The Principles of Mathematics (1903)

Post by johnbender » Mon, Jun 12, 2017, 10:22 PM UTC

hackinthebochs wrote: post #17

orangutango wrote: post #13

Earlier quoted context omitted.

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 mathema…

>My main point is I disagree with the view of mathematics as nothing more than some axiomatic program I just don't see how this follows from Godel. It gives us a more expansive view of math, but I don't see how any fundamental understanding is overturned. I don't see how this takes away from the connection between axioms and theorems. The characterization of math as discovering the logical consequences of axioms is j…

Cousin comment helped me out a bunch:

https://news.ycombinator.com/item?id=14540054

Post by yequalsx » Tue, Jun 13, 2017, 2:10 AM UTC

johnbender wrote: post #20

yequalsx wrote: post #19

Earlier quoted context omitted.

That is not quite correct. Some axioms systems are categorical. This means that there is only one model up to isomorphism. For instance, take the collection of all statements about the Natural numbers (using the standard model of them under the first order Peano Axioms). This collection forms an axiom system for the Natural numbers and all true statements are provable in this system. Indeed, every true statement is a…

I will try to translate my understanding: > This collection forms an axiom system for the Natural numbers and all true statements are provable in this system. Indeed, every true statement is an axiom. You have defined the axiom system as the set of all true statements about natural numbers so of course all true statements are provable! But crucially ... > The problem with doing this is that the collection of all true…

You're welcome. I think your understanding is correct.

I will add one more thing for completeness sake. There are statements that are true of the natural numbers, the natural numbers that you and I think of when see this term, that can't be proven in the first order theory. This means that there is a non-standard model in which there is a non-standard integer that is a counterexample to the statement. The second order axioms are categorical so there is a proof of such a statement using the second order axioms. One may not be able to find such a proof but there is one.

Re: The Principles of Mathematics (1903)

Re: The Principles of Mathematics (1903)