I will start with the ontological argument.
Something that is impossible to exist, cannot exist by definition and so doesn't exist.
Something that is possible to exist has two possibilities, it exists or doesn't exist.
Something that exists necessarily, cannot but exist by definition and so it exists.
Descartes argument from what I understand unlike what is taught in Academia goes something like:
There are levels of existence.
The highest level type is necessary.
God is defined to be so great or perfect and so as far as this issue goes, it would be a necessary being.
If God is properly defined to be necessary, it follows it exists in the real world.
That is we can see by merely remembering God is necessary by the concept of necessary, that it exists.
Some proofs to the above.
God is life to the absolute to the extent there can't be more possible life/existence then it (by definition).
If any life/existence is possible without God (any independent aside from God is possible) in any possible world, then God (not a Creator or lesser god is meant here, but the big absolute being) is not possible.
God is possible.
Therefore any life/existence is impossible without God.
It follows then God exists.
In fact, it's easy to see:
If God exists, he would be a necessary being.
If a necessary being exists, it would be possible for us to recognize that as an aspect or trait of it.
The predicate contention doesn't make sense:
(1) It's a red herring if true since those categories exist anyway.
(2) A dependent existence is lower then an independent or necessary existence as far existence attribute goes.
The bold is purposeful and self-explanatory.
(3) It would make necessary existence incoherent but then the same can be said about impossible to exist, and both are coherent and are directly related to the issue of existence.
If a necessary being can be conceived, it definitely has to exist.
When we think of God not only is it a candidate for necessity in definition, but it's in fact impossible any other thing exists by necessity but it.
At the end, the only faith premise is: "God is possibly conceived to be possible". If this is true, then he will be proven to exist by reflecting over what absolute existence implies.
I will be discussing more proofs.