# Strange but True: Infinity Comes in Different Sizes

If you were counting on infinity being absolute, your number's up

A LARGER INFINITY: Mathematically speaking, some infinities are bigger than others, such as the infinity of numbers with decimals in them exceeding that of counting numbers (1,2,3,4...). Image:

• ### Neutrino Hunters

In the 1995 Pixar film Toy Story, the gung ho space action figure Buzz Lightyear tirelessly incants his catchphrase: "To infinity … and beyond!" The joke, of course, is rooted in the perfectly reasonable assumption that infinity is the unsurpassable absolute—that there is no beyond.

That assumption, however, is not entirely sound. As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.

Take, for instance, the so-called natural numbers: 1, 2, 3 and so on. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the "reals." (This set comprises all numbers that can be represented as a decimal, even if that decimal representation is infinite in length. Hence, 27 is a real number, as is π, or 3.14159….)

In fact, Cantor showed, there are more real numbers packed in between zero and one than there are numbers in the entire range of naturals. He did this by contradiction, logically: He assumes that these infinite sets are the same size, then follows a series of logical steps to find a flaw that undermines that assumption. He reasons that the naturals and this zero-to-one subset of the reals having equally many members implies that the two sets can be put into a one-to-one correspondence. That is, the two sets can be paired so that every element in each set has one—and only one—"partner" in the other set.

Think of it this way: even in the absence of numerical counting, one-to-one correspondences can be used to measure relative sizes. Imagine two crates of unknown sizes, one of apples and one of oranges. Withdrawing one apple and one orange at a time thus partners the two sets into apple-orange pairs. If the contents of the two crates are emptied simultaneously, they are equally numerous; if one crate is exhausted before the other, the one with remaining fruit is more plentiful.

Cantor thus assumes that the naturals and the reals from zero to one have been put into such a correspondence. Every natural number n thus has a real partner rn. The reals can then be listed in order of their corresponding naturals: r1, r2, r3, and so on.

Then Cantor's wily side begins to show. He creates a real number, called p, by the following rule: make the digit n places after the decimal point in p something other than the digit in that same decimal place in rn. A simple method would be: choose 3 when the digit in question is 4; otherwise, choose 4.

For demonstration's sake, say the real number pair for the natural number 1 (r1) is Ted Williams's famed .400 batting average from 1941 (0.40570…), the pair for 2 (r2) is George W. Bush's share of the popular vote in 2000 (0.47868…) and that of 3 (r3) is the decimal component of π (0.14159…).

Now create p following Cantor's construction: the digit in the first decimal place should not be equal to that in the first decimal place of r1, which is 4. Therefore, choose 3, and p begins 0.3…. Then choose the digit in the second decimal place of p so that it does not equal that of the second decimal place of r2, which is 7 (choose 4; p = 0.34…). Finally, choose the digit in the third decimal place of p so that it does not equal that of the corresponding decimal place of r3, which is 1 (choose 4 again; p = 0.344…).

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1. 1. imuruk 04:29 AM 11/14/07

It is not clear cut that there are different kinds of infinity. See the present debate led by Rodych on Wittgensteins criticisms of transfinite set theory.

2. 2. Nani77 02:39 PM 11/20/07

My head's about to explode. I'm not an expert and I'm still trying to recover from that explanation. Can someone please explain it in simple terms?

3. 3. Larry in Tampa 12:57 PM 11/22/07

I suspect that the wiley Cantor could have performed the 'trick' in reverse (more naturals than reals) with equal validity.

4. 4. Rich_G 05:09 PM 12/7/07

Am I missing something? The infinite series 1,2,3,... is clearly larger than the infinite series 2,3,4,... , isn't it?

5. 5. ggiann 08:18 PM 12/11/07

Cantor s argumentation is probably one of the first allusions to fractals: An infinite series contains in an, even infinitesimal, part of it an infinite number of elements similar to its entirety. Climax is everything. Interesting, uh?

