# Which concept is more primary in modern formal math: set, function or natural number?

My reasons to ask this question are the following:

1) formal definition of a function is "a subset of cartesian product of two sets" so it is defined with the set and natrual number as basic concepts.

2) in axiomatic set theory we need to be able to use logical stetements, so we need to be able to say that the current statement is true or false. As i understand we need a set $\{ 0, 1\}$ or $\{ T, F \}$ or some other set (SET!) consisting of two (NUMBER!) different elements, one of which is considered to represent truth value, the other - false value.

3) some formal definitions of natural numbers which i met were made in set theory (so we need set as a basic concept) and when we want to speak about algebraic operations we need a function concept.

I've just began to learn foundations, i.e. logic, set theory and so on. So i apologize for the question which many of you may consider to be stupid. And i'll be very grateful for any answers and ideas. Thanks in advance!

## 2 Answers

Set is the primary concept in modern mathematics. We should be able to define everything in terms of sets; in particular it is standard to define both functions and numbers in terms of sets.

In our somewhat postmodern era, the primacy of sets in the foundations of mathematics has been questioned by some. In particular there has been work to put categories at the foundations of mathematics. The categorical point of view definitely is in spirit aligned with the idea that functions ("morphisms") are more important than sets ("objects"), but they are not necessarily more primary in a logical sense: indeed the standard definition of a category begins with a collection of objects and a morphism is something associated to a pair of objects, just like a function is associated to a pair of sets.

Let me also say that most working students and practitioners of mathematics are not concerned with what is "logically primary". We are all too familiar with the following situation: you want to make a formal definition for a certain kind of mathematical object, and you want to choose your definition so that certain properties, or theorems, hold concerning those objects. Thus the definition is in truth inextricably linked to its logical consequences, the most basic of which are probably (temporally at least) "prior" to the definition. Then too one often finds characteristic properties of a given class of objects, which amounts to saying that they could equally well have been taken as the definition. As long as one knows that certain properties are equivalent it doesn't matter at all which one of them is taken as the definition. It would be a fairly serious mistake to make too much of the distinction between a definition and a characteristic property proved using that definition: really what we have is different descriptions of the same class of objects. For instance if you go around asking various people the definition of determinant of a matrix, you will soon acquire half a dozen different-looking definitions. "Which one is the real definition of a determinant?" is not a mathematically fruitful question.

It depends who gives you the answer, which is not absolute. Set theorists would obviously answer SET!!! but some finitist number theorist would say NUMBER!(and define sets as collections of numbers)