Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity

From: Timothy Little (tim-via-n.i.net_at_little-possums.net)
Date: 01/25/05 

Date: 25 Jan 2005 06:16:59 GMT

Ross A. Finlayson wrote:
> if you have a sequence with infinitely many ones and zero you can
> generate most irrationals, and half of the possible sequences.

Actually, you can get all real numbers if you're have a wide enough
deefinition of "reorder".

For example, define a "reordering" to be an injective map f:N->N where
f(n) is the position in the old sequence from which you get the n'th
element in the new. For any finite set of labels, such a "reordering"
is bijective. For infinite sets of labels, it need not be.

> That consideration of sampling real numbers is digression from the
> point about measure theory and probability: that the utility is
> primarily about the cardinality of the continuum and continua, and thus
> the basically geometric nature of the continuum instead of its
> cardinality, and that it could be explained that way.

Not at all. Many of the functions I talk about aren't even defined
over the reals, let alone dependent upon their geometric structure.

> So, aside from that digression, if I want to support measure
> theoretical results or provide alternate mechanisms for correct results
> using my little theory where infinite sets are equivalent, then it
> would lead to some retrofitted underpinnings of measure theory as
> necessary

Definitely. All of analysis to begin with.

> I wonder: is there any use in meaure of transfinite cardinality
> besides the cardinality of the continuum?

Cardinality of the powerset of the continuum is certainly used; I
don't personally recall encountering explicit use of cardinalities
larger than that.

> Second: does probability theory use transfinite cardinals besides
> using measure theory?

As I was taught it, probability theory *is* measure theory for a
particular class of measures.

- Tim

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