Re: infinity

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.

.

Relevant Pages

  • Re: infinity
    ... > The empty set is less than any set with one or more members. ... > 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 ...
    (sci.math)
  • Summary (partial): again problems with login to cluster
    ... the rpc.statd processes on all 4 members works correct. ... The problem with the caa-Application o20_spooler.. ... are solved after removing ... Postfach 1312, D-85741, Garching bei Muenchen, Germany ...
    (Tru64-UNIX-Managers)
  • Re: MSN Group has had too little activity...
    ... additional members & removing the occasional off-topic post). ... Yes, I'm an assistant manager there as well, and I tried to do the ... re-activation thing that MSN groups provided, ...
    (comp.sys.ibm.ps2.hardware)
  • Re: Ruby observer module
    ... On May 23, 2008, at 12:57 PM, Ryan Davis wrote: ... How will the rest of the members ... removing itself from the list might also hose the iteration. ...
    (comp.lang.ruby)
  • Re: infinity
    ... >>> corresponding members of the two sets. ... Number of elements is the value of the inverse function at the top of the range ... >>> correspond to any members of the other set. ... >> first set will have no corresponding members in the second set. ...
    (sci.math)