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Edited by ggiann at 12/11/2007 12:27 PM

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Edited by ggiann at 12/11/2007 12:29 PM

6. 6. yp172894913 02:40 AM 12/21/07

Howdy, The described apples-oranges matching establishes equality only for finite sets: E.g., If apples and oranges are each labeled by natural numbers (so they are co-numerous), but apple N is removed with orange 2N (i.e., apple 1 with orange 2, apple 2 with orange 4, etc.), then the apple crate is "emptied" while infinitely many odd-numbered oranges remain. The correct statement is that 2 sets are co-numerous, or have the same cardinality, if and only if there EXISTS a one-to-one pairing that exhausts both sets. For infinite sets, there always exist non-exhausting matchings whether or not there is a true one-to-one correspondence.
Cantor's diagonal argument does indeed establish that the cardinality of the reals (or even those between 0 and 1) exceeds that of the natural numbers by showing that no one-to-one correspondence can exist.
Be well, Dave Kelly

7. 7. RubenV 12:13 AM 12/23/07

Somebody noted that the infinite series of 1, 2, 3, ... is obviously larger than that of 2, 3, 4, ...

I must agree, isn't this a much easier proof of different-sized infinities?

If:
t(n) = n+1
T(n) = n
(with n a natural number)

Then we can pair up t0 (=1) with T1 (=0), t1 (=2) with T2 (=2), t2 with T3, etc... Pairing up like this, we would have no element of the t(n) series to pair up with T0 (=0), so even though both are infinite, T(n) always has at least one element more.

Why isn't this mathematically acceptable?

8. 8. RPost1 07:26 PM 12/29/07

IMHO that the reasoning presented is far more complex than necessary and that simpler is better.

Let each natural number, say 27, be paired with itâ€™s equivalent real-number partner which is 27.000 . . . (an infinite number of zeros)

Then clearly all other real numbers with a whole number component (those digits left of the decimal point) of 27 will be unpaired with any natural number.

27 <--> 27.0000 . . .
? <--> 27.1 - no possible match
? <--> 27.2 - no possible match

?
28 <--> 28.0000 . . .

There are, of course, an infinite number of real numbers of the form: 27.nnnnnn . . .

Which means for every natural number . . .n. there are an infinite number of real numbers of the form . . .n.nnn. . .

Thus there are an infinite number of natural numbers and an (infinite times infinite) number of real numbers.

There isnâ€™t any need for â€˜diagonalizationâ€™.

Nani77 wanted a simpler explanation so here is another one:

Natural numbers can have an infinite number of digits to the left of the decimal point but have nothing but an infinite number of zeroes to the right of the decimal.

Real numbers can have an infinite number of digits to the left of the decimal and can also have an infinite number of digits to the right of the decimal.

You donâ€™t have to be a rocket scientist (I mean, mathematician) or be â€˜wilyâ€™ to see that for each natural number (all zeroes to the right of the decimal) there will be an infinite number of real numbers (non-zero digits to the right of the decimal) that have the same digits to the left of the decimal point.

Read on for an interesting twist to the articleâ€™s conclusion.

Can anyone comment on this from my high-school days?

We were taught that the natural number 1 is identical in value to the real number 0.9999. . .

That is, the real number that is zero to the left of the decimal and an infinite number of nines to the right of the decimal is equal to the natural (and real) number 1.000000

Is it true that these two numbers are equal? I never believed it.

If true then when doing the mapping suggested in the article, or my own simpler mapping, havenâ€™t we mapped two real numbers to the natural number 1.00000?

But wait! What about 2.000 and 1.9999 . . . â€“ these numbers must be identical also.

Since there are at least an infinite number of these real-number pairs donâ€™t these pairings reduce the level of infinity of the real numbers?

Summary:

I think there are two things that cloud this type of analysis:

The first is the perceived difference between pairing solutions that are performed â€˜seriallyâ€™ and those performed in parallel.

The article appears to correctly state that you canâ€™t pair the natural numbers with the real numbers in parallel.

But can the author dispute that it is possible to pair the natural numbers with the real numbers serially? For every real number you give me I can associate it with the next sequential natural number. And I can do this in perpetuity (not me personally, you understand).

Is the author saying that â€˜in the limitâ€™ I have not paired all of the numbers?

The second â€˜cloudâ€™ is the notion of â€˜the next highest real numberâ€™. Given the real number 1.0000 what is the next highest real number? What is the real number that is next lowest? Is the next lowest number 0.9999 . . . or is that the same value as 1.0 per the discussion above.

What is the next real number lower than 0.9999 . . .? Is there any mathematical way to express this number?

9. 9. RPost1 07:43 PM 12/29/07

One other point I wanted to make is that the decimal part of real numbers (the digits to the right of the decimal) has the same level of infinity as the natural numbers.

Consider the set of real numbers that have only zero to the left of the decimal point. This set of real numbers DOES have a one-to-one correspondence with the natural numbers.

This can be demonstrated easily by showing that the two sets are mirror images of each other.

