Re: An uncountable countable set

mueckenh@xxxxxxxxxxxxxxxxx wrote:

Franziska Neugebauer schrieb:
mueckenh@xxxxxxxxxxxxxxxxx wrote:

An infinite sum of 1's is not infinite?

n
lim sum 1 = lim n =def L
n -> oo i = 1 n -> oo

There is no such L in N.

Correct.

The antecedent is true.

Therefore there are not infinitely many difference[s] of 1
between natural numbers.

Your consequent is proven false (see below). Therefore your implication
is false, too.

A difference of two numbers b and a is usually denoted as b - a. We
introduce the difference operator "-" action upon ordered pairs:

-(a, b) def= b - a

"How many differences there are" means the cardinality of the set
of all pairs {(a, b)}.

Restricting a and b to omega and to "difference[s] of 1" one gets

P def= {(a, b) | a, b e omega & -(a, b) = 1}
= {(a, a + 1) | a e omega }

Since there is a bijection between P and omega, namely

B: P x omega def= {((a, a + 1), a) | a e omega},

it follows that P ~ omega, meaning P is of same cardinality as omega.

Thus there are "as many difference[s] of 1 between natural numbers as
there are natural numbers". Since the cardinality omega is infinite
there *are* "infinitely many difference[s] of 1 between natural
numbers".

F. N.
--
xyz
.

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