Therefore the statement: "I am a liar" is a proposition which cannot be proven in any propositional formal logic, it can only be negated (inverted).
No, the statement "I am a liar" is not problematic. What you want is "this statement is false." That one is where the issue lies.
Well, I guess that's the same thing if you define "liar" to mean "someone who always and only makes false statements." But that's not the normal definition of "liar," and the point is more clear if you just refer to the statement directly.
Therefore it is a tautology that no formal logic system is complete.
It's not a tautology, because it only applies to systems of formal logic sufficiently powerful to encode the statement "this statement is false." I.e., you need to be able to make statements about the truth value of other statements in the system. Systems that are too weak to do that don't suffer from this flaw - they can be shown to be consistent and complete.
Also, even for strong systems, your statement isn't true as written. I can most certainly construct a formal logic system that is complete by simply adding the required statements ("this statement is false," etc.) to it. The problem I'll encounter if I do that - and this is the meat of Godel's theorem - is that the resulting system will be inconsistent. I'll be able to both prove
and disprove any statement I can come up with.
Godel's theorem goes like this: any system of formal logic powerful enough to model the natural numbers cannot be both consistent and complete.
Note that there exist interesting systems below the power threshold in question. Euclidean geometry is known to be both consistent and complete. To the thread title: Boolean algebra is not subject to Godel, and has been proven to be complete.
And you can always construct a system of formal logic that escapes Godel and still proves any first-order statement you want it to, simply by making the axioms an exhaustive list of all first-order properties you want to be true. Granted, that's kind of trivial (and bloated), but there it is.
just my opinion of course.
