Re: Well Ordering the Reals

David R Tribble said:
> David R Tribble said:
> Tony Orlow wrote:
> >> When you say a number like x = 1:1010...1010 has N digits,
> >> where N is supposedly infinite, this means that 2x has N+1
> >> digits and x^2 has 2N digits, etc. What good is N for, then?
> >
>
> Tony Orlow wrote:
> > For comparing sets formulaically, when they cannot be directly measured due to
> > infinity.
>
> We can define a correspondence between every member of a set A and
> every member of a set B, for example. This tells us that sets A and B
> have exactly the same number of members. Notice I didn't mention
> whether the sets are finite or infinite, because it works for all sets.
That is the assumption behind cardinality, but it doesn't satisfy our
intuitions well about the behavior of infinite sets, and violates some basic
universal rules about sets, like that adding elements increases the size of the
set, and that proper subsets are smaller.
>
> You are saying that this is not true, but that you have to compare the
> sets "formulaically" instead. What if you can't do that? How do you
> determine the size of the sets then?

How do you define a mapping betwene infinite sets without some sort of formula?
You can't.

>
> Consider the set
> Y = {0} u {x+r, where x is in Y and r = random(1,10)}
> So Y is a set of random naturals, starting with 0 and increasing to
> each successor member by a random integer increment r, where
> 0 < r < 11.

That's some sort of formula. Let's see if we can work with it. You seem to be
defining this as a linear set, starting with zero, and each successor being
between 1 and 10 units greater. So, if this is truly random, over an infinite
number of iterations the average increment is 5.5. So, the set has N/5.5
elements in N. How hard was that?

>
> How many members are in Y? I can answer that question (using a
> simple correspondence wiht N). Can you determine the size of Y
> formulaically using your "range comparisons"?
See above.
>
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

Relevant Pages

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