Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: Proof 0.999... is not equal to one.
Replies: 194   Last Post: Feb 16, 2017 5:56 PM

 Search Thread: Advanced Search

 Messages: [ Previous | Next ]
 Rupert Posts: 3,810 Registered: 12/6/04
Re: Proof 0.999... is not equal to one.
Posted: May 31, 2007 3:02 AM
 Plain Text Reply

On May 31, 4:16 pm, chaja...@mail.com wrote:
> I have written a proof that 0.999... cannot be equal to one in the
> system of real numbers.
>
> While at the end of it all you may not fully agree with my proof, much
> I as have never seen a proof asserting they were equal that I was able
> to consider valid, I'm sure you will agree that the
> ideas I present are not a simply rehashing of basic objections of
> others before me.
>
> It is available in several formats:http://www17.brinkster.com/chajadan/Math/Proofs/Proof1.dochttp://www17.brinkster.com/chajadan/Math/Proofs/Proof1.odthttp://www17.brinkster.com/chajadan/Math/Proofs/Proof1.txt
>
> --Charles J. Daniels
> chaja...@mail.com

I can't gain access to that webpage.

Here are the generally accepted axioms for the real numbers:

(1) For any real numbers a, b, and c, a+(b+c)=(a+b)+c
(2) For any real numbers a, b, a+b=b+a
(3) There exists a unique number 0 such that for all numbers a, a+0=a
(4) For all numbers a there exists a unique number -a such that a+(-
a)=0
(5) For all real numbers a, b, c, a.(b.c)=(a.b).c
(6) For all real numbers a, b, a.b=b.a
(7) There exists a unique number 1, different from 0, such that for
all
numbers a, a.1=a
(8) For all numbers a different from 0 there exist a unique number
a^(-1) such that a.a^(-1)=1
(9) For all numbers a, b, c, a.(b+c)=(a.b)+(a.c)
(10) For all numbers a, b, if a>0 and b>0, then a.b>0
(11) For all numbers a, b, if a>0 and b>0, then a+b>0
(12) For all numbers a, b, a<b if and only b>a
(13) For all numbers a, b, c, if a>b, then a+c>b+c
(14) For every set S of real numbers, if S is nonempty and there
exists
a number a such that for all x in S, a>x or a=x, then there exists a
number b with that property such that, for every number a with that
property, b<a or b=a. The number b is called the least upper bound of
S.

We define 10 to be 1+(1+(1+(1+(1+(1+(1+(1+(1+1))))))))).

We define S to be the set of all real numbers x which are in every set
T
with the property that 10^(-1) is in T, and whenever y is in T, y.
10^(-1)
is also in T. Then we define S' to be the set of all numbers x such
that
1-x is in S. Thus S={0.9, 0.99, 0.999, ...}.

We can prove from the above axioms that there exists a unique number x
such that x is an upper bound for S (i.e. is greater than or equal to
every
member of S) and is less than or equal to 1. We call this number
0.9999....

It can be proved from the axioms that this number is also equal to 1.

Thus 0.9999...=1. There is certainly no doubt that this follows from
the axioms given using second-order logic. I can show you the details
if you like.

If your proof also uses these axioms, then you've shown that the
axioms
are inconsistent, but I don't think this is very likely. It's a shame
I
can't see your proof and show you the mistake.

