[roywiggins]

> Godel proved that any formal system of logic can be shown to be logically inconsistent at at least one point.

He proved they can be inconsistent, or incomplete. The ones that mathematicians work with are incomplete and assumed to be consistent[0], namely that there are statements which are true, but not provable, within that system. Consistent-but-incomplete systems don't have any contradictions or logical holes; they just can't determine the truth of every possible statement.

From an incomplete system you can build a "more powerful" system- one with another axiom that makes more things provable without contradicting anything in the original one.

In inconsistent systems, everything is provable, even statements' own negations, so they aren't very useful.

[0] one statement that you can't prove within a (sufficiently powerful) system is "this system is consistent."