1 <-> .1
2 <-> .2
. . .
9 <-> .9
10 <-> .01
11 <-> .11000 . . .
. . .
19 <-> .91000 . . .
20 <-> .02000 . . .

and so on.

The decimal portion of every possible real number can be created by the reflection of a natural number around the decimal point.

Thus, there is nothing â€˜magicâ€™ about the infinity of the decimals. It is a difference of perception. We create natural number sequences by incrementing the right-most digit. This is always possible for the natural numbers but not always possible for real numbers.

Consider the real number consisting of an infinite number of threeâ€™s (3) to the right of the decimal. This number is easy to imagine: simply divide one by three.

But what is the next highest real number? Is there a mathematical way to express the next highest real number? There is no â€˜right-mostâ€™ digit to increment!

10. 10. billy bob 07:49 AM 1/5/08

No Sh*t, I was calulating how much to withold for my federal taxes needless to say 36 went into infinity............................

11. 11. nevets101 07:51 AM 1/5/08

"(Cantor) assumes that these infinite sets are the same size". That is an absurd assumption and the fatal flaw in his work. Size is not a property that can be measured with regard to infinity and is thus undefined.

12. 12. codemnky21 12:10 PM 1/11/08

The assumption of the example is flawed. He tries to prove there are more real numbers between 0 and 1, than there are natural numbers. But to do this, he restricts the set of natural numbers (here 3 sets.) After the completion it is proclaimed that the new real number is not in the natural number set. However, to demonstrate the larger set of 0-1 infinity of real numbers to natural infinity, you would need to include both full sets. By doing this, any generated decimal will be included in the natural set. (i.e. if you came up with .344344344 ad infinitum, this would correspond with 344344344....., which only goes to demonstrate the equality of the infinities.

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Edited by codemnky21 at 01/11/2008 4:25 AM

13. 13. Bill Shcaller 05:58 AM 1/14/08

There are other ways of pairing numbers such that there are more integers than reals. For example if each integer is paired with a real number between 0 and one (to match the set used in the example), by just removing the decimal point, then the real number that was not in the set of real numbers, would have a match in on the integer side. However 10, 100, 1000... have not pair in the real side, so there are more integers than reals. What really happened is that be careful picking of the pairing process we make one appear larger than the other, like an optical illusion. It is not really bigger.

14. 14. khaley 03:42 AM 1/17/08

> Strange but True: Infinity Comes in Different Sizes

I have to say that the SciAm article was a bit disappointing. The conclusion is correct, but it isn't explained fully. I'll try to explain it a little differently, and address the objections raised by RPost1 and others:

First of all, I think everyone would agree that if you take two finite sets of things and put them into a 1-1 correspondence, you've shown pretty convincingly that the two sets have the same number of elements; that is, they're the same size. You don't have to count them, you just need to match every element in one with an element in the other, and when you're done, if there's nothing left unmatched, you can safely conclude that they have the same number.

Now, with infinite sets, it's a little weird. Notice that you can put the set of all natural numbers into a 1-1 correspondence with the set of even numbers as follows:

1 <-> 2
2 <-> 4
3 <-> 6
4 <-> 8
5 <-> 10
6 <-> 12
etc.

Now, clearly, there are natural numbers in the left set that aren't in the right set, but since the 1-1 correspondence exists, the two sets are considered to be the same size! Understand this point: When we're "counting" infinite sets, it doesn't matter that one set is a proper subset of the other (all the elements in one are in the other but not vice versa). If they can be put into a 1-1 correspondence, we'll consider them equal in "size." (actually, the word Cantor used was "cardinality". The cardinality of a finite set is simply the number of members in it. But since "infinity" isn't technically a number, we have to expand the meaning of cardinality for infinite sets.)

OK, fine. Maybe we can do the same with ALL infinite sets, and just declare that all infinities have the same cardinality (or size); namely infinity. Big deal.

Well, this is what mathematicians believed up until the time of Cantor. Cantor pondered a little further, asking himself if all the positive rational numbers could be put into a 1-1 correspondence with the real numbers. A rational number is simply the ratio of any two natural numbers. After all, rational numbers have this "dense" property that natural numbers do not: Between any two rational numbers there's always at least one more. Just average the two. If you think about that, you'll quickly come to the conclusion that between any two rational numbers there's an INFINITE number of rationals between them--just keep on taking the average of the lower number and the previous average. Wow! That is a dense set!

