Question

Can someone explain me why there are more real numbers between 0 and 1 than the set of natural number

Answer 1

wikipedia:
Our intuition gained from finite sets breaks down when dealing with infinite sets. In the late nineteenth century Georg Cantor, Gottlob Frege, Richard Dedekind and others rejected the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. One example of this is Hilbert's paradox of the Grand Hotel.

The reason for this is that the various characterizations of what it means for set A to be larger than set B, or to be the same size as set B, which are all equivalent for finite sets, are no longer equivalent for infinite sets. Different characterizations can yield different results. For example, in the popular characterization of size chosen by Cantor, sometimes an infinite set A is larger (in that sense) than an infinite set B; while other characterizations[which?] may yield that an infinite set A is always the same size as an infinite set B.

For finite sets, counting is just forming a bijection (i.e., a one-to-one correspondence) between the set being counted and an initial segment of the positive integers. Thus there is no notion equivalent to counting for infinite sets. While counting gives a unique result when applied to a finite set, an infinite set may be placed into a one-to-one correspondence with many different ordinal numbers depending on how one chooses to "count" (order) it.

Additionally, different characterizations of size, when extended to infinite sets, will break different "rules" which held for finite sets. Which rules are broken varies from characterization to characterization. For example, Cantor's characterization, while preserving the rule that sometimes one set is larger than another, breaks the rule that deleting an element makes the set smaller. Another characterization may preserve the rule that deleting an element makes the set smaller, but break another rule.[citation needed] Furthermore, some characterization may not "directly" break a rule, but it may not "directly" uphold it either, in the sense that whichever is the case depends upon a controversial axiom such as the axiom of choice or the continuum hypothesis. Thus there are three possibilities. Each characterization will break some rules, uphold some others, and may be indecisive about some others.

If one extends to multisets, further rules are broken (assuming Cantor's approach), which hold for finite multisets. If we have two multisets A and B, A not being larger than B and B not being larger than A does not necessarily imply A has the same size as B.[citation needed] This rule holds for multisets that are finite. Needless to say, the law of trichotomy is explicitly broken in this case, as opposed to the situation with sets, where it is equivalent to the axiom of choice.

Dedekind simply defined an infinite set as one having the same size (in Cantor's sense) as at least one of its proper parts; this notion of infinity is called Dedekind infinite. This definition only works in the presence of some form of the axiom of choice, however, so will not be considered to work by some mathematicians.

Cantor introduced the above-mentioned cardinal numbers, and showed that (in Cantor's sense) some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers (ℵ0).
Source(s):
http://en.wikipedia.org/wiki/Cardinality …

Answer 2

Look up "cantor's proof" and Cantor's diagonal counting

Answer 3

Did u get that now!

Answer 4

@Rohangrr your citation contains NO PROOF, and NO HINT TO THE proof, and NOT EVEN THE GUIDANCE WHERE to LOOK FOR THE PROOF.

Answer 5

Hey I am not WIKIPEDIA I posted the link where you'll get the proof to prove @sauravshakya 's answer

Answer 6

I repeat CANTOR'S DIAGONAL METHOD will supply the proof

Answer 7

I repeat CANTOR'S DIAGONAL METHOD will supply the proof

Answer 8

I repeat CANTOR'S DIAGONAL METHOD will supply the proof

Answer 9

were can I find that @Mikael

Answer 10

http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

Answer 11

Thanx

Answer 12

http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

Answer 13

Thanks @sauravshakya

Answer 14

Thanks @experimentX

Answer 15

no probs .. just supporting right argument.

Answer 16

Thanks @mathslover

Answer 17

Sorry to say..... I didnt get it

Answer 18

Well the intuition is thus: one builds a meandering line which passes through ALL the points of an infinite matrix (which contains all the raional numbers)

Answer 19

mathematicians are sadists

Answer 20

they make simple things too complicated

Answer 21

Then , one CONSTRUCT an "exceptional" number, which will be different from every rational number in at least one digit. Such "exceptional" number will not be rational.

Answer 22

Actually only my last remark is needed. The first is for something else - the countability of rationals

Answer 23

It isn't anything to do with our intuition breaking down, it all starts once you allow the creation of a completed infinite set ie a ZFC axiom of infinity.
(After that you can prove nonsense like Banach Tarski....)

Answer 24

natural numbers you can count .. real numbers you can't count.
if you try to count real number ... you will construct an exceptional number which would be made up of diagonal digits which is not equal to any one of them.

http://upload.wikimedia.org/wikipedia/commons/thumb/0/01/Diagonal_argument_2.svg/250px-Diagonal_argument_2.svg.png
only possible when you can't count them.

Answer 25

The set of even number and natural number are equal right?

Answer 26

As one famous great Frenchman once said , when people called his profetic math "nonsense":
Con- jugerais

Answer 27

Not equal - equal POWER

Answer 28

Can someone explain me why there are more real numbers between 0 and 1 than the set of natural number

Or between 0 and 0.1 or between 0 and 0.01 or between....

Answer 29

Isn't it the whole discussion here about ?????

Answer 30

I think I got it....... THANX @experimentX for that pic

Answer 31

Ya that number will be very different

Answer 32

Aand , if we ontinue in such manner @estudier - why not ask , whether 5 cubes, and 5 spheres are the same number ? If you want answer read Peano . It is not an easy readung i must warn you

Answer 33

sets of both numbers is infinite ... one you can count the other you cannot count.

Answer 34

Thanks @ParthKohli

Answer 35

Mikael, I am completely familiar with all it, have studied it many times over years and concluded it is complete nonsense...
I am a constructivist, haven't any time for this classical position (probably also if you can't compute it, its not worth doing anyway)

Answer 36

Now, what about set of natural number and set of even number

Answer 37

Are they equal

Answer 38

By the way
A) there are middle paths between these approaches
B) Some things in analysis are unconstructible

Answer 39

"Are they equal" What do you mean by @equal@

Answer 40

Do both sets have equal number of elements

Answer 41

you can't exactly say they are equal ... equal implies same elements. perhaps you meant they have same cardinality.

Answer 42

Sorry Didnt know that

Answer 43

define a mapping by
f:x->2x, you will get one to one relationship between those two numbers.

Answer 44

In effect, they have been defined as "equal" in a certain sense.

Answer 45

So, cant I do the same for all real numbers between 0 and 1...... and the set of natural number

Answer 46

no you can't ... that's the whole argument of cantor diagonel

Answer 47

but you can do for real rational numbers.

Answer 48

U mean fractions

Answer 49

Or repeating decimals...

Answer 50

Cant all repeating numbers be expressed in fractions

Answer 51

yep

Answer 52

Maybe you can have some good idea from here
http://en.wikipedia.org/wiki/Continuum_hypothesis

Answer 53

http://upload.wikimedia.org/wikipedia/commons/thumb/8/85/Diagonal_argument.svg/429px-Diagonal_argument.svg.png

Answer 54

What I did the same:
1/10^1 1/10^2 1/10^3 1/10^4 1/10^5.....
2/10^1 2/10^2 2/10^3 2/10^4 2/10^5....
3/10^1 3/10^2 3/10^3 3/10^4 3/10^5...
.
.
.

Answer 55

So cant we list all the real numbers between 0 and 1 like that

Answer 56

Do the same thing

Answer 57

yes, you can do that only to get the rational numbers ... there are irrational numbers to consider and there are transcendental numbers ....
the set of irrational numbers is uncountable.

Answer 58

Most people forgot that Cantor did 2 proofs, the other is more interesting (IMO) although not considered as "pretty" or "elegant"....