In most mathematics, it is already assumed that the natural numbers are embedded in the reals. However, this?

Question

In most mathematics, it is already assumed that the natural numbers are embedded in the reals. However, this has never been proven or done. What are the typical rigorous methods to get around this problem?

Answer 1

Using the standard "construction" of the natural numbers, they ARE NOT embedded in the real numbers. The real numbers are like a completion of a splitting field of a class of polynomials over the natural numbers. (Existance of transcendental numbers, such at pi and e, show that the real numbers themselves are not the splitting field of the set of polynomials over the naturals. See any good textbook on abstract algebra for definitions of splitting fields, transcendental numbers, ... ) As such, the natural numbers and real numbers are indeed disjoint sets. HOWEVER! Consider the function f which takes 1_N ( 1 in the natural numbers ) to 1_R ( 1 in the real numbers ), 2_N to 2_R, ... etc.

f( m_N + n_N ) = m_R + n_R
f( m_N * n_N ) = m_R * n_R

These properties show that the map f is a homomorphism. Moreover, the only element which maps to 0_R is 0_N showing that the map is, indeed, an embedding.

While the sets themselves may be disjoint, the set of natural numbers is "isomorphic" to what we consider the natural numbers sitting inside the reals. As we typically consider such sets only "up to isomorphism", we can thus assume that the natural numbers do indeed sit inside the reals.

Answer 2

Usually, the integers are defined using the natural numbers, the rationals are defined using the integers, and the reals are defined using the rationals.

At each stage it is shown that the previous set of numbers can be considered as part of the set of numbers it defines. Each member of the defining set, say the integers, is associated with a rational number; the rational numbers are equivalnce classes of ordered pairs of integers. Since it must be shown that the operations, multiplication etc. pass over to the next set of newly defined numbers, this is a tedious, unrewarding task and I seldom did it in class. I have never taught the Jordan Curve Theorem either. Indeed I have never read it.

There are some things that are true because a just and merciful God would not have it otherwise.

Having said this, I think the questions such as you asked need consideration. I have written something about these ideas at
mathematicsteacher.org

Answer 3

I guess we all just assume that imaginary or complex numbers are not natural. They certainly don't feel natural to me. But I have no idea how you would prove such a thing. Will be interested to see the other anwsers.

Answer 4

I was under the impression (from various math classes) that the natural numbers are defined as the positive counting numbers. I certainly don't know anyone who counts using complex numbers.

Answer 5

I don't understand what's your question