Can this set be put into a one-to-one correspondence with the natural numbers? Well, it turns out the answer is yes. Suppose you arrange the positive rational numbers in a grid where the columns of the grid are numbered from 1 to infinity (going to the right), and the rows are numbered from 1 to infinity (going down). Picture a very large Excel spreadsheet. In each cell, put the number you get by dividing the column number by the row number. So the fraction 13/39 would be found in cell at column 13 and row 39. Now, if we can number these cells so that every natural number is assigned to one and only one cell in this "infinite" grid, we've accomplished our correspondence. To do it, just number the cells as follows:
1 3 6 10 15
2 5 9 14
4 8 13
7 12
11

and so on, numbering the diagonals from the lower left to the upper right and continuing in this way ad infinitum. This way, we assign a unique natural number to every positive rational number, completing the 1-1 correspondence. So, you say, big deal. All infinities are the same size--and this just proved it again.

The next question Cantor pondered is this: Is there an infinite set that CANNOT be put into a 1-1 correspondence with the natural numbers. And, if so, how could we prove that? Well, if you consider the REAL numbers between zero and one, it's possible to show that no such correspondence can exist. This is what the SciAm article showed. If you go back and re-read it, you should be convinced that NO MATTER HOW you attempt to pair up the natural numbers with the real numbers, there will always be real numbers missing from your list. The diagonalization argument shows how you can produce at least one, no matter what the list looks like. So, since no 1-1 correspondence is possible, and since every effort to create one will always leave out at least on real number (actually it will leave out an infinity of them), we say that the set of real numbers must have a HIGHER cardinalilty than the set of natural numbers. Or in simple terms, it's a bigger set. The infinity of reals is larger than the infinity of natural numbers.

This brings up an interesting question. Why doesn't the above argument work with rational numbers, and contradict the proof above that we DO have a 1-1 correspondence between the natural numbers and the rational numbers? Can't I use the diagonalization trick to produce a rational number that's not in my list above? The answer is no, because sooner or later you have to allow the decimal expansion of your rational number to begin repeating, as all rational numbers do. Once that happens, you no longer have control of the remaining digits in the number and, sooner or later, the rational number you have in mind will indeed appear in the list we constructed above.

Now let's address the questions raised by RPost1:

He says

"IMHO that the reasoning presented is far more complex than necessary and that simpler is better.

Let each natural number, say 27, be paired with itâ€™s equivalent real-number partner which is 27.000 . . . (an infinite number of zeros)

Then clearly all other real numbers with a whole number component (those digits left of the decimal point) of 27 will be unpaired with any natural number.

27 <--> 27.0000 . . .
? <--> 27.1 - no possible match
? <--> 27.2 - no possible match

?
28 <--> 28.0000 . . .

"

The fact that you've shown a 1-1 correspondence that leaves out some real numbers doesn't accomplish what the proof in the article showed...which is that THERE IS NO POSSIBLE 1-1 correspondence which maps the reals to the natural numbers.

The odd thing about infinite sets is that there may be many ways to map them to each other in a 1-1 correspondence, and some of these mappings may leave out a lot of members of one of the sets or the other. But that's not sufficient to prove that one set is BIGGER, when you're talking about infinite sets. You have to show that there's no possible mapping that consumes all the members of both sets. If you do that (which Cantor did, as the article shows), then you've shown that one set is truly bigger. (Again, you have to accept this definition of "bigger", but it seems reasonable, and modern mathematicians have indeed accepted it.)

In another post, RPost1 said,

"
This can be demonstrated easily by showing that the two sets are mirror images of each other.

1 <-> .1
2 <-> .2
. . .
9 <-> .9
10 <-> .01
11 <-> .11000 . . .
. . .
19 <-> .91000 . . .
20 <-> .02000 . . .

and so on.

The decimal portion of every possible real number can be created by the reflection of a natural number around the decimal point.
"

Not true. Consider the simple rational number 1/3 = .3333... What natural number does it map to? There's no natural number with an infinite number of 3's to the left of the decimal. The same is true of any rational number that doesn't end with an infinite string of 0s, not to mention irrationals.

I'd like to comment on the other items RPost1 raised ("are 1.000 and 0.999.. the same number?" and "what's the next real number lower than 0.999...?"), but they're beyond the scope of the point I'm making here. I'll comment separately.

Cheers,

kenhaley

15. 15. RPost1 08:25 PM 1/20/08

Thanks, Khaley, for responding and providing a more mathematical point of view on some of the points raised above by myself and others.

