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From: Marchal <marchal.domain.name.hidden>

Date: Tue Feb 15 06:58:04 2000

Fred wrote:

*>Hmmm...I would have thought the set of even numbers is much smaller than
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*>the set
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*>of reals, because between any two even numbers, you have infinitely more
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*>reals.
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This is false reasoning because though you have an infinite set of fractions

(rationals) between any two natural numbers, the cardinality of fractions is

still the same as the cardinality of natural numbers.

N = \omega = {0, 1, 2, 3, 4, 5, 6, ...} is as big as

Z = [... -6, -5, -4, -3, -2, -1, O, 1, 2, 3, 4, 5, 6, ...} which is too

as big as Q the set of rationals: Q = {p/q : p q in Z, q ‚ 0}

It is indeed easy to see that Q is embeddable in NxN,

which is itself easy to show equinumerous with N by dovetailing or zigzaging.

R, the set of real numbers is NOT in bijection with N, Z, or Q.

The proofs relies on Cantor's diagonalization. R is much bigger than N, Z, Q which are

equivalent (with respect to cardinality).

A more general theorem is card(2^B) > card(B), where 2^B = the set

of functions from B to 2 (2 = {0 1}).

The relation between cardinality, measure and topologies are intricate to

work with. In Theoretical Computer Science (Recursion Theory), it is

known that there are some "competition" between measure like notion and

topological notions.

Bruno

Received on Tue Feb 15 2000 - 06:58:04 PST

Date: Tue Feb 15 06:58:04 2000

Fred wrote:

This is false reasoning because though you have an infinite set of fractions

(rationals) between any two natural numbers, the cardinality of fractions is

still the same as the cardinality of natural numbers.

N = \omega = {0, 1, 2, 3, 4, 5, 6, ...} is as big as

Z = [... -6, -5, -4, -3, -2, -1, O, 1, 2, 3, 4, 5, 6, ...} which is too

as big as Q the set of rationals: Q = {p/q : p q in Z, q ‚ 0}

It is indeed easy to see that Q is embeddable in NxN,

which is itself easy to show equinumerous with N by dovetailing or zigzaging.

R, the set of real numbers is NOT in bijection with N, Z, or Q.

The proofs relies on Cantor's diagonalization. R is much bigger than N, Z, Q which are

equivalent (with respect to cardinality).

A more general theorem is card(2^B) > card(B), where 2^B = the set

of functions from B to 2 (2 = {0 1}).

The relation between cardinality, measure and topologies are intricate to

work with. In Theoretical Computer Science (Recursion Theory), it is

known that there are some "competition" between measure like notion and

topological notions.

Bruno

Received on Tue Feb 15 2000 - 06:58:04 PST

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