Consistency of real number algebra

[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

 

[honestrosewater]

1) I've seen matt grime talk about this several times, so a search might turn up your answer.
2) Yes. (Well, since I count logic, set theory, and other such foundational subjects under Mathematical Logic.) How far have you gotten with these?

 

[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.

 

[Hurkyl]

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)

 

[matt grime]

1) I've seen matt grime talk about this several times, so a search might turn up your answer.
2) Yes. (Well, since I count logic, set theory, and other such foundational subjects under Mathematical Logic.) How far have you gotten with these?

Really? Cos I have no idea about consistency things.

 

[honestrosewater]

Really? Cos I have no idea about consistency things.

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.

 

[selfAdjoint]

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.

 

[Hurkyl]

Any system of axioms that can derive arithmetic can be mapped into itself by Goedel's procedure and proved incomplete.

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 was thinking of Hurkyl.

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

 

[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.

 

[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?

 

[Hurkyl]

ZF is indeed not complete.

Incidentally, removing axioms cannot make an incomplete theory complete.

 

[C0nfused]

Thanks for your interest and your answers.

I noticed that consistency and completeness are mentioned in such a way that they seem to be similar things. As far as I know they are not at all referring to the same thing. I haven't studied Godel's work (simply because I lack much much knowledge) but I have heard of his incompleteness theorems. The second is the one that chronon states-I didn't know that but checked in mathworld and found that completeness and consistency of arithmetic can be proved using transfinite induction
http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html

I am not sure i understand any of it though. However, I noticed that the first theorem states that if there is a consistent axiomatic system of arithmetic then it includes undecidable propositions. The second states that we cannot prove consistency within a system. So how do we know that the consistent system to which the first one refers really exists? Generally, is there any consistent system, or have we proved that at least one such system exists? Eucleidian Geometry must be consistent but we cannot find a proof of this because of Godel's theorem?

And the axiom of infinity is so important for ZF? I have heard much about this axiom but not why it causes so many problems!

Sorry for contributing to the discussion only with questions and not with answers but I am really confused with these things, maybe because I haven't seriously studied these topics.(or maybe because my mind refuses to understand any of these things!)