After reading your detailed response I think a large part of the reason that we disagree on several statements is because I am using a layman's definition of some key terms that probably have a very precise mathematical meaning.

So let me try to illustrate the meaning that some of the terms used in the article have for me (and probably many other lay persons) and why my perception of these terms may be causing some of the miscommunication.

1. size. I do not believe that the term 'size' has any meaning for infinite sets; but rather any set with a 'size' attribute is not infinite since an infinite set is unbounded.

2. all; as in 'the set of all natural numbers'. I don't think the term 'all' has any meaning for infinite sets; and, to layman at least,is misleading. The phrase 'the set of natural numbers' is fine but not with the word 'all'. See item 4 below. At best the term 'all' would be redundant since the term 'set' would define what can and cannot be a member. Using the term 'all' somehow implies containment and the potential for enumeration; attributes that, it seems to me, are more appropriate for finite sets.

3. the term 'set' often has a dual meaning. One meaning is the use of the term 'set' to define the requirements that elements must meet to be a member of a set. For example, 'the set of natural numbers' when used to define the requirements for the elements of the set.

The second meaning is often used to actually enumerate the elements of a set; or suggest how an enumeration can be performed. The 'set of natural numbers less than 4' not only defines the requirements for membership in the set but implicitly suggests that the set is exactly defined as {1, 2, 3}. (Not being a mathematician I don't know if 0 should be in there or not).

For finite sets a dual meaning may be appropriate and useful.

My opinion is that, for infinite sets, the term 'set' can only be the definition of the membership requirements and cannot have the meaning that implies an enumeration.

4. one-to-one correspondence. My perception of a 1-1 correspondence is that the correspondence must be, for lack of a better word, bi-directional (commutative perhaps?).

KHaley states that for finite sets; you just need to match every element in one with an element in the other. My opinion is that you must also match in the reverse direction. Thus, Set A is in a one-to-one with Set B if each and every element in A can be matched with one and only one element of B and conversely that each and every element in B can be matched with one and only one element of A.

MHaley also states that 'Notice that you can put the set of all natural numbers into a 1-1 correspondence with the set of even numbers'. That isn't what I notice, again for the reasons stated above. The respondent even agrees 'there are natural numbers in the left set that aren't in the right set'. This statement to me proves that the two sets can't possibly be in a 1-1 correspondence. It is no more valid to allow 'extras' in one of two infinite sets than it is to allow them for finite sets.

To a layman like myself the example would seem to imply that the set of natural numbers is exactly twice as large as the set of even numbers. Is this true? Is it valid to make such a statement for these two infinite sets? As intuituve as it might seem I don't think '2 times infinity' is meaningful.

As I stated, IMHO that if one set has numbers that aren't in the other set then the sets cannot be in a one-to-one correspondence. I think this opinion may be contrary to the mathematical point of view and may contribute to the confusion expressed by myself and others.

The closest one can come to a one-to-one correspondence for infinite sets would be a correspondence between the odd natural numbers and the even natural numbers. Even though one cannot pair up, or exhaust, 'all' the members (all has no meaning for infinite sets in my view) logically one can show that the pairings can be continued indefinitely and that both sets are exhausted at the same rate.

Mr. Haley's last point really has me flummoxed. I stated that 'the decimal portion of every possible real number can be created by the relection of a natural number around the decimal point' and he ask what natural number does '.333...' (an infinite number of threes) map to.

For me, it is no harder to imagine an infinite number of threes to the left of the decimal than it is to imagine them to the right of the decimal.

Simply start with the number 3, and then perform the following operation an infinite number of times: multiply the number by 10 and add 3. Eventually you will create the proper number (though not in my lifetime).

Just as there are languages that are read left-to-right and others that are read right-to-left perform the following mental exercise.

Imagine an esoteric group of people (perhaps Mensans) that perform mathematical operations like ours except their number representation assigns the decimal portion of numbers to the left side of the decimal and the whole number portion to the right. Their system is a mirror image of ours.

Their version of 1/3 creates exactly the mirror image of our systems '.333...'.

16. 16. SteveC 08:47 AM 2/18/08

RPost1:

You seem very confused about several things in regards to the sizes of infinities. Hopefully I can clear some of it up for you.

You say that you don't believe that 'size' has any meaning for infinite sets. All of mathematics is founded on definitions. In this case, your intuition is simply incorrect. As was pointed out, the definition of size or cardinality used here is exactly that two sets have the same size if there is a one to one correspondence (called a bijection) between the two sets.

