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The question asked here is, suppose we are given a a conjecture to prove in number theory (with numerical evidence showing its true). Say an important well studied conjecture that most will believe is true-though it may have remained unproven for more than 200 years or more (a long time). How do we know it is true that the axioms of the real number field are enough to prove that conjecture? One reason I'm asking is axioms are unproven statements which seems to be one possible reason to me that we maybe unable to prove a conjecture to be true. Is it possible that a new axiom could be discovered for the field of real numbers in the future? thus proving such conjectures described in this question and adding to the axioms which we feel are so far intuitively correct.

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    $\begingroup$ A short answer - we do not know in advance. It can happen that some statement is independent of particular logical system. I fell there exist such statements for the axioms of reals, but I don't know one for sure. $\endgroup$ – lisyarus Feb 4 '16 at 21:43
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    $\begingroup$ In fact, many conjectures have been found to be false. $\endgroup$ – ajotatxe Feb 4 '16 at 21:47
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    $\begingroup$ You prove it is true under the conditions that the axioms are true. You do not prove it is true in objective axiom free universe. And we don't know something is true in an axiom-free objective universe. .... psst I'll let you in on a little secret. the real numbers don't exist in the physical "real" world. Take parallel lines. Real space is warped; parallel lines don't exist. In math; Euclidean, Spherical and Hyperbolic system all exist equally validly but separately in different differently axiomed "universes". $\endgroup$ – fleablood Feb 4 '16 at 21:48
  • $\begingroup$ You ask this question as if the only doubt about our ability to "prove" a theorem is the fact that the axioms we use are unproven. In actual mathematics, there are many reasons we might be unable to prove a fact after seeing numerical evidence for it, but "the axioms might not be true" is not a reason for doubt. Questioning the axioms is a possible reason for some very smart person to invent a new kind of mathematics with a new set of axioms; it has happened before in some fields of math, though I can't say how it might have happened in number theory. $\endgroup$ – David K Feb 4 '16 at 22:22
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    $\begingroup$ Let me point out that you seem to be conflating two things: (a) whether or not the axioms are enough to prove a conjecture; and (b) whether or not we believe the axioms themselves. The existence of a proof hinges only on (a); question (b) determines whether such a proof establishes the conjecture's truth. The latter is a question more for mathematical philosophers than for mathematicians themselves, in my opinion, although there certainly is overlap. $\endgroup$ – Brian Tung Feb 4 '16 at 22:47
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Set aside the reals for the moment. As some of the comments have indicated, a distinction must be drawn between a statement being proven, and a statement being true. Unless an axiomatic system is inconsistent or does not reflect our understanding of truth, a statement that is proven has to be true, but the reverse is not necessarily the case.

For instance, Fermat's Last Theorem (FLT) wasn't proven until 1995. Until that moment, it remained conceivable that it would be shown to be undecidable: that is, neither FLT nor its negation could be proven within the prevailing axiomatic system (ZFC). (Such a possibility was especially compelling ever since Gödel showed that any sufficiently expressive system, as ZFC is, would have to contain such statements.) Nevertheless, that would make it true, in most people's eyes, because the existence of a counterexample in ordinary integers would make the negation of FLT provable. So statements can be true but unprovable.

Furthermore, once the proof of FLT was established, that proof was not relative to some absolute notion of truth, but only relative to the axiomatic system in which it was proven. One could imagine axiomatic systems in which FLT was false; that there were integers $x, y, z \geq 1$ and $n \geq 3$ such that $a^n+b^n = c^n$. However, we are generally not interested in such systems for ordinary arithmetic (as opposed to, say, modular arithmetic), since addition and multiplication would have to work differently from what we're used to.

In the field of geometry, it was long believed that the parallel postulate (viz., given any line ${\mathcal l}$ and any point $P$ not on $\mathcal l$, it is possible to draw exactly one line through $P$ that doesn't intersect $\mathcal l$) could be proven from the other axioms of Euclidean geometry. Various proofs were devised through the centuries that were later shown to have more or less subtle flaws. There had to be flaws, because it was eventually demonstrated that no such valid proof could be constructed.

One of the more interesting attempts at such a proof was made by Giovanni Saccheri (1667–1733). He attempted to prove the parallel postulate by contradiction: He assumed that something other than exactly one parallel line could be drawn (to be more precise, he assumed that the angles in a triangle added up to more or less than $180$ degrees, but the two statements were equivalent), and came up with a contradiction. That is to say, he came up with what he found to be so counter-intuitive that he felt it had to be a contradiction, published his results, and soon thereafter expired.

But in actual fact, the parallel postulate is undecidable in the sole presence of the other axioms, and therefore, either it, or some negation of it, may be asserted as an axiom. Note that we don't have to assert any other axiom. The remaining axioms of Euclidean geometry form a perfectly consistent system, which is occasionally called absolute geometry.

However, certain things that we like to know about Euclidean geometry rely on the parallel postulate, and so we have to follow Euclid and assert it as an axiom. We can also do the opposite and assert one of its negations: If we assert that there are no parallel lines (and also make a modification to another of the axioms), we get elliptical geometry; if we assert that there are many—in fact, infinitely many—parallel lines, we get hyperbolic geometry, which includes as some of its proven propositions the counter-intuitive results that Saccheri found so repugnant.

So, in short (tl;dr), we don't, in fact, know that our conjectures can always be proven (or disproven). Sometimes, neither can happen, and we have to assert something extra as an axiom to decide which.

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  • $\begingroup$ No, the answer doesn't really have anything specifically to do with ZFC (which stands for Zermelo-Fraenkel set theory plus the Axiom of Choice). In fact, one point of my (admittedly long-winded) answer is that it doesn't really matter what the axiomatic system is; the proof is relative to that system, and not to some Platonic truth. All you need to know about ZFC is that it's currently our consensus axiomatic basis of mathematics. Things that we prove are generally implicitly relative to ZFC (even those that don't mention ZFC at all). $\endgroup$ – Brian Tung Feb 4 '16 at 23:45
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We don't. In any consistent system of axioms strong enough to express the natural numbers there will be true theorems that are not provable within that system. The is due to Godel's incompleteness theorem.

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The idea behind axioms is that, like definitions, they set the framework of what you are working with. It makes no sense to prove a definition; it's completely made up! Of course it's not that simple with axioms, as they don't necessarily encompass all of what you are working with (in your case, real numbers). You can't necessarily prove everything about the real numbers with the field axioms (ordering is an example of something you can't prove with them), but you can do quite a lot. And that's what branches of math such as algebra try to do: see what you can do using certain sets of rules.

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