Plantiga's ontological argument.

[toro]

Now axiom S5 can be derived from the proposition "Possibly P, implies necessarily possibly P".

The argument that it's circular doesn't seem that convincing because it's arguing that if it's possible, then it's necessarily based on this axiom, but it's not simply stating it's necessarily.

Now the controversial axiom seems to be if God is possible in a possible world P. But I don't see how this is a disputed premise, how can it be that God is not possible in any possible world?

The premise is disputed because it is an assumption with no validity that affects the direction of the argument. One can equally presuppose God doesn't exist in some possible world. This leads to two premises one can make:

1. God exists in a possible world P1.
or...
2. God doesn't exist in a possible world P2.

There is no reason one should presuppose one or the other.

If we follow the logic of premise 1 (Prem:1), we have the above

1. A being is defined as one that if it exists, exists necessarily. (Premise)
2. It is possible that there is a being that exists necessarily. (Prem:1)
3. Therefore, there is a world P1 in which this being exists. (from 2)
4. Therefore, there is a world P1 in which it is necessarily true that this being exists (from 2, 3)
5. Therefore, it is necessarily true that this being exists. (from S5 and 4)
6. Therefore, since necessarily true propositions are true, this being exists (from 5)

However, we have no reason that has yet been stated to choose between the initial proposal that God exists in one world or God doesn't exist in another before we follow the logic.

If we propose premise 2 (Prem:2), we get:

1. A being is defined as one that if it exists, exists necessarily. (Premise)
2. It is possible that there is not a being that exists necessarily. (Prem:2)
3. Therefore, there is a world P2 in which this being (that must exists necessarily) doesn't exists. (from 2)
4. For this being to exist in another world P3, its existence must be necessary in P3. (from 1).
5. This being cannot exist necessarily in P3, as its non-existence in P2 precludes it from existing necessarily in all other worlds. (from 4, 3)
6. Therefore, it is necessarily true that this being does not exist. (from s5 and 5)
7. Therefore, since necessarily true propositions are true, this being does not exist. (from 5)

What we see is that our initial assumption of possibility determines the outcome of the argument. So either we are left not knowing the possibility, and therefore not being able to say anything further. Or, we are left with defining a priori what the outcome is going to be by defining our assumption.

If we do the former, we learn nothing: "God either possibly exists, or possibly doesn't exist."

If we do the latter, we are answering our own question: "God exists. Therefore, God exists." "God doesn't exist. Therefore, God doesn't exist."

... to try and put it sucinctly anyway.

There are a number of other issues with the argument, but I think this one is the most obvious and it is one which explains the uncertainty both theological sides have about its validity. (I don't mean to be appealing to authority here, just pointing out an observation I've had.)

The conclusion doesn't seem to follow then. However it can be argued a necessary being is possible in a world p. What is meant by necessary is not simply necessary for that world, but necessary in all worlds. But to argue that as possibly true, you have to know that God is necessary in this world and necessary for all possible worlds. So it seems rather circular to do so.

I don't see the definition as particularly problematic. I mean, the definition does nothing more than define what you're talking about. The problem is that the circularity of the argument comes from the assumption of possible existence over possible non-existence.