Modelling the continuum, Part II @ unwanted capture

Modelling the continuum, Part II

Posted June 19, 2009 @ 2:52 pm. spencer No Comments »

In this post I gave a brief sketch of what I claim is a better approach to modelling the continuum. In this post I am going to develop the ideas a little bit more thoroughly and also explain why I this is a better approach than the usual set-theoretic definitions of the real numbers.

Here are a few of the points which I think distinguish this approach:

$\begin{array}{l} \bullet\ \ \textrm{New entities to work with.}\\ \bullet\ \ \textrm{Forces a consideration of formal logic.}\\ \bullet\ \ \textrm{Better reflects our physical intuitions and practice outside mathematics.}\\ \bullet\ \ \textrm{More efficient presentation.}\\ \end{array}$

Beneath the fold I will hint at a few of these points, and give some small detail for the first. Although the details are by no means worked out, I am certain that most of our methods for the real numbers (e.g., calculus, topological groups, measure theory) will carry over in a natural way. A significant exception may be descriptive set theory (and other theories of the fine logical detail of $\mathbb{R}$ ); in fact it seems to me that my notion of the continuum doesn’t even have a well-defined cardinality. Rather than a failing of this approach, I see this as a sign that set theory has injected unnecessary baggage into our theory of the number system.

These ideas are mostly idle musings, not in active development, and I’m releasing them to the wind. If anyone who happens by wants to develop these ideas, go ahead. You can take them for yourself (with an attribution) or I would be happy to work on a joint project. Otherwise, I’ll continue a haphazard development in these pages from time to time.

Let’s start by recalling the set-up for my model. The underlying machinery of the model will be the nonstandard theory of the natural numbers. A very nice introduction to these ideas can be found in the fourth chapter of Edward Nelson’s book Radically Elementary Probability Theory. Roughly speaking, a nonstandard model of $\mathbb{N}$ has two nested models of the natural numbers. The raw ingredients are a pair of nested sets $N_0\subseteq N$ , an element $0\in N_0$ and a bijective successor function $s:N\to N\setminus\{0\}$ which also acts as a successor $N_0\to N_0\setminus\{0\}$ . We call the elements of $N_0$ the standard elements; this is an important property, so we’ll give it a special name: $\textbf{std}(n)\iff n\in N_0$ . All the rest of the elements are non-standard. Corresponding to these two different types, we have two notions of induction. “External” (true) induction says that the standard elements form a model for the natural numbers:

$\varphi(0)\ \ \&\ \ (\varphi(n)\to\varphi(sn))\ \ \Rightarrow\ \ \forall n\in N_0\ \varphi(n)$

The internal induction axiom will apply to all of $N$ ; we want to say that $\textbf{std}$ and the compound propositions which include it are the only ones which are not inductible relative to the entire set. Thus for any proposition $A$ which does not involve $\bf{std}$

$A(0)\ \ \&\ \ (A(n)\to A(sn))\ \ \Rightarrow\ \ \forall n\in N\ A(n)$

What all this means is that the standard numbers $N_0$ make up an initial segment of $N$ , and both sets act like the natural numbers. I like to think about the standard numbers as corresponding to the actions in our day-to-day life (like counting or dividing) or in our formal system (lengths of derivations), whereas the rest of $N$ consists of the huge numbers like astronomical distances or the number of atoms in an everyday object. At a practical level, if I can count to $n$ I can count to $n+1$ , but I’ll never be able to count the grains of sand on a beach even though there are a finite number of them. This non-inductible character to our actions is exactly the intuition which non-standard analysis captures.

Since $s$ and $0$ are inductible, the entire arithmetic/order theory of $\mathbb{N}$ comes for free, and we can use all our usual intuitions about these operations. Just to emphasize this, we’ll start referring to the non-standard model with the bold notation: $\mathbb{N}_0\subseteq \mathbb{N}$ . Now we want to recover the theory of the continuum. To do this, we rescale until the gaps in the natural numbers are too small to see. Specifically, we fix a non-standard number $N\in\mathbb{N}$ , and we want the number $2^N$ to be our new unit. This doesn’t effect the addition or order operations at all, since these are indifferent to scaling. Multiplication, however, must be modified so that the new unit squares to itself. Using $\times$ to distinguish this new operation, we would like to say

$n\times m=\frac{nm}{2^N}$

Of course, division by $2^N$ is not defined in $\mathbb{N}$ , so really we should put a ceiling or floor operator on this definition.

$n\times m=\left\lfloor\frac{nm}{2^N}\right\rfloor$

But wait, the approximation is going to screw up our axioms like associativity and commutativity; sometimes we’ll end up off by one or two. To keep our operations well-behaved, we have to mod out by infinitesimal distances. Non-standard analysis to the rescue! We can define an equivalence relation by

$n\sim n'\ \ \iff\ \ \textbf{std}(|n-n'|).$

The we can define the half-ray $R=[0,\infty)$ by $R:=\mathbb{N}/\sim$ . In fact, we’ll be focusing on the compact interval

$I=[0,1]:=\{n\in\mathbb{N}|0\leq n\leq 2^N\}/\sim.$

It is a bit of work, which I will leave to another day, to see that addition, multiplication, order and (truncated) subtraction are well-defined on these equivalence classes. Division is a bit trickier, but I would argue that we don’t even really need it; it will be enough to work out division by two, and this is easily accomplished. Moreover, this squares with elevated position that is accorded to dyadic rationals in the theory.

Notice that the definition of $I$ involves the $\bf{std}$ predicate. This means that we may need to be careful when using propositions defined from elements of $I$ . In some sense this is like taking limits, as it pushes us from a relatively simple domain (the internal theory of the natural numbers) into a more complicated situtation (the nonstandard theory). Because of this, we will usually try to make all our definitions and conduct our calculations before passing from $\mathbb{N}$ to $I$ .

It is worth noting here that the structure I’ve just described cannot be the real numbers to which we are accustomed. To see this, consider the element $[N]\in I$ . We assumed at the beginning that $N$ is a non-standard element, so $N\not\sim 0$ and $[N]$ is not the zero element of $I$ . If we try to think about distances in $I$ , we must rescale so that $\tilde{d}(o,1)=1$ . Then

$\tilde{d}(0,[N])=\left\lfloor\frac{N}{2^N}\right\rfloor=0/$

Thus $N$ is a true infinitesimal. Moreover, there are lots of others, $\sqrt{N}, N^2, N^3, N^{13/2},\ldots$ , all prearranged in a nice arithmetic hierarchy. Indeed, any real number $\alpha>0$ defines a different scale of infinitesimal $N^\alpha$ . These are precisely the “new entities” referred to above the fold. Although I haven’t worked though the details yet, I believe these infinitesimal scales ought to help us provide a more concrete picture of local properties, especially different degrees of differentiability.

I’m going to leave it at that for now. The next time I revisit this topic I’ll say some words about the function theory of my non-standard interval.

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Modelling the continuum, Part II

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