Wednesday, October 18, 2017

From the finite to the countable

Causal finitism lets you give a metaphysical definition of the finite. Here’s something I just noticed. This yields a metaphysical definition of the countable (phrased in terms of pluralities rather than sets):

  1. The xs are countable provided that it is possible to have a total ordering on the xs such if a is any of the xs, then there are only finitely many xs smaller (in that ordering) than x.

Here’s an intuitive argument that this definition fits with the usual mathematical one if we have an independently adequate notion of nautral numbers. Let N be the natural numbers. Then if the xs are countable, for any a among the xs, define f(a) to be the number of xs smaller than a. Since all finite pluralities are numbered by the natural numbers, f(a) is a natural number. Moreover, f is one-to-one. For suppose that a ≠ b are both xs. By total ordering, either a is less than b or b is less than a. If a is less than b, there will be fewer things less than a than there are less than b, since (a) anything less than a is less than b but not conversely, and (b) if you take something away from a finite collection, you get a smaller collection. Thus, if a is less than b, then f(a)<f(b). Conversely, if b is less than a, then f(b)<f(a). In either case, f(a)≠f(b), and so f is one-to-one. Since there is a one-to-one map from the xs to the natural numbers, there are only countably many xs.

This means that if causal finitism can solve the problem of how to define the finite, we get a solution to the problem of defining the countable as a bonus.

One of the big picture things I’ve lately been thinking about is that, more generally, the concept of the finite is foundationally important and prior to mathematics. Descartes realized this, and he thought that we needed the concept of God to get the concept of the infinite in order to get the concept of the finite in turn. I am not sure we need the concept of God for this purpose.

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