Re: infinity

Virgil said:
> In article <MPG.1db04e511fd10f9098a423@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
>
> > David R Tribble said:
> > > Ross A. Finlayson wrote:
> > > >> So: well-order the reals.
> > > >
> > >
> > > Jonathan Hoyle wrote:
> > > > Sigh. Didn't I do this one already? Here it is again <grin>:
> > > >
> > > > Take any arbitrary 1-1 mapping F from the reals R to the power set of
> > > > natural numbers P(N). By the Axiom of Choice, we know we can
> > > > well-order P(N), so take any such well-ordering, <=. Define a <=
> > > > operation (obviously different from the standard one) on R such that
> > > > for a,b in R, a <= b whenever F(a) <= F(b). You have now well-ordered
> > > > R. Creating F and well ordering P(N) are left as exercises. :-)
> > >
> > > To well-order P(N), we can define a '<' relation between members
> > > of P(N):
> > >
> > > a. {} < {i, ...} for all i in N;
> > > The empty set is less than any set with one or more members.
> > >
> > > b. {i, ...} < {j, ...} if i < j for all i,j in N;
> > > Any set with a least element i is less than any other set with a
> > > least element j if i < j.
> > >
> > > c. {i, j, ...} < {i, k, ...} if {j, ...} < {k, ...}
> > > for all i,j,k in N and i < j and i < k;
> > > For two sets both having a least member i, the first set is less
> > > than the second if the set formed by removing i from the first set
> > > is less than the set formed by removing i from the second set
> > > (this is a recursive definition that makes use of the previous rules).
> > >
> > > So all the the members of P(N) can be ordered using the '<' relation
> > > defined above.
> > >
> > >
> > Isn't this essentially the same as using binary strings to denote memebrship
> > in
> > each element of P(N), and ordering them in the natural way, as they are for
> > the
> > elements of N? It seems that if a set can be enumerated in a linear fashion,
> > then its power set can also.
>
> If TO thinks that the above described ordering is a well-ordering, he
> should be able to give a general rule for finding the successor of each
> element in P(N) under that ordering.
Given the binary string describing the subset, take it as a binary whole
number, with the first element represented by the bit in the 1's column, etc,
and add 1 in the normal fashion. This gives you the binary string describing
the successive subset.
>
> What, for example, would be the successor to the set of even naturals
> under the ordering described?
.....010101010+1=......01010101011
Whew! That was hard. So we have the union of the set of evens E, with {1}.

You might want to try a harder one next time. How about the successor to the
set of primes? Virgil, I leave this to you as an exercise. Have fun!
>

--
Smiles,

Tony
.

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