Consequences of Godel's Theorems

Galteeth

Taken to an extreme, do Godel's incompleteness theorems imply that the consistent mathematics we know (i.e, 2+2=4) can not encode all of reality? That certain aspects of reality do not obey conventional mathematics?

Preno

No .

Galteeth

Care to elaborate?

blkqi

In particular, Godel constructs a true statement of reality for which no sufficiently powerful formal system can account for. It's construction is simply for the reason of stumping formal systems.

But you should know something else: I am a liar.

Hurkyl

This is almost trivially false. Consider the formal system consisting of a single axiom: the statement so constructed. (And Godel's theorems has nothing to do with "reality")

blkqi

Pay attention: A formal system of a single axiom would not qualify as sufficiently powerful! The system should capture at least the notions of the Peano axioms.

blkqi

Reality in this context would refer to what we accept as common knowledge. E.g. clearly Godel's statement is a true statement. You agree that I agree that you agree... etc. So we have some sort of "reality" here.

Hurkyl

Do I need to spell out a formal system that captures Peano's axioms and proves whatever particular statement you are considering?

Hurkyl

No that is not clear. Such clarity would contradict Gödel's theorem which says that, roughly speaking, every "clearly true" statement of any first order formal system is provable.

(the precise meaning of "clearly true" here is that it is true in every interpretation of the formal system)

blkqi

OK I misunderstood you. You can include Godel's statement in your axioms, awkward as it may be. But the Godel method can be applied to this system as well. You can add an infinite number of Godel's statements to your system but the method will never fail.

blkqi

But completeness reflects exactly a system's ability to prove as theorems every "clearly true" statement. The statement G:="G is not a theorem of system X" is true if it is not provable in system X. But it is false if it is provable in system X. A dichotomy: consistency or completeness?

Hurkyl

It will fail if I add statements in a way that isn't recursively enumerable.

But that's not the point. That Gödel constructs an "unprovable truth" is a myth -- a misinterpretation of what the theorem states. Or, possibly, a biased interpretation of what the theorem states.

For a (sufficiently strong, but not too strong) formal theory, Gödel constructs a statement that can not be proven within that theory.

Whether that statement is true or false, something in-between, or even whether it's meaningful to ask that question is a matter of semantics -- of interpretation.

One thing's for certain: there exists a set-theoretic model of the theory in which the statement is false. (also there also exists a set-theoretic model of the theory in which the statement is true)

HallsofIvy

Godel still isn't saying anything about "reality", even in terms of "common knowledge" (which, I think, you would have difficulty defining).

Godel says that in any consistent axiom system, strong enough to include the natural numbers, there must exist a statement that can neither be proved nor disproved.

Galteeth

This is what I am asking. At it's bare bones, isn't conventional math an axiomatic system?

tauon

godel's theorems are about decidability in certain formal systems.
they have nothing to do with "encoding reality"... whatever that means.