Finitable Mathematics

The word finitable doesn’t exist in the dictionary, so I had to invent it: it is an adjective which means “can be approximated arbitrarily well by finite things”. At first I wanted to use the word pro-finite, but that word already has other, more restrictive meanings, in mathematics (e.g. profinite groups and profinite spaces — they are totally discontinuous). Another possibility is the word “countable”, but it doesn’t fit too well either.

So why finitable mathematics ? The idea is that, since our world is finite (even the number of molecules in the universe is finite, or so they say), so for mathematical objects to have any relevance to the real world, they must be either finite, or approximations of finite objects. Objects which cannot be (arbitrarily well) approximated by finite objects simply don’t matter, and mathematicians (at least the applied ones) can safely forget about them. I will call the objects which can be approximated by finite objects, finitable. Finitable mathematics deals only with finitable objects. My claim is that most of the mathematics that we know (and all which matters to the real world) can be developed using only finitable objects, and that’s enough for applications. Moreover, finitable mathematics will save us from “useless” paradoxes of the set theory (e.g. a set which contains itsel as an element and which at the same time doesn’t contain itself). No more worry about mathematics being built on “shaky” grounds :-) And who cares about logical statements which are true but which cannot be proved.

A countable set, e.g. the set N of natural numbers, is finitable: just forget the “tail” of the set, and you get an approximation. The set R of real numbers, and the interval [0,1], are also finitable (an exercise for you). Likewise, paracompact differentiable manifolds are finitable. But now, the interesting thing is that, there are subsets of the interval [0,1] which are not finitable ! Why ? Because the cardinal number of the set (or the family) of all finitable objects is the continuum, while the set of all subsets of [0,1] has the cardinal number of 2^continuum, which is strictly greater than the continuum, according to the classical set theory. In fact, it is easy to see that, a subset of [0,1] “matters” (i.e. is finitable) if and only if it is a closed subset ! So we recover an important notion of topology, namely closed subsets, by asking the finitable property.

What about open subsets then ? Well, they don’t appear in finitable mathematics (except for the trivial ones). Sounds strange, doesn’t it, because the complement of a closed subset is an open subset. But the complement in finitable mathematics is not the same as the complement in the usual set theory. In fact:

The finitable complement of a subset A = the closure of the set-theoretic complement of A

So the finitable complement of a finitable subset is also closed (and is finitable). You can see that the intersection of a subset and its finitable complement is not empty in general, but is equal to its boundary. Can one do topology with only closed subsets (without open subsets) ? You bet ! Most important notions (e.g. compactness, connectedness) will remain essentially the same. Also, though I didn’t try to prove it yet, it seems that all finitable topological spaces will automatically be metrizable (i.e. it is really OK for mathematicians to assume that all topological spaces “which matter” are metrizable — for example, the Fréchet vector spaces that one often encounters in analysis and geometry are metrizable, and it seems that they are finitable)

Did I mention that in finitable mathematics the greatest cardinal number is the continuum ? So one doesn’t have to worry about sets whose cardinal number is greater than the continuum.There is this crazy set-theoretic open problem: does there exist a set whose cardinal number is stricly smaller than the continuum but strictly greater than countable ? In finitable mathematics this crazy problem simply dissapears: one can easly show that if the cardinal number of a finitable set is strictly greater than countable, then it is the continuum.

What about functions ? Since finitable functions must be defined as limits of finite functions (functions on finite sets), they are automatically continuous ! The only problem is that they may be multi-valued (even though their finite approximations are single-valued). Take a function from R to R for example. If it’s continuous, then its finitable version is itself. If its discontinuous, then the value of its finitable version at a point is the set of all limit values at that point. For example, if the function is constant by pieces, then it admits 2 values at each boundary point of the pieces (= intervals on which the function is constant). …

People working in probability theory must probably be happy with finitable mathematics, because now many abstract existence theorems in probability (e.g. Kolmogorov theorem ?) will now become a piece of cake ?! So it becomes unnecessary to remember the proof existence theorems (say of martingale measures), because things which matter “automatically exist”

etc… (I will add if I find something amusing)

OK, I will copyright, trademark and register the word finitable mathematics. You learned it from here first :D

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