How is it possible that to prove theorems about some set (say, the naturals) we have to introduce elements from outside that set?

Question

Take as an example, Analytic Number Theory. As far as I have read, it used the tools of complex numbers to reach conclusions about Number Theory (the set here would be the naturals). However, assuming that there exist Number Theory theorems without elementary proofs, how can it be that we don't have enough information about the naturals in the naturals themselves for us to introduce new elements (Reals and i) which don't have to do with the naturals? Why can't we prove them using only the naturals and nothing else?

This question also extends to theorems about the rationals and the reals.

Also, I'm sorry if this is a stupid question, or if the tags are not appropiate. Thanks.

Answer 1

I'm not fully qualified to answer this question, but I will give it a shot anyway.

Studying objects is hard, but not because of a reason you may suspect. Studying objects is hard because objects are too general. If I can't tell you what those objects do or how they interact with each other (or with other things) then you basically have no hope of understanding them beyond the simple fact that they are objects.

So now consider the natural numbers. I can tell you what they are, I can tell you what you can do with them (add them and multiply them), I can tell you how certain properties of natural numbers behave under those operations, but truly your tools are limited. The "language" of Peano Arithmetic is, in a certain sense, incredibly narrow. There are so very few things we can talk about.

But, natural numbers don't just show up in Peano Arithmetic. We know how to find them in much more robust "languages", like ZFC. And in this wider world of mathematical objects the natural numbers have a chance to interact with so many more types of objects, and certain truths that used to be inaccessible can now be proven. We can now say more about the naturals because our expressive power has increased.

Does that mean that every proof which leverages some object outside of Peano Arithmetic is necessarily revealing a non-elementary truth? No. There are often times equivalent proofs in more restrictive "languages", but they are almost always "harder" to find, or lack a certain sense of "beauty". Indeed, there may even be questions that cannot be phrased without the expressive power of "stonger languages".

A full accounting of the core idea here, that there are true statements that cannot be proven without finding a "language" with more "power" than the one you currently have, is very deep. You can whet your appetite with this aritcle on Goedel's Incompleteness Theorems.

Answer 2

Just to clarify: you are asking why we can't prove all statements concerning the natural numbers using only facts about natural numbers?

The integers $\mathbb{Z}$ are created using natural numbers (i.e., it's $\mathbb{N} \times \mathbb{N}$ modulo an equivalence relation), the rationals $\mathbb{Q}$ are constructed from $\mathbb{Z}$, the reals $\mathbb{R}$ are constructed from subsets of $\mathbb{Q}$, and complex numbers $\mathbb{C}$ is, formally, $\mathbb{R}^2$ (with $i = (0,1)$).

By this, you can say that $\mathbb{C}$ is constructed from $\mathbb{N}$, and any result using $\mathbb{C}$ is a result based on $\mathbb{N}$. Adding in a field-theoretic structure is again, still using $\mathbb{N}$ (e.g., multiplication on $\mathbb{C}$ is, formally, simply a subset of $\mathbb{C}^3$, a set that is based on $\mathbb{C}$ (and hence based on $\mathbb{N}$)).

Apologies if the above is unclear; I'm unsure if I understood your question correctly.