Re: Proof 0.999... is not equal to one.

On Jun 1, 2:31 am, hagman <goo...@xxxxxxxxxxxxx> wrote:
On 1 Jun., 08:55, chaja...@xxxxxxxx wrote:





If you insist, here is the first obvious mistake in the paper:

-------------------------------------------
Let S be the set of all real numbers in the interval [0,1]
Let T be the set of all real numbers in the interval (-1,0]

If you applied an operator '+' to these two sets that sums the elements
of both sets, you would get: S + T = 1
-------------------------------------------

You are trying to sum an uncountable set of numbers. Please define what
this means. Also consider: if you pair the numbers differently, can you
make the sum come out to something else?

FInally, I didn't notice anything looking like a proof in your article,
though I admit I haven't read all of it.

--
Eric Schmidt

--
Posted via a free Usenet account fromhttp://www.teranews.com

Hi Eric,

what I am attempting to point out in the S + T result is that any
element of the infinite set T corresponds to exactly one element in
the infinite set S of equal value and opposite sign, except for the
positive 1 in set S. Should you hold the infinity of values in your
awareness, all cancel except for one that cannot.

You would not be able to have a final result upon summing the entire
infinity of elements in S union T other than 1.

--charlie

Oh, so you want to define sum(X) for certain subsets X of the reals?
This should be easy if X is finite: just sum the elements (fortunately
addition is associative and commutative).
It should also be easy to define sum(X) if sum(Y) is defined
where Y = X\(-X): just set sum(X)=sum(Y).

I don't understand what the backslash means.

One should investigate any further definitions as to whether all that
stuff yields a consistent definition of sum.

Your example amounts to obtaining sum( (-1,1] ).
With X = (-1,1], I would calculate Y = X\(-X) = {1} and obtain
sum(X)=sum(Y)=1.

I would agree sum(X) = 1, I don't understand the construction of Y to
say.


Nothing has been said yet about sum(S) if S\(-S) is infinite, esp.
this has no relevance yet if S is the set of all rationals of the form
9/10^n.

However, we can form the set union of -S and 10 times S, i.e.
let C = (-S) u (10*S).
It turns out that C\(-C) is {9}.

I don't believe this would be accurate. If C were defined as (-S) u (S
+ {9}) I would agree.


We are forced to conclude sum(C)=9.

If t is a non-zero number and sum(S) is defined, can sum(t*S) be
anything
but defined and equal to t*sum(S)?

I wouldn't think so.


If A and B are disjoint and both sum(A) and sum(B) are defined,
can sum(A u B) be anything but defined and equal to sum(A)+sum(B) ?


I wouldn't think so, and I would tend to use a multiset when doing
this kind of thing, so I wouldn't require they be disjoint.

In fact, the last two issues are what motivates the way to
calculate sum(X) from sum( X\(-X) ) used above and introduced by you.

I'm still not sure what \ means.


Thus sum(-S) = -sum(S), sum(10*S)=10 *sum(S)
Agreed

and finally
sum(-S u 10*S) = 9*sum(S).

It should

Hence sum(S) = sum(C)/9 = 1.

I disagree sum(C) = 9. I find it strictly less.

hagman- Hide quoted text -

- Show quoted text -

--charlie

.

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