If open theism is true and there is an infinite future afterlife full of free choices, then some of the puzzling cases involving non-measurable sets and probability that I like to discuss on this blog are faced by God. For any set A of sets of natural numbers, there is the proposition pA that the set of days in heaven on which David will dance a jig is a member of A. But it seems likely that some sets A will be non-measurable relative to the relevant probability measure.
So, open theists should have motivation to work on highly technical formal epistemology. The more working on that, the merrier. :-)
6 comments:
Objective chance raises similar issues.
Think about a countably infinite collection of independent fair coin flips. If you take an epistemic view of probability, non-measurable sets of outcomes may not be a problem – it is enough that you can assign a probability to any event you can construct. But if you see the coins as objectively chancy, the existence of (even non-constructible) non-measurable events is a worry. It seems to show that the description ‘independent fair coin flips’ is at best incomplete. Further, it cannot be made compete, even by arbitrary extensions. I’m not sure what to make of this.
If open theism is true, then there is a very simple resolution of Cantor's Paradox, namely that God creates numbers (and associated possibilities). Conversely, if open theism is not true, then formal epistemologists need a different resolution, and what could that be? Good luck with that ;-)
The current favorite is that there are no numbers, and no collections, and that formal thinkers must make do with formal replacements; but, they must also claim that there are no things, e.g. no letters, no formal definitions, no formal rules, and so forth. Were there such things, there would be different ways of selecting some of them, which would give us Cantor's Paradox (God's omnipotence would give theists all the different ways needed for paradox).
I am not clear on how this solves Cantor's paradox. Is the idea that God keeps on creating bigger and bigger sets, but at any given time there are only finitely, or countably, many sets, or something like that? But if at any given time there are only finitely, or countably, many sets then much of modern math doesn't work. But I may not be clear on exactly how you propose to solve the paradox.
Ian:
I think it's OK to have things that are chancy but have no numerical probabilities. I suspect that free choices by people are like that, actually.
If many free choices are immeasurable, like that, then presumably such propositions as the one about future jig-dancing will be neither true nor false, which is what many open theists take them to be anyway.
If open theism is true, then God might be creating future days in the future, so that such future days do not already exist, not even in the future. That might just be the way things are, which is to say that it might just be a very good description of reality; to say that any future future days must already be future days might just be a bad description of reality. (Similarly, to say that each point of sky must either be a point in a cloud or else not a point in a cloud may well be a bad description of reality.)
Such a God could possibly create mathematical possibilities by doing mathematics (the bizarre possibility is that humans could do that, and yet there is Constructivism in mathematics), which solves such paradoxes; any modern math that is lost was bad math, in my opinion (but why do mathematicians not care that the current mainstream has lost all actual numbers!
...and not just numbers, but also natural collections. Suppose you have some things, and you want to consider some of them; there are different ways of selecting some of them, and one obvious way of distinguishing such ways from each other is via the things they select: two ways are of the same (obvious) kind when, and only when, they select the same things. For a God, the ways of selecting would come from omnipotence, and would give us natural collections (and associated actual numbers).
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