The word "unbounded" has a precise mathematical meaning and it is not as you are using it here. If you consider the real numbers between 0 and 1 they are bounded in the sense that they are all at most 1. This is a bounded infinite set.

You don't think that "all" has any meaning either. Your comment that it is redundant is correct. The set of all natural numbers is the same as the set of natural numbers. However, using the word all does not imply anything about enumeration. We have another word for that (actually several equivalent words): countable. We say that a set is countable if it is either finite or has the same size (in the mathematical sense above) as the natural numbers. By definition, the natural numbers--an infinite set--is countable.

To rephrase Cantor's conclusions: the set of rational numbers is countable while the set of real numbers is not.

The word set never implies countability (enumeration).

Commutative is the wrong word to describe a one to one correspondence--the word is invertible--but you have the right intuition, here. If there is a 1-1 correspondence between sets A and B, then for every element of A there is one and only one corresponding element of B. Similarly, for each element of B there is one and only one corresponding element of A.

Your objection with the set of natural numbering not being in 1-1 correspondence with the set of even natural numbers is not correct. If you give me a natural number n, then I can give you the corresponding even natural number 2*n. Similarly, if you give me an even natural number 2*m, then I can give you the corresponding natural number m. There are no "extras" as you call them.

You are correct that 2 times infinity is not meaningful since infinity is not a number. However, you are incorrect in asserting that the natural numbers are twice as large as the even natural numbers. The correspondence has nothing to do with containment. You would consider {1,2,3,4} and {a,b,c,d} to have the same size even though one is not contained in the other--one is a set of numbers and the other a set of letters--since both have 4 elements. An analogous thing happens with infinite sets.

I _hope_ that clears up some of your misconceptions about sets. If not, wikipedia's article on cardinality might be enlightening.

As for the comments about infinite numbers of 3s on either side of the decimal point, what does it mean to have an infinite decimal number? Well, the answer has to do with limits from elementary calculus. The point is that we can define an infinite decimal in terms of an infinite sequence of numbers (example below). The definition makes sense because the series is "convergent" which I am not going to define here but basically it gets closer and closer to a number.

For example, consider:
0.3
0.33
0.333
0.3333
and imagine this sequence of numbers continuing. The more 3s we tack onto the end, the closer this number comes to the rational number 1/3, but never quite makes it. However, we can get arbitrarily close with a (potentially large) but finite number of digits. It is precisely this property that lets us talk about the infinite decimal expansion 0.333....

If you consider the the case of
3
33
333
3333
and so forth, this sequence of numbers does not approach any number but grows without bound. As a result, we would say that the sequence is not convergent and thus we have no definition for what a number such as ...333 would mean.

Your esoteric group of people example doesn't actually show anything, merely that one can have different representations. We could easily have decreed that one do math in your representation but it wouldn't have changed the above. It isn't the order in which the numbers are written, that is merely convention, it is the meaning assigned to the numbers.

- Steve

17. 17. Gerry63 12:00 AM 4/20/08

A cute way to show what number 0.999... is equal to is simply to multiply by 10 (giving 9.999...) and then subtract 0.999... giving 9.

So what number, which when multiplied by 10 and subtracted away leaves 9?

That is if 10x-x=9 what is x?

x=1, try it!

18. 18. sleekmason 01:31 PM 7/1/09

Always go to the larger pattern to find answers within the current pattern. For each set of numbers include the present pattern, but are not allowed to include the larger. meaning 1.0< will never be larger than 1.1< , as 1.1< allows for 1.0, but 1.0 does not allow for 1.1. By this reasoning, .99999999< does indeed allow for equality to 1, as 1 is the hidden, but accepted part of .9999999. in order for .99999999 to exist, there must be a starting point of 1. You must also understand that just because it is allowed, does not make it true for us, as we can not break through to 1 from .9999999 We can only create another .999999< within the current .99999999

19. 19. Mat 08:48 AM 10/4/09

I still think 'Infinity' is one and an absolute 'characteristic'. What differs is density.
Fill a balloon with air and the other with sand. They can occupy the exact same volume, but have different densities. I think this analogue better explains the phenonmenon; otherwise, we have 'clever' mathematics encroaching on semantics and painting an untrue picture of the World.

20. 20. jack.123 06:25 AM 4/14/10

Infinity is only in those who are legends in their own minds trying to prove their own minds can comprehend the infinite.

21. 21. Xannicus in reply to Bill Shcaller 08:49 PