Can this Theorem be disproved?

Page 1 of 2 [ 22 posts ] Go to page 1, 2 Next

Ahaseurus2000
Veteran
Veteran

Joined: 21 Sep 2007
Gender: Male
Posts: 1,546
Location: auckland

01 Apr 2008, 1:23 am

for any pair of Real Numbers, there is a Real Number between them.

So: 1.0 and 1.1 have 1.05 between them, for example.

_________________
Life is Painful. Suffering is Optional. Keep your face to the Sun and never see your Shadow.

twosheds
Raven
Raven

Joined: 23 Feb 2007
Age: 44
Gender: Male
Posts: 108

01 Apr 2008, 1:58 am

A set with that property is said to be densely ordered. The real numbers are such a set, as are the rational numbers.

http://www.abstractmath.org/MM/MMRealDensity.htm

SqrachMasda
Sea Gull

Joined: 29 Aug 2004
Gender: Male
Posts: 211
Location: not sure

01 Apr 2008, 3:12 pm

sounds like what i like to call
Cantor's Banter

twoshots
Veteran
Veteran

Joined: 26 Nov 2007
Age: 39
Gender: Male
Posts: 3,731
Location: Boötes void

01 Apr 2008, 10:43 pm

SqrachMasda wrote:

sounds like what i like to call
Cantor's Banter

I don't know what you mean, but +1 :lol:

...and twosheds... there is a great disturbance in the force...

_________________
* here for the nachos.

abstrusemortal
Sea Gull

Joined: 7 Feb 2007
Age: 35
Gender: Male
Posts: 218
Location: DC/VA area

02 Apr 2008, 10:23 am

Ahaseurus2000 wrote:

for any pair of Real Numbers, there is a Real Number between them.

So: 1.0 and 1.1 have 1.05 between them, for example.

I think that if this could be disproved than PI would actually be finite........

_________________
Uninvention Convention

EvilTeach
Pileated woodpecker

Joined: 14 Mar 2007
Gender: Male
Posts: 196

13 Apr 2008, 2:37 pm

sure it is true.

pick two different values as end points

average them

that is a new value between the two.

replace either end point with the new value.

repeat until the end points are the same value.....

i wonder if you could do it formally with mathematical induction....

GoatOnFire
Veteran
Veteran

Joined: 22 Feb 2007
Age: 38
Gender: Male
Posts: 4,986
Location: Den of the ecdysiasts

13 Apr 2008, 3:48 pm

Ahaseurus2000 wrote:

for any pair of Real Numbers, there is a Real Number between them.

So: 1.0 and 1.1 have 1.05 between them, for example.

Worded that way it can be disproved. It said any pair of real numbers, so if you pick the same number twice...

_________________
I will befriend the friendless, help the helpless, and defeat... the feetless?

twoshots
Veteran
Veteran

Joined: 26 Nov 2007
Age: 39
Gender: Male
Posts: 3,731
Location: Boötes void

14 Apr 2008, 11:19 am

EvilTeach wrote:

i wonder if you could do it formally with mathematical induction....

No. Mathematical induction only deals with integers. Besides the average 'em out method works quite well as a constructive proof.

_________________
* here for the nachos.

Dhp
Veteran
Veteran

Joined: 30 Jul 2007
Age: 51
Gender: Male
Posts: 538

11 May 2008, 12:41 am

First of all, I don't know much of mathematics besides that I love the subject.

Well, in Rudin's "Principles of Mathematical Analysis", on page 41, there is a proof that shows exactly what you are stating, but in different words. A Corollary to this theorem states that Every interval [a,b] (a,b) is uncountable. In particular, the set of all real numbers is uncountable. I unfortunately cannot prove this at the current moment in time due to lack of mathematical knowledge, but look up that book. It is an excellent book on introductory analysis. Hopefully, in a few years of study, I might be able to explain it to others. I hope this helped you a little.

Escuerd
Raven
Raven

Joined: 1 May 2008
Age: 41
Gender: Male
Posts: 101

11 May 2008, 3:32 am

abstrusemortal wrote:

Ahaseurus2000 wrote:

for any pair of Real Numbers, there is a Real Number between them.

