Of course God is unfalsifiable

God's existence cannot be proven false.  Since God is unfalsifiable by definition, being unfalsifiable is a property of God's existence.  Since the statement God is unfalsifiable and cannot be disproven is accepted and supported by scientists and Atheists alike, then Occam's Razor shows the simplest explanation to be valid, since a defined and logical statement with scientific and academic consensus is more likely to be true than an argument against it. 

Calculus cannot be proven to be false.  Calculus doesn't wait for some physical evidence to appear and disprove it since Calculus is already logically defined.  We can't prove Calculus exists as a physical thing, but we know Calculus is valid.  Calculus uses self supporting axioms based on the unfalsifiable laws of logic, where abstract theorems do not require physical testability to be valid.  There are many invalid proofs by amateurs who try to disprove Calculus based on a limited understanding of Calculus and a desire to prove Calculus wrong, yet it doesn't matter if you choose not to believe in Calculus or if you declare all of it invalid simply because you don't like it, as such arguments will not make mathematicians everywhere drop the study of Calculus because you don't think it has sound logic.  No matter how you argue, Calculus cannot be falsified. 

Replacing Calculus with God's existence shows the same properties and the same reasons why arguments against them do not work.  Something can be both logically valid and non-falsifiable.

>>21
Explicitly, Goedel's theorem says:
"Theorem VI. For every ω-consistent recursive class κ of FORMULAS there are recursive CLASS SIGNS r, such that neither v Gen r nor Neg(v Gen r) belongs to Flg(κ) (where v is the FREE VARIABLE of r)"

The common interpretation that "no system describing number theory can be both consistent and complete" is reasonably accurate when you're trying to understand what the theorem says. However, it's absolutely horrible if you're trying to apply it somewhere. Mathematical theorems are razor-sharp, and you shouldn't apply them in a more vague sense.

One of the conditions was ω-consistency of the system in question. A system is ω-inconsistent if for every formula A(x) such that A(n) is true for all natural numbers, the statement "for all x, A(x) is true" is false. A system is ω-consistent if it is not ω-inconsistent.

The natural numbers are certainly such a system, so Goedel's theorem applies. However, the real numbers fail this condition. For example, x^2 = 2 fails for every natural number, yet in the reals Sqrt(2) works. I can even weaken it to the rational numbers if I like: 2x = 3  fails for all natural numbers, but x = 3/2 is an object which satisfies it.

Goedel's theorem tells us that any system describing the natural numbers and *only* the natural numbers cannot be consistent and complete. It has no bearing on any sorts of weird(or straightforward) extensions of the naturals I might come up with.