"[A]ll pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles ..."
All pure math can be deducible from axioms doesn't mean that all math can be deducible from a single set of axioms. Rather, it means that for each mathematical proposition there is a set of axioms from which one can deduce it.
For any set of axioms you take, if they are consistent then it is incomplete. You can enhance the axiom set to extend its reach, but an adversary can always find true, unprovable statements.
My main point is I disagree with the view of mathematics as nothing more than some axiomatic program-- in 1903 many were hopeful that a system (like Russell's formal logic in Principia) could simply generate the truths of mathematics. Gödel shattered that dream.
Incompleteness means there are true statements that can't be proven. Given that any standard set of "fundamental logical concepts" is probably sound and as long as "all pure mathematics" means "that which can be proven" then there's nothing wrong with saying that "all its propositions are deducible" from those principles.
I don't think this is a correct view of what the Incompleteness Theorem says. The Incompleteness Theorem says that there are statements that are true in the standard model of the integers that are not provable in the first order Peano axioms. This does not mean that such statements are not provable. They just aren't provable in the first order axioms. The second order axioms are categorical and this means using the second order axioms any true statement can be proven.
The second order Peano axioms are a superset of the first order axioms. There is one small change in one of the axioms and that is the difference between the two systems.
hackinthebochs|8 years ago
orangutango|8 years ago
My main point is I disagree with the view of mathematics as nothing more than some axiomatic program-- in 1903 many were hopeful that a system (like Russell's formal logic in Principia) could simply generate the truths of mathematics. Gödel shattered that dream.
unknown|8 years ago
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unknown|8 years ago
[deleted]
johnbender|8 years ago
yequalsx|8 years ago
The second order Peano axioms are a superset of the first order axioms. There is one small change in one of the axioms and that is the difference between the two systems.