Date: Jan 13, 2013 1:02 PM
Author: Zaljohar@gmail.com
Subject: Re: A intuitive notion of set size.
On Jan 13, 3:13 pm, Butch Malahide <fred.gal...@gmail.com> wrote:

> On Jan 13, 4:20 am, Zuhair <zaljo...@gmail.com> wrote:

>

> > For any two sets A,B:

> > [1] A is bigger than B iff |A| > |B| Or A,B are sets of naturals and

> > there exist a non empty set C of naturals such that for every element

> > n of C:

> > |A(n)| > |B(n)| and |A(n*)| / |B(n*)| >= |A(n)| / |B(n)|.

> > where X(n) = {y| y in X & y <' n};

> > <' stands for natural strict smaller than relation;

> > | | stands for cardinality defined after Cantor's.

> > n* stands for the immediate successor of n in C with respect to

> > natural succession.

> > [2] A is smaller than B iff B is bigger than A.

> > [3] A is equinumerous to B iff ~ A bigger than B & ~ A smaller than

> > B.

> > /

>

> Example:

>

> For natural k, let n_k = 3*2^{k-1} + 1;

> so n_1 = 4, n_2 = 7, n_3 = 13, n_4 = 25, and so on.

>

> Partition N into two seta A and B as follows:

>

> A = {1} u [n_1, n_2) u [n_3, n_4) u [n_5, n_6) u ...

> = {1, 4, 5, 6, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 49,

> 50, ...};

>

> B = {2, 3} u [n_2, n_3) u [n_4, n_5) u [n_6, n_7) u ...

> = {2, 3, 7, 8, 9, 10, 11, 12, 25, 26, 27, 28, 29, 31, 32, ...}.

>

> In other words: the first natural, 1, is in A: the next 2 naturals are

> in B; the next 3 are in A, the next 6 in B, the next 12 in A, the next

> 24 in B, and so on.

>

> By your definition, A is bigger than B.

> Hint: let C = {n_2, n_4, n_6, n_8, n_10, ...}.

>

> By your definition, B is bigger than A.

> Hint: let C = {n_1, n_3, n_5, n_7, n_9, ...}.

>

> So A is both bigger and smaller than B. B is both bigger and smaller

> than A. Since A is equinumerous to itself, the relations "bigger than"

> and "smaller than" fail to be transitive.

>

> This does not seem very intuitive to me.

Definitely not. You are right! Thanks.

Zuhair