http://friam.383.s1.nabble.com/Model-of-induction-tp7588431p7588474.html
Glen,
Okay, given some of the later postings against the original question, I am thinking that your question may have morphed or that I have completely misunderstood what you are asking. Not sure. For example, somehow we have gone from probability theory and its ontological status to the Banach-Tarski Theorem and the Axiom of Choice. This seems like a non-sequitur, but not sure. First off, a theory is inductive, whereas, a theorem is deductive; so that is my first disconnect. So I don't understand how we got here ... but this often happens to me. :-(
Then we go to what I think is a refinement of the original question. Yes? (I am just trying to navigate the thinking to get to the core issue, that I seem to be missing):
But what is this "set of events"? That's the question that is being discussed on this thread. It turns out that the events for a finite space is nothing more than the set of all possible combinations of the sample points. (Formally the event set is something called a "sigma algebra", but no matter.) So, an event scan be thought of simply all combinations of the sample points.
So, the events already have probabilities by virtue of just being in a probability space. They don't have to be "selected", "chosen" or any such thing. They "just sit there" and have probabilities - all of them. The notion of time is never mentioned or required.
An event is not all the combinations of the sample points. As Grant has said, an event [outcome] has probability depending on how it is arbitrarily configured from the event space by the researcher. Moreover, there is an important distinction to be made between the distribution of values [e.g., the numbers on each side of a dice being equally likely] and the sampling distribution that is dependent on how the event is composed in a trial sequence. The sampling distribution is the mathematical result of the convolution of probabilities when choosing N independent, usually identically-distributed random picks from the parent distribution.
Another example might be helpful: I think you are trying to define the sample space like with an urn of 10 balls with three red and seven white. An event, in that case, would be something like picking three balls all red. We could easily compute the probability of this event by using hypergeometric arithmetic; this is because of the sample space changing if you do not replace any balls after each pick. But, there is a finite number of other possible events in this scenario of picking three things from a bin of ten things. To be sure, though, this statistical problem does not relate at all to the paradoxical Axiom of Choice ... unless I am still missing something. We are not interested in slicing and dicing [no pun intended] a probability space of a certain size in a way for coming up with, say, two identical but mutually exclusive probability spaces of the same size. This would make no sense, IMHO.
Events are just the outcome(s) one is interested in computing the probability for. They don't exist--as selections, in the way that I think you mean--until they are formulated by the researcher ... not trying to conjure up anything spooky here between the observer and the experiment as at the quantum level. :-) Nor are these events--not being mathematical entities of any type--something to be discovered in some platonic math sense [I mistakenly called you a Platonist, but on rereading the thread, I think you are not. Sorry. But the world wouldn't be as interesting without Platonists. :-) ].
For example, there is the possible event of being dealt four aces in one hand of five cards and for which I can assign a probability given the conceptual structure of the probability space: a deck of cards. This is nothing more than laying out the number of possible [combinations--so order doesn't matter] of hands (a sample) and determining how many ways I could be dealt four aces [just one] ... then dividing the latter by the former. This is an example of a categorical probability space, where the events are all the various ways [combinations] one can be dealt five cards from a deck of 52. We could go on to define these into categories like two of a kind, three of a kind, and so forth. Each of those events can be then assigned a probability.
and then:
Perhaps it's helpful to think about the "axiom of choice"? Is a "choosable" element somehow distinct from a "chosen" element? Does the act of choosing change the element in some way I'm unaware of? Does choosability require an agent exist and (eventually) _do_ the choosing?
The Axiom of Choice is a paradox that seems to get into trouble with set-cardinality, where it comes to infinite sets. To me is nothing more than a mathematical curiosity that has no impact on the practical world. So I don't think this is helpful to your cause. But I would be more than curious to see how you think it might be. I am more an applied mathematician|statistician than anything like a theoretical mathematician; though, I have happily worked with many of the latter ... and hopefully the reverse was true. :-)
Okay, back to your observation: the fact that it is possible to choose a particular event from the set of all possible events in the event space is a trivial requirement. I cannot, for example, pick a black ball--an impossible event--from the previous urn of only red and white balls. So being able to choose three red balls from that urn makes the event "choosable." Is that event then distinct from that same event that has been "chosen?" At the classical level--as opposed to the quantum level--I cannot see any meaningful distinction EXCEPT to say that the former event is a possibility and the second event is a realization ... and that the way such events get discussed in practical probability and statistics. There is no spooky agent that needs to get factored into the calculus, IMHO.
Somehow, I still feel I am missing something. Maybe you can figure it out, but it may not be all that important, and your question may have already been addressed satisfactorily by the other responses posted to the thread.
Cheers
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