Consistency of real number algebra

1. Jun 8, 2005

C0nfused

Hi everybody,
I have recently read some things about what consistency of a system of axioms is and it really seems an important matter to me. So I would like to ask 2 things:

1)Have we proved the conistency of the real number algebra? I have read that some of the axioms of ZF-Set Theory have been proved to be consistent but is it necessary to connect arithmetic with Set Theory?

2)What knowledge is needed in order to get to understand such kind of proofs? Is this a topic of Mathematical Logic?

Thanks

2. Jun 8, 2005

honestrosewater

2) Yes. (Well, since I count logic, set theory, and other such foundational subjects under Mathematical Logic.) How far have you gotten with these?

3. Jun 8, 2005

chronon

Godel tells us that it's impossible to prove the consistency of the axioms of arithmetic from within those axioms - in fact he proved that if such a system of axioms could prove its own consistency then it would be inconsistent. So in the end there's no point in looking for formal proofs of the consistency of arithmetic or set theory (how would you know the axioms used for the formal proof were consistent?)

Some axioms of set theory such as the axiom of choice and the continuum hypothesis have been shown to be relatively consistent - that is if the other axioms are consistent then adding the new axioms won't make the system inconsistent.

4. Jun 8, 2005

Hurkyl

Staff Emeritus
There is a point in proving one theory consistent relative to another. E.G. one can prove that if set theory is consistent, then the theory of real numbers is consistent.

I believe that when left unqualified, people usually mean "consistent relative to ZF" when they speak about consistency.

Some mathematical logic would probably be useful, not necessarily for understanding the proofs, but for what the proofs mean. The proof that the theory of the reals is consistent doesn't need any fancy logic. In fact, you may have even seen it already -- the proof is to construct a model of the reals from the rational numbers, by defining a real number to be a Dedekind cut of the rationals. (Or, maybe, an equivalence class of Cauchy sequences)

5. Jun 8, 2005

matt grime

Really? Cos I have no idea about consistency things.

6. Jun 8, 2005

honestrosewater

Eh, actually I decided to search and couldn't find them either. I thought some were in the prove addition thread. When people are talking about Gödel or the continuum hypothesis and ZF and ZFC. Maybe that geometry was consistent? Maybe I was thinking of Hurkyl. You people say a lot of things.

7. Jun 8, 2005

Staff Emeritus
Any system of axioms that can derive arithmetic can be mapped into itself by Goedel's procedure and proved incomplete. On the other hand, geometry has been proven complete, including a completeness axion, so the real number system apart from arithmetic appears to be complete. This can be extended to measure theory.

8. Jun 8, 2005

Hurkyl

Staff Emeritus
Correct, if you say integer arithmetic.

It's quite perplexing at first, one cannot recover the theory of integer arithmetic given the theory of the real arithmetic.

The trick is that the theory of integer arithmetic has one very important thing that is overlooked -- integers. There's no way, using just real arithmetic, to define what is an integer, and what is not an integer.

(Though you could if you had an appropriate induction axiom)

So, in particular, there's no way to tell if a system of equations has an integer solution, because there's no way to tell if an arbitrary real number is an integer.

Maybe -- I find this topic very interesting, and I like to talk!

9. Jun 8, 2005

mathwonk

well i know nothing at all about thsi topic but i also like to talk. so to me this problem boils down the the existence of a model for the reals. so it boils down for me to whether i believe infinite decimals exist, and whether i believe the arguments used to prove they satisfy the axioms for the reals are valid.

i have been through these arguments in great detail with a junior high and high school class at a private school here in atlanta, and it went ok, modulo the usual mysterious proofs by contradiction.

10. Jun 8, 2005

honestrosewater

So, if someone wants to talk about it, the axiom of infinity jumps out immediately. What does it do to ZF with respect to completeness? I haven't done anything with it that I can think of, but it gives you the set of integers, yes? And union and intersection give addition and multiplication. So is ZF doomed?

11. Jun 8, 2005

Hurkyl

Staff Emeritus
ZF is indeed not complete.

Incidentally, removing axioms cannot make an incomplete theory complete.

12. Jun 8, 2005