So: 1.0 and 1.1 have 1.05 between them, for example.

I think that if this could be disproved than PI would actually be finite........

Pi is finite. It's just not rational. But this property holds for rational numbers as well as real numbers. They're both dense sets.

wolphin
Velociraptor

Joined: 15 Aug 2007
Age: 37
Gender: Male
Posts: 465

11 May 2008, 5:53 am

Dhp wrote:

I don't have that book so I can't be exactly sure, but it sounds like something in a section on Bolzano-Weierstrass or Cantor's 1st uncountability proof. If so then you don't really need all that thanks to a later, much simpler argument by Cantor: http://planetmath.org/encyclopedia/CantorsDiagonalArgument.html

Also, density in and of itself doesn't imply uncountable. The rationals satisfy the same property as mentioned.

And as GoatOnFire said, just pick the same real number twice. No real number is between that pair.

Orwell
Veteran
Veteran

Joined: 8 Aug 2007
Age: 35
Gender: Male
Posts: 12,518
Location: Room 101

11 May 2008, 10:03 am

Ahaseurus2000 wrote:

for any pair of Real Numbers, there is a Real Number between them.

So: 1.0 and 1.1 have 1.05 between them, for example.

Disproving that would imply that there is some type of discrete, fundamentally smallest unit that comprises real numbers, akin to the atomic theory of matter. Because of infinitesimals and the way we have defined our number system, this is held to be false.

_________________
WAR IS PEACE
FREEDOM IS SLAVERY
IGNORANCE IS STRENGTH

D1nk0
Veteran
Veteran

Joined: 11 Dec 2007
Age: 46
Gender: Male
Posts: 1,587

14 May 2008, 6:32 pm

twoshots wrote:

EvilTeach wrote:

i wonder if you could do it formally with mathematical induction....

No. Mathematical induction only deals with integers. Besides the average 'em out method works quite well as a constructive proof.

errrr, ever heard of Transfinite Induction?

Orwell wrote:

Ahaseurus2000 wrote:

for any pair of Real Numbers, there is a Real Number between them.

So: 1.0 and 1.1 have 1.05 between them, for example.

The Reals are infinite, which means that there is NO smallest non-zero real number. And they are uncountable so between any 2 distinct reals there are uncountably infinitely many reals between those 2 numbers. Disproving such a theorem would imply that the Reals are countable which was disprove (first)by George Cantor in the 19th century.

twoshots
Veteran
Veteran

Joined: 26 Nov 2007
Age: 39
Gender: Male
Posts: 3,731
Location: Boötes void

14 May 2008, 7:17 pm

D1nk0 wrote:

errrr, ever heard of Transfinite Induction?

Actually I hadn't. Thanks.

_________________
* here for the nachos.

wolphin
Velociraptor

Joined: 15 Aug 2007
Age: 37
Gender: Male
Posts: 465

14 May 2008, 7:51 pm

D1nk0 wrote:

Disproving such a theorem would imply that the Reals are countable which was disprove (first)by George Cantor in the 19th century.

Not necessarily, actually. Of course, neither density nor uncountability can actually be disproven with the reals cause they're true, but in a more general sense, it depends on the type of space you're working in.

For example, consider the real numbers as a metric space and add an extra "ghost zero" member to that space and define the metric as you would expect (i.e., the "ghost" zero behaves like zero, except it's not equal to zero), "ghost zero" and "real zero" are then two distinct points of the metric space but there is no point "between" them ("between" defined appropriately as well). Yet the space is still "uncountable" - the absence of density within specific subsets of a space doesn't necessarily imply some countability.

D1nk0
Veteran
Veteran

Joined: 11 Dec 2007
Age: 46
Gender: Male
Posts: 1,587

15 May 2008, 12:26 am

wolphin wrote:

D1nk0 wrote:

Disproving such a theorem would imply that the Reals are countable which was disprove (first)by George Cantor in the 19th century.

I wanna see some citations/referrences for this wolphin.............

Page 1 of 2 [ 22 posts ] Go to page 1, 2 Next

Jump to:

Can this Theorem be disproved? Post Reply New Topic

Can this Theorem be disproved?