“What is our intuitive understanding of the natural numbers? Surely this being the firmest of all our mathematical ideas, should have a definite, transparent meaning. Let us examine a few attempts to make this meaning clear:
(1) The natural numbers may be thought of as symbolic expressions: 1 is |, 2 is ||, 3 is |||, 4 is ||||, etc. Thus, we start with a vertical stroke | and obtain new expressions by appending additional vertical strokes. There are some obvious objections with this approach. First, we cannot be talking about particular physical marks on paper, since a vertical stroke for the number 1 may be repeated in different physical locations. The number 1 cannot be a class of all congruent strokes, since the length of the stroke may vary; we would even acknowledge as a 1 a somewhat wiggly stroke written by a very nervous person. Even if we should succeed in giving a sufficiently general geometric characterization of the curves which would be recognized as 1's, there is still another objection. Different people and different civilizations may use different symbols for the basic unit, for example, a circle or a square instead of a stroke. Yet, we could not give priority to one symbolism over any of the others. Nevertheless, in all cases, we would have to admit that, regardless of the difference in symbols, we are all talking about the same things.
(2) The natural numbers may be conceived to be set-theoretic objects. In one very appealing version of this approach, the number 1 is defined as the set of all singletons {x}; the number 2 is the set of all unordered pairs {x, y}, where x =/= y; the number 3 is the set of all sets {x, y, z} where x =/= y, x =/= z, y =/= z; and so on. Within a suitable axiomatic presentation of set theory, clear rigorous definitions can be given along these lines for the general notion of natural number and for familiar operations and relations involving natural numbers. Indeed, the axioms for a Peano system are easy consequences of the definitions and simple theorems of set theory. Nevertheless, there are strong deficiencies in this approach as well.
First, there are many competing forms of axiomatic set theory. In some of them, the approach sketched above cannot be carried through, and a completely different definition is necessary. For example, one can define the natural numbers as follows: 1 = {∅}, 2 = {∅, 1}, 3 = {∅, 1, 2}, etc. Alternatively, one could use: 1 = {∅}, 2 = {1}, 3 = {2}, etc. Thus, even in set theory, there is no single way to handle the natural numbers. However, even if a set-theoretic definition is agreed upon,it can be argued that the clear mathematical idea of the natural numbers should not be defined in set-theoretic terms. The paradoxes (that is, arguments leading to a contradiction) arising in set theory have cast doubt upon the clarity and meaningfulness of the general notions of set theory. It would be inadvisable then to define our basic mathematical concepts in terms of set theoretic ideas.
This discussion leads us to the conjecture that the natural numbers are not particular mathematical objects. Different people, different languages, and different set theories may have different systems of natural numbers. However, they all satisfy the axioms for Peano systems and therefore are isomorphic. There is no one system which has priority in any sense over all the others. For Peano systems, as for all mathematical systems, it is the form (or structure) which is important, not the "content". Since the natural numbers are necessary in the further development of mathematics, we shall make one simple assumption:Basic Axiom There exists a Peano system.“
Elliott Mendelson, Number Systems and the Foundations of Analysis