To clarify what this question's about: First I state the title and some stuff I believe, then something which contradicts that and my personal feelings about this contradiction. Then I pose my question.
Full title which is too long to fit: Complete axiomatization of the real numbers vs We can't prove every statement about the real numbers vs Every statement about the real numbers is either true or false – how can these all be true/why do people say them?
How well do the formal axioms of real numbers actually capture our human informal knowledge of them? From an informal viewpoint, there are algorithms for addition, subtraction, multiplication, and division, and we expect statements about them (even those which are quantified) to be either True, or False, since they are exact statements, to the human mind anyways.
Take this above paragraph to be true. I don't think anyone will quarrel with it. Now add this, too:
People say that (and this is all in my experience) – that the axioms of an ordered field that is closed under taking LUBs of subsets is the complete axiom system for the reals from which we can derive all of our known statements about them. Otherwise, they wouldn't be the real number axioms, right? I mean what else would you expect of a complete axiomatization? Students trust teachers that any proof for a statement about real numbers can be made from these axioms; for if they cannot, it would be malicious to teach students a standard of proof that may both limit their discoveries and imagination by telling them that those axioms are all there is to know about the natural and real numbers.
Ok. Cool. I believe that most people would agree with all that. Some might question the last sentence, which I agree is the least well founded assumption. That's my argument though. Anyways, well then how on earth is this not a contraction to my own argument:
As per Goedel's Incompleteness theorems (or something – the details are largely irrelevant I believe) there are statements about N and R which cannot be proven from the axioms.
And since this theorem or whatever is supposed to be a mathematically rigorous result, I feel logically forced to conclude that teachers are just being malicious by purporting such proof systems, which as I said was the least well founded assumption in my argument. My heart tells me that this is wrong, and I hope it is, but hopefully you can understand my reasoning. There are many mathematicians who are far more capable than I and I trust that their methods of abstraction by axioms are superior and useful (and very much worthwhile to learn at any cost!), but I cannot understand why I as a student must follow the same footsteps. Critically, for me it appears, it all depends on weather our axioms for the real numbers can or actually cannot tell us everything there is to know about them, about which I am still unclear. Moreover, since math is so black and white and precise, appealing to the actual algorithms for the reals and meta-theorizing about what may be true about them ought to be considered proof.
Question: What critical information I am missing that people will claim all of the following:
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Axioms blah-blah-blah describe the real numbers totally and should be the foundation of all proofs involving them.
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Since it is math, and math is a precise subject, we know that there are always answers to things – True and False, to be sure.
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There are statements about the reals and the naturals that can't be proven from the axioms.
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Nothing of the items on this list contradict each other.
That concludes the asking of this question. Thank you very much for taking the time to read it!
Best Answer
Questions about axiomatizations really depend on what language we're talking about. Specifically, an axiomatization of the reals needs to prove all true statements in that particular language.
So, for instance, if we are restricted to the first order logic of the reals, a language which only allows:
There is a complete axiomatization. All our questions can be answered easily! (Mainly does this polynomial have a real root, but we can also talk about $<$, so we can ask a few questions about least upper bounds)
Contrast with the first order logic of the natural numbers, and you run into Gödel's incompleteness theorem. Generally speaking (and a bit counterintuitively!) real numbers are nice and decidable, and natural numbers invoke Gödel's incompleteness theorem.
So why are there statements about $\mathbb{R}$ which invoke Gödel's incompleteness theorem? Well, if we add in some operation that allows us to talk about integers! Say, for instance, $\sin(\pi x)$, which $=0$ exactly when $x$ is an integer.
As a result, we get Richardson's Theorem: https://en.wikipedia.org/wiki/Richardson%27s_theorem
No matter what computable axiom system we use for the reals with $\sin$, there is a function involving $x, \sin , +, -, *, \pi,$ for which it cannot be proved from the axioms whether or not the function has a root.