Re: Calculus XOR Probability

Mike Kelly wrote:
Tony Orlow wrote:
Mike Kelly said:

<snip>
Do all sets have a unique ordering, then?



Sets can be reordered, but reordering sets of reals out of natural quantitative
order breaks the IFR method of comparison, because it depends on the number of
elements being determined by the value range.

I didn't ask about sets of reals. I asked about sets in general. Must a
set have an order to have a size? Does "looking at it in a different
order" change the size? Does that make any sense to you?

Finite sets need not have order.
Infinite sets must be generated somehow by inductive rules along some metric in
order to be measured.

Are you saying "infinite sets must be generated by inductive rules" or
"in order to be measured, infinite sets must be generated by inductive
rules"? If it's the former then that is simply untrue. If it's the
later then your idea of "size" only applies to ordered sets, which seem
a bit bizarre. Are you saying you can't determine the size of a set
just by looking at what elements it has? Do sets without a "natural
quantitativeve ordering" not have a "size"?

I think we have established (months if not a year or so ago), that Tony
cannot give a Bigulosity for the following set of strings over the
alphabet {"0", "1"}:

{"0", "1", "10", "11", "100", "101", ... }

Or just possibly it's that he cannot give a Bigulosity, or even gives a
different Bigulosity, for the (identical) following set of strings over
the alphabet {"0", "1", "V", "X"}:

{"0", "1", "10", "11", "100", "101", ... }

Or possibly he claims this set is not well-defined, though he has never
been able to exhibit a string for which the membership or not of this
set is at all unclear.

Informally, his Bigulosity looks remarkably like the notion of limiting
density of a subset of the naturals (if I have the terminology
correct). *** Winter gave him a set with no limiting density, and he
went quiet very briefly, then carried on anyway.

Brian Chandler
http://imaginatorium.org

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