Formality fades away in the air

Question

I'm trying to study by myself mathematics, but I realized that I have only a naive notion of certains building blocks of mathematics; certain parts of the formalism. So I tried to start with logic, but this uses the notion of set, then i get confussed, and i searched, then I found that this sets are intuitive and are the metatheory, but this is informal, and if we want a formalization of that we need a metametatheory,and so on. Because that I ask the following, is there a way of formalise all without leave the most basic notions to the intution, or is this impossible?

Answer 1

This is impossible for the same reason that it is impossible to write a dictionary of the English language in which every word will be defined in such a way that no circularity is present.

More formally, for every defined thing say that its deflexity (standing for 'definition complexity') is $1+$ the largest of the deflexities of all terms appearing in the definition. Assuming that every definition is finite, this is a well-defined notion. Basically, the deflxity of something is largest of all deflexities defining it, plus 1. Now, look at all the things you define at any given point in time and consider the set of deflexities of these things. Its a subset of the natural numbers, so has a smallest element. Let $D$ be the thing that corresponds to the smallest deflexity. By assumption that everything is defined without leaving any basic notions to intuition, $D$ is defined in terms of other things. By the definition of deflexity, $D$ is defined in terms of something of strictly smaller deflexity than $D$, but that is impossible by minimality of the deflexity of $D$. QED.

It should be noted that the common approach in mathematics is the axiomatic approach. We usually do not define what something is, but rather what one can do with that something. So, we are being very honest. We say, I have no idea what that thing really is (in fact I don't even want to claim that it has any objective existence or meaning at all), but I know what I can do with it. At the same time, the axioms are intended to be interpreted in some universe of sets, of which we have no proof of existence (nor will we ever have one, unless none exists).

Just as an example, when you think of the real numbers you may be thinking of a very concrete description of the real numbers. However, there are several different explicit constructions that will give different models of the real numbers from, say, the rationals. Which one is actually the reals? The answer is that it does not matter. A healthier way to think of the reals is that the reals form a complete ordered field. Thus, instead of saying what the reals are, I say what you can do with the reals. There are many different models of the reals (in fact so many, the all models form a proper class) but they are all essentially the same and there is absolutely no canonical choice.