Wednesday, May 21, 2014

A metaphysical argument for a version of the Axiom of Choice

I will argue for the Axiom of Choice for sets of real numbers (ACR). ACR states that:

  • Given a set U of non-empty sets of real numbers, there is a function f on U such that f(A)∈A for every AS.
ACR is sufficiently strong to generate all the interesting paradoxes about the Axiom of Choice such as the ones linked here.

  1. There is a physically possible causally isolated situation in which there is a physical process R generating exactly one maximal semi-infinite (with beginning and no end) sequence of independent fair indeterministic coin tosses, and such that every combination of coin toss results can occur.
  2. For any possible physical process, and any cardinality K, it is possible that there are K causally isolated situations in each of which an instance of that process runs.
  3. If there is a set S of causally isolated situations, a set E of event types, and a function g from S to T such that for each sS an event of type g(s) is causally possible in s, then it is metaphysically possible that for each sS, an event of type g(s) occurs in s.
  4. If U is a set of non-empty sets of real numbers and possibly possibly there is a function f on U such that f(A)∈A for every AU, then there is a function f on U such that f(A)∈A for every AU.

Premise (1) is highly intuitive. Premise (2) is plausible, though finitists will deny it.

Premise (3) has the dubious form of saying that if each of some set of propositions—in this case, the propositions that g(s) as s ranges over S—is possible, then these propositions can all be true at once. Of course, this is false in general. But (3) limits this claim by making the propositions not just be metaphysically possible, but causally possible, and by saying that the propositions report what happens in different causally isolated situations. And individually causally possible events in different causally isolated situations should be compossible.

Finally, premise (4) says that the truth of mathematical propositions about the existence of numerically valued functions on sets of sets of real numbers does not vary across possible worlds, or even possibly possible worlds (given S4, the two would be be the same).

Now on to the argument. Let r be a function from semi-infinite sequences of heads-tails to real numbers, such that every real number is in the range of r. (For instance, one can let the coin toss sequence define a binary fraction between 0 and 1, and then apply some function to scale that up to all of (−∞,∞).) Let K be the cardinality of our set U of non-empty sets of real numbers. By (2) there is a possible world w1 containing a set S of K causally isolated instances of our random toss process R. Suppose now that w1 is actual. Let h be a bijection from S to the set U in ACR. Let g(s) be the event type of the random toss process in generating a sequence x of tosses such that r(x)∈h(s). Then g(s) is a causally possible event type, since every heads-tails sequence can occur by means of R and every real number can be generated by applying r to some heads-tails sequence. By (3), there is a possible world w2 at which all of this happens and each event g(s) occurs. But at w2, we can then let f(A) for AU be equal to r(x) where x is the result of R in the unique situation s such that h(s)=A. Then f(A)∈A since g(s) occurs.

We have thus shown that at w2, there is a choice function f. Since w2 is possible at w1, and w1 is actually possible, by (4) there is a choice function f, and the argument is complete.

I've given a version of this argument before, but this version identifies the assumptions more clearly, especially premise (3) about the conglomeration of causal possibilities across isolated scenarios.

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