Re: An uncountable countable set

mueckenh@xxxxxxxxxxxxxxxxx wrote:

Franziska Neugebauer schrieb:
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.

You are in error.

Where precisely is the error?

You just proved it to be true.

You may "just" as well prove the opposite. Please do so.

The set of natural numbers (i.e., finite numbers n, i.e., numbers
with finitely many differences of 1 between 1 and n) does not yield
infinitely many differences of 1.

This is a reiteration not a proof.

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".

Correct, but your final conclusion is the typical mistake of set
theorists.

Do you agree that P ~ omega?
Do you agree that card (omega) /e omega?

1) The set exist.
2) The elements do not exist.

What type of comment do you expect here?

You state it in the form:

1) There are infinitely many differences of 1.

Do you agree that P ~ omega is the correct formalization of this
statement?

If not: What _is_ the correct formalization of the statement?

2) The sum of these is not infinite. Why not?

You have agreed to that this limit does not exist ("There is no L in
N"). So the sum is at least _not_ _finite_ (not in omega).

You should try to comprehend what "existence" means.

Do not confuse me with your therapist.

F. N.
--
xyz
.

Relevant Pages

  • Re: An uncountable countable set
    ... lim sum 1 = lim n =def L ... Since there is a bijection between P and omega, ... The sum of these is not infinite. ...
    (sci.math)
  • Re: An uncountable countable set
    ... lim sum 1 = lim n =def L ... "How many differences there are" means the cardinality of the set ... Since there is a bijection between P and omega, ...
    (sci.math)