Arithmetic on real numbers

Question

So far, I have studied elementary set theory and I have some questions.

1. I know how to add or multiply natural numbers and ordinals, but how do I subtract or divide or root or log? Is there any theorem such as finite and infinite recursion theorem?

2. Should I treat a set of real numbers completely distinct from $\omega$? That is, how do I define arithmetic on real numbers? Since the sets I have learnt have 0 as the least element then next, 1, not 1/2,1/3..., I don't feel free to treat real numbers. Imaginary numbers too.

3. Please suggest me a nice book which has contents about what I wrote above.

Answer 1

Set theory, in particular ZFC, is a foundational theory. This means that we can create "most" mathematics inside a model of ZFC (and by most I mean at least real analysis, basic algebra, etc.).

What does that mean? It does not mean that we have explicit formulas depicting the functions $x\mapsto\sin x$ or $\theta\mapsto e^{i\theta}$. We could write something like this, but it would be of extremely little use. Instead what we do is prove that the basic tools of analysis (the real numbers, their basic properties, etc.) are true in some interpretation of the real numbers inside the model.

What does that mean? Well, we prove that there is a set of cardinality $2^{\aleph_0}$ and there exists a binary relation, and two binary operations, such that all these have the same properties as $\mathbb R$ with addition, multiplication and order. From there, we prove that we can define the rest using these as parameters.

Why parameters? Well, there are many isomorphic ways to define addition on the real numbers, given $f\colon\mathbb R\to\mathbb R$ which is a bijection we can define $x\oplus y=f(x+y)$; similarly for the rest of the operations and relations. So we know that if there is one way to define these things, then there are many ways defining them. So what do we do? We simply say something like "If $A$ is the function describing addition then $0^A$ is the unique element $x$ such that $A(x,x)=x$...". From these we can define convergence, continuity, then we can define exponentiation, roots and logarithms, etc.

Okay, so we agreed that if we can show that there are sets which describe something which behaves like the real numbers then we are in the clear. For that we can take any interpretation of the natural numbers, e.g. $\omega$ with its very canonical addition and multiplication, from there define the integers, the rationals, and the real numbers along with addition, multiplication and order. We do this each step of the way, and we end up with a decent way to realize the real numbers as sets. However there is no canonical way to do that, as there is with defining the finite ordinals and showing they behave like $\mathbb N$.

The real numbers can be realized as an equivalence relation over rational Cauchy sequences; as one-sided Dedekind-cuts or as two-sided Dedekind-cuts. Each of these steps is inherently different and the results are different as sets. However we know that the operations we define are isomorphic to one another so the result, whatever it is, still behaves just like the real numbers would.

Answer 2

While it is nice to observe that the complete ordered field $\mathbb{R}$ can be constructed set-theoretically--as it allays concerns that all theorems regarding complete ordered fields are only vacuously true--it isn't really necessary. One can consider $\mathbb{R}$ as the set of left sides of Dedekind cuts of $\mathbb{Q}$; $\mathbb{Q}$, as a quotient of $\mathbb{Z}\times(\mathbb{Z}\smallsetminus\{0\})$ under an appropriate equivalence relation; $\mathbb{Z}$, as a quotient of $\omega\times\omega$ under an appropriate equivalence relation. There are ready embeddings $\omega\hookrightarrow\mathbb{Z}\hookrightarrow\mathbb{Q}\hookrightarrow\mathbb{R}$. However, in most cases, it is more useful (or at least less cumbersome) to consider the elements of $\mathbb{R}$ simply as points on a line, with no intrinsic set-theoretic properties, and work from there.

If you're really anxious to define things set-theoretically, consider how they are developed in the context of real analysis. For example, exponentiation by natural numbers is generally defined recursively--$a^0=1, a^{n+1}=a^n\cdot a$--and negative integer powers of non-0 reals are defined as reciprocals of corresponding positive powers. Next, $n$th roots of non-negative reals are defined in terms of inverse functions of $n$th powers of non-negative reals. Using roots and integer powers, we can define rational powers of non-negative numbers. We define arbitrary real powers in terms of limits of rational powers. Finally, we define logarithms of positive numbers in terms of inverse functions of real exponential functions on a given positive base.

Does that help?

Answer 3

Structures isomorphic to the real numbers $\mathbb{R}$ can be constructed set-theoretically, for example as sequences of rationals. Arithmetic operations can then be defined on these structures in terms of the usual set-theoretic operations. The complex plane is simply the cartesian product of the representation of $\mathbb{R}$ with itself. Such structures will be distinct from $\omega$, although defining an embedding from $\omega$ to the set-theoretic representation of $\mathbb{R}$ is straightforward.

Answer 4

A nice book that goes into these kinds of things is Goldrei's Classic Set Theory.

In regards to the specific issue you're having, a detour into the history of set theory will be instructive.

For a long time, humanity toyed with one number system, namely $\mathbb{N} \setminus \{0\}$. Then we discovered $0,$ which allowed us to discover $\mathbb{Z}.$ Then more number systems started coming along; indeed, the process eventually went into overdrive. By the beginning of the 1900's, mathematicians were looking into all sorts of crazy algebraic and relational structures. Now, they knew some ideas for mathematical structures simply didn't make sense (think: a number system where $0 \neq 1$ and $0 = 1$ both hold). So, they became concerned with the existence of structures. The idea is this: if make use of a structure $X$ to prove something about a structure $Y$, well this proof is only valid if $X$ actually exists.

The problem then becomes, how do you decide which structures are valid, and which are not? Set theory came to the rescue, because people realized that it serves as a sort of all-purpose Lego kit for building mathematical structures. (My thanks go out to Peter Smith for that wonderful turn of phrase). But the point is, once you've shown that a mathematical structure exists, the actual details of its construction cease to matter.

This has implications for the issue you are having.

Once you've shown that $\mathbb{N}$ and $\mathbb{R}$ exist, it ceases to matter whether you constructed them in such a way that $\mathbb{N} \subseteq \mathbb{R}$. The point is, there is a function $f : \mathbb{N} \subseteq \mathbb{R}$ such that $f(1_\mathbb{N} + \cdots + 1_\mathbb{N}) = f(1_\mathbb{N}) + \cdots + f(1_\mathbb{N})$, where the number of entities being summed on the left equals the number being summed on the right, and this function turns out to be the unique embedding of the natural numbers into the reals that preserves all the usual operations. We call it the 'natural embedding' (of the naturals into the reals). Whether or not $f$ is the inclusion map is entirely up to you. Indeed, you can make this choice anew, in every theorem you state and every proof you write. Just write: 'without loss of generality, assume the natural embedding $\mathbb{N} \rightarrow \mathbb{R}$ is the inclusion map.'

Thus, for example, you don't have to try to define logs of natural numbers. You do it for the reals, and whenever you need to take the log of a natural number, you make sure the reader knows that the natural embedding $\mathbb{N} \rightarrow \mathbb{R}$ is the inclusion map in this context.

Now admittedly, I do not know a formal proof that you can just simply assume, without loss of generality, that the natural embedding $\mathbb{N} \rightarrow \mathbb{R}$ is the inclusion map, or even how to formulate this concept as a theorem. But, it makes intuitive sense.

Anyway, the point is this.

You get to choose.