Date Subject Author
5/31/07 chajadan@mail.com
5/31/07 karl
5/31/07 chajadan@mail.com
5/31/07 karl
5/31/07 chajadan@mail.com
5/31/07 karl
5/31/07 chajadan@mail.com
5/31/07 Virgil
6/1/07 chajadan@mail.com
6/1/07 Richard Tobin
5/31/07 pomerado@hotmail.com
5/31/07 chajadan@mail.com
5/31/07 Glen Wheeler
5/31/07 The Ghost In The Machine
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
5/31/07 Glen Wheeler
5/31/07 chajadan@mail.com
5/31/07 chajadan@mail.com
5/31/07 Glen Wheeler
5/31/07 chajadan@mail.com
5/31/07 David W. Cantrell
5/31/07 chajadan@mail.com
6/5/07 Michael Press
5/31/07 Dr. David Kirkby
5/31/07 mensanator
5/31/07 chajadan@mail.com
5/31/07 mensanator
5/31/07 Jesse F. Hughes
5/31/07 Dik T. Winter
5/31/07 Rupert
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
2/2/17 JÃÂ¼rgen R.
2/2/17 abu.kuanysh05@gmail.com
5/31/07 William Hughes
5/31/07 chajadan@mail.com
5/31/07 chajadan@mail.com
5/31/07 Virgil
5/31/07 chajadan@mail.com
5/31/07 quasi
5/31/07 chajadan@mail.com
5/31/07 quasi
5/31/07 chajadan@mail.com
5/31/07 quasi
5/31/07 William Hughes
5/31/07 chajadan@mail.com
5/31/07 William Hughes
6/1/07 hagman
6/1/07 chajadan@mail.com
6/1/07 William Hughes
5/31/07 T.H. Ray
5/31/07 Jesse F. Hughes
5/31/07 T.H. Ray
5/31/07 Jesse F. Hughes
5/31/07 T.H. Ray
5/31/07 Jesse F. Hughes
5/31/07 Denis Feldmann
5/31/07 T.H. Ray
5/31/07 chajadan@mail.com
5/31/07 T.H. Ray
5/31/07 Dave Seaman
5/31/07 T.H. Ray
5/31/07 William Hughes
5/31/07 Jesse F. Hughes
5/31/07 chajadan@mail.com
6/1/07 Eric Schmidt
6/1/07 chajadan@mail.com
6/3/07 T.H. Ray
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
2/2/17 bassam king karzeddin
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
2/2/17 JÃÂ¼rgen R.
5/31/07 William Hughes
5/31/07 chajadan@mail.com
5/31/07 Dave Seaman
5/31/07 chajadan@mail.com
6/1/07 Glen Wheeler
5/31/07 William Hughes
6/1/07 chajadan@mail.com
6/1/07 William Hughes
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
5/31/07 Glen Wheeler
5/31/07 Marshall
6/5/07 Michael Press
5/31/07 bassam king karzeddin
5/31/07 Glen Wheeler
5/31/07 bassam king karzeddin
5/31/07 bassam king karzeddin
5/31/07 neilist
5/31/07 tommy1729
5/31/07 neilist
5/31/07 tommy1729
5/31/07 neilist
5/31/07 tommy1729
5/31/07 chajadan@mail.com
5/31/07 Dave Seaman
5/31/07 quasi
5/31/07 chajadan@mail.com
5/31/07 quasi
6/1/07 Dr. David Kirkby
6/1/07 quasi
6/1/07 chajadan@mail.com
6/1/07 hagman
6/1/07 chajadan@mail.com
5/31/07 hagman
5/31/07 chajadan@mail.com
6/1/07 Dr. David Kirkby
6/1/07 hagman
6/1/07 chajadan@mail.com
6/1/07 Eric Schmidt
6/1/07 chajadan@mail.com
6/1/07 hagman
6/1/07 chajadan@mail.com
6/2/07 hagman
6/18/07 chajadan@mail.com
5/31/07 Richard Tobin
5/31/07 mathedman@hotmail.com.CUT
5/31/07 Richard Tobin
5/31/07 William Hughes
5/31/07 Jesse F. Hughes
5/31/07 Brian Quincy Hutchings
5/31/07 Brian Quincy Hutchings
6/1/07 chajadan@mail.com
6/1/07 Richard Tobin
6/1/07 chajadan@mail.com
6/1/07 Jesse F. Hughes
6/1/07 Richard Tobin
6/1/07 Dik T. Winter
6/1/07 Jesse F. Hughes
6/1/07 Brian Quincy Hutchings
5/31/07 Dr. David Kirkby
5/31/07 chajadan@mail.com
5/31/07 quasi
5/31/07 quasi
5/31/07 quasi
6/1/07 Dr. David Kirkby
6/1/07 Virgil
6/1/07 Dr. David Kirkby
6/1/07 chajadan@mail.com
6/1/07 Dr. David Kirkby
6/1/07 Dik T. Winter
6/1/07 bassam king karzeddin
6/1/07 Dr. David Kirkby
3/22/13 John Gabriel
3/22/13 John Gabriel
6/1/07 Dr. David Kirkby
6/1/07 chajadan@mail.com
6/1/07 Denis Feldmann
6/1/07 chajadan@mail.com
2/7/13 Brian Q. Hutchings
2/8/13 JT
2/8/13 Virgil
2/8/13 JT
2/8/13 Virgil
2/8/13 Virgil
2/8/13 JT
2/8/13 Virgil
2/21/13 John Gabriel
6/1/07 JEMebius
6/1/07 bassam king karzeddin
2/2/17 bassam king karzeddin
6/1/07 mike3
9/26/07 JEMebius
9/26/07 mike3
9/27/07 Brian Quincy Hutchings
6/2/07 OwlHoot
6/3/07 jsavard@ecn.ab.ca
6/5/07 zuhair
6/10/07 Brian Quincy Hutchings
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
2/2/17 JÃÂ¼rgen R.
2/2/17 Robin Chapman
2/2/17 JÃÂ¼rgen R.
2/2/17 R.J.Chapman
2/2/17 JÃÂ¼rgen R.
2/2/17 JÃÂ¼rgen R.
2/3/17 R.J.Chapman
2/8/17 George Cornelius
2/8/17 abu.kuanysh05@gmail.com
2/13/17 Dan Christensen
2/13/17 bassam king karzeddin
2/13/17 bursejan@gmail.com
2/15/17 William Hughes
2/15/17 netzweltler
2/15/17 William Hughes
2/15/17 William Hughes
2/15/17 netzweltler
2/15/17 William Hughes
2/15/17 netzweltler
2/15/17 Peter Percival
2/16/17 bassam king karzeddin
2/16/17 Peter Percival
2/15/17 William Hughes

© The Math Forum at NCTM 1994-2017. All Rights Reserved.