Re: Well Ordering the Reals

David R Tribble said:
> David R Tribble said:
> >> It's your math, Tony. How can you draw any conclusions from it if it's
> >> not established yet?
> >
>
> Tony Orlow wrote:
> >> The set is established and some properties are established, but all the
> >> arithmetic rules of this number system have not been explored. I assign you the
> >> task of figuring out how to add two H-riffic numbers in their native format. I
> >> await your findings with great anticipation. Good luck!
> >
>
> David R Tribble said:
> >> Do your own homework. Why would I waste my time on
> >> something that I know is wrong?
> >
>
> Tony Orlow wrote:
> > Wrong? You think the H-riffic numbers are "wrong"? What is wrong with them?
> > They are a number system with a structure that works. They may not be great for
> > everything, but they are certainly not "wrong". That's just a dumb remark that
> > says you aren't interested in whether there is anything interesting to
> > discover.
>
> Okay, prove something simple like f+1 > f for all infinite f.
> Show that the sum of the digits for f+1 is greater than the sum of the
> digits for infinite f.
>
> I admit that I can't because I can't create a workable definition for
> infinite numbers, so that addition, comparison, etc. work properly.
> But your definitions so far are no better.
>
>

define x>y if the most significant digit where they differ is a 1 in x and a 0
in y.
prove: n+1>n

case 1: If n has a 0 in the least significant digit, then n+1 will have a 1,
and this will be the most significant digit where they differ, making n+1
bigger.

case 2: If n terminates in a string of n (n caln be infinite) 1's preceded by a
0, then adding a 1 will cause carries over all the 1's, and replace the end of
the string with a 1, followed by n 0's, and the rest of the strings will be the
same. So, that 1 is the most significant digit where n and n+1 differ, and n+1
has a 1 there, so n+1 is larger.

Does that do it for you? If not, why not?
--
Smiles,

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

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