Does a mathematical object that does not contradict itself have to exist?

Question

I have recently finished the chapter on constructing the real numbers in my Analysis textbook (via Dedekind cuts). At first the natural numbers, then the whole numbers and the rational numbers were constructed and after that we used Dedekind cuts to prove the existence of the reals.
However what suddenly struck me is that we never constructed a specific set. From what I could see we always just proved that something exists via the fact that it does not contradict itself. For example, when proving that the Dedekind cuts form a field, I do not actually prove the existence of the field but only that the objects being Dedekind cuts and the objects being the elements of a field are properties that "play along".
So given a set of properties which are consistent among themselves and consistent with our system, say ZFC, is the set of mathematical objects that fulfill such properties always non-empty?
So to rephrase it: If one can prove that a set of properties formulated using our axiomatic system does neither contradict itself nor the system, does an object that has this set of properties have to exist?
By "exist" I mean that there is an actual mathematical object that can be constructed with these properties. We for example know that the existence of a vector space does not contradict ZFC, but do we automatically know from that, that there is also a concrete vector space?

Answer 1

In short, I will try to explain to you why I think you misunderstood your book, and are mistaken about your understanding of « non-contradiction » and « existence ».

When you say

From what I could see we always just proved that something exists via the fact that it does not contradict itself.

and

We for example know that the existence of a vector space does not contradict ZFC

you are wrong. If ZFC is contradictory, then the sentence « there exists a set that is a vector space » contradicts ZFC. And we do not know if ZFC is contradictory or not. Moreover, the book is highly unlikely to contain a proof that something doesn’t contradict ZFC, because this would be a proof of the consistency of ZFC, which would, in turn, from Gödel’s theorem, prove us that ZFC is contradictory, and this would be breaking news.

Now, turning to the set of real numbers and all this, let me first define the following abbreviations: we say that a mathematical object is N-like if it has all the properties that are axiomatically required to define the set of natural numbers (to have a zero, an addition, a multiplication, satisfying some properties); a mathematical object is said to be Z-like if it has all the properties axiomatically required to define the set of integers (an abelian group generated by a unique element of infinite order). Similarly, we define Q-like and R-like objects for « rational » and « real ».

What the book actually does is something else than proving « if N exists, then Z exists » (which is kind of a silly sentence): it provides several recipes:

• one that allows us to make a Z-like object from an N-like object
• one that allows us to make a Q-like object from a Z-like object
• one that allows us to make an R-like object from a Q-like object.

Note that this does not assume existence of anything anywhere! In the same way, I can give you a recipe of dark matter ice cream without having to prove you that dark matter actually exists.

Also note that I never spoke about what integers, rational numbers and real numbers are. And I did not speak about « the » set of natural numbers, « the » set of integers, etc. Inside ZFC, the axioms allow us to prove that some particular set is N-like, so you could call this « the » set of the integers. Now, one can, and usually does, call « the » set of integers the Z-object resulting from the first recipe, but there is no obligation to do so. Similarly, one call « the » set of rational numbers the thing obtained after following the second recipe. But historically, there are two recipes to build an R-like thing from a Q-like thing: Dedekind cuts and Cauchy sequences. So, is a real number a Cauchy sequence or a Dedekind cut? This question does not bother the vast majority of mathematicians, but some people working in topos theory (some very specific area of logic) do bother.

Turning now to the general question you ask, Mauro Allegranza wrote in his or her comment that a technical answer is given by the model existence theorem. See also the wiki page. This is a correct answer to the question you are asking, but what I just tried to show you is that the general question you asked has more or less nothing to no with the the construction of sets of numbers.

Answer 2

we used Dedekind cuts with the rational numbers to prove the existence of the reals

No, we define "Dedekind cuts" and their operations, then we rather prove that these form a field that we then call the field of real numbers (hoping I have not simplified too much, anyway the point is): the definition already brings those objects into existence.

How many are there? That is not granted, though I suppose it is trivial to prove that the set of real numbers is not empty, e.g. once proved that every rational number is a real number.

So given a set of properties which are consistent among themselves and consistent with our system, is the set of mathematical objects that fulfill such properties always non-empty?

If I read you correctly, I'd say no: take this definition for example: {x in U | x <> x} (the set of individuals in some universe of discourse that are not equal to themselves). There is nothing contradictory about it, it is a definition of the empty set: and there you have a property, the property of not being equal to oneself, that is satisfied by no individuals in that universe of discourse.