Re: *** T. Winter says: Definition: sum{i in N} i = 0

On 11 Jun., 21:19, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
On 11 Jun., 04:30, "*** T. Winter" <***.Win...@xxxxxx> wrote:

In article <1181383051.610959.236...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> WM <mueck...@xxxxxxxxxxxxxxxxx> writes:
> On 9 Jun., 01:13, "*** T. Winter" <***.Win...@xxxxxx> wrote:
> > By the axiom of infinity there exists a set without last element...

> Yes, that is our basis of discussion. But the truth of
> sum{n = 1 to oo} 1/2^n is not larger than 1
> is prior to set theory (and not in contradiction with the axiom of
> infinity) , because
> 1) it has been proved without any formalization or axiomatics of
> mathematics
> 2) it can be proved by pure logic.

I do not understand how you can state that something that has not
been defined can be compared with a number.

sum{n = 1 to oo} 1/2^n has been defined. It is the abbreviation of 1/2
+ 1/4 + 1/8 + ... and these expressions also have been defined in the
elementary school.



> Also the truth of
> sum{n = 1 to oo} n is not less than 1
> is prior to set theory (and, as H&J show, not in contradiction with
> the axiom of infinity) , because
> 1) it has been proved without any formalization or axiomatics of
> mathematics
> (and one must have a very scabby brain to doubt this)
> 2) it can be proved by pure logic.

I do not understand how you can state that something that has not
been defined can be compared with a number.

How do you compare something undefined with a natural number? What
are the rules?

sum{n = 1 to oo} n has been defined. It is the abbreviation of 1 + 2 +
3 + ... and these expressions also have been defined in the elementary
school.

> By using the same logic we can also conclude that it is false to state
> |{2,4,6,...}| is a number which is larger than any natural number.
> This is prior to set theory and in contradiction with set theory,
> though not in contradiction with the axiom of infinity. But it is in
> contradiction with the assumption that every set must have a cardinal
> number. Don't know why this silly idea contaminated the brains of some
> people.

Again comparing something that is undefined outside set theory with
a number. How do you compare those undefined things?

See elementary school.



> It is as undefinable as sum{n = 1 to oo} 1 which,
> however, according to set theory is aleph_0.

According to H&J with *their* definition of infinite sums, and using
cardinal arithmetic. I am neither using their definition, nor
cardinal arithmetic. I do in the definition not even use arithmetic
at all.

sum{n = 1 to oo} 1 is larger than any real number. See elementary
school.



> But IF we accept this
> result, namely sum{n = 1 to oo} 1 = |{1,2,3,...}| = aleph_0, THEN
> sum{n = 1 to oo} n > 1, ndependent of any further definition. That
> means you need only the assumption |{1,2,3,...}| = aleph_0 to conclude
> by the majorant criterion that sum{n = 1 to oo} n > 1.

Wrong. H&J *define* sum{n = 1 to oo} 1 as |{1,2,3,...}|, it is not
a result, but a definition (or, if you want to stress it, the result
of their definition about the addition of infinitely many cardinal
numbers). So outside the context of *that* definition that result
does not exist.

That is nothing but a very cheap attempt to save set theory.



> This theorem relies upon the limit of a Cauchy sequence which is the
> logical continuation of the finite case.

Wrong. The theorem relies upon the definition of the real numbers. Pray
learn some very basic mathematics. That Cauchy sequences converge follows
from the definition of real numbers.

Again just the wrong approach. Real numbers can be defined as
equivalence classes of convergent Cauchy sequences _because_ Cauchy
sequences converge.

Regards, WM

So where does the Cauchy sequence 1 -1/2 +1/3 -1/4 +-... converge to
*before* you have defined real numbers?

.

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