premises.

The basic structure of the GOA is as follows:
1. We define what is a positive property.
2. Using the definition of positive properties, we prove that positive properties are always consistent.
3. We prove that consistent properties are always possible.
4. The property of being God is taken to be positive.
5. Therefore God is possible.
6. Therefore God is necessary (this part of the proof is identical to Plantinga's argument).

If you're interested in the particulars, I wrote out the entire ontological argument years ago.  The goal in this series is to explain the steps in more depth rather than detail.

2. Existence is a predicate

Earlier in this series, I wasted much time with an in depth discussion of whether existence is a predicate. In a first analysis, it cannot be a predicate, or at least not a very useful one.  In later analysis, I note that existence can be a predicate after all, under certain conditions.  One possible condition is if we use second order predicate logic (SOPL), and that is what the GOA does.

In more detail, the GOA defines necessary existence as follows:
$$E(x) \Rightarrow \forall\phi~ (\phi~ess~x \Rightarrow \square \exists y~ \phi(y) )\tag{1}\label{ref1}$$
In plain language, necessary existence (E) means that in all (accessible) possible worlds, there exists an object with all exactly the same properties.  This requires SOPL since it applies a quantifier to a predicate ($\phi$).  In my opinion, this completely answers Kant's objection to the ontological argument.

You might notice that \ref{ref1} includes the phrase "$\phi~ess~x$" which means "$\phi$ is the essence of x".  This simply means that $\phi$ is the conjunction of all properties of x.

3. God's definition is no longer fluff

Earlier in the series, I complained that many version of the ontological argument have completely pointless augmentations.  They define God as a maximally great being, or a being with all the perfections, and this uses far more assumptions than is necessary.

In the GOA, God is defined as a being which has the conjunction of all positive properties, but this is actually a crucial component of the proof!  The entirety of the proof hinges on the definition of "positive properties", and there isn't an obvious way to reformulate the argument without it.

In fact, it almost seems too powerful.  One of the objections to the GOA is that God would even have positive properties that describe the world that God lives in, rather than God himself.  For instance, God would have the property that "trivialknot likes cake."  And since God necessarily exists, God would have this same property in all possible worlds, implying that I like cake in all possible worlds.

Because Gödel cleverly addressed so many of the standard problems with other ontological arguments, I will have to devote several more posts to the particular flaws in his argument.