So you want to count an infinite power set ?
From: HERC777 (herc777_at_hotmail.com)
Date: 11/20/04
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Date: 20 Nov 2004 13:52:39 -0800
What *is* the power set?
A power set is a transform from subset space to element space.
I run a banner exchange and to improve the targeting of the
banner placements, everyone nominates the CATEGORY of their site.
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NAME : Herc
WEBSITE : www.entrenchederrorsinmaths.com
BANNER : www.entrenchederrorsinmaths.com/ban1.gif
CATEGORY :
Categories are : sci, rec, alt, public_service_announcements
So you can see the problem, I have 2 categories I want to use,
sci and p_s_a. Now there's 2 ways to implement multiple categories.
You maintain a list of several categories for each site, or...
use a powerset of categories.
CATS = {s, r, a, p}
P(CATS) = {{s}, {r}, {a}, {p}, {s,r}, {s,a}, {s,p},
{r,a}, {r,p}, {a,p}, {s,r,a}, {s,r,p}, {r,a,p}, {s,r,a,p}}
Now I can select the "single" category "Sci & Publi_Service_Announcement".
We wanted a SUBSET, but instead used an ELEMENT of the power set.
Is this process feasable for an infinite list? Mathematics texts will
say no, as Cantor's proof is useless balony of course it's possible.
Say the set is the digits of pi, augmented with their digit place
so each member is unique.
SET = { <0,3>, <1,1>, <2,4>, <4,1>, <5,5> ... }
Now we want to make a single selection of that set but containing
multiple members!
1st Attempt:
P_1(SET) = {{1000000000000.. AND SET},
{0100000000000.. AND SET},
{1100000000000.. AND SET}...}
We use logical AND as a filter on the set with a reverse binary
number and get:
P_1(SET) = {{ <0,3> }, { <1,1> }, { <0,3>,<1,1> } ... }
We get all the combinations of the SET but the members are all
finite sets. Need something better.
What can we use as a generator of *infinite* binary expansions?
2nd Attempt:
P_2(SET) = An, {UTM(neN, 0) AND SET}
UTM(neN,0) will emulate every computer program from 1 to infinity
and output a (potentially infinite) sequence of 1s and 0s for every
natural. Every possible permutation is present.
Hence we can generate countable power sets of infinite sets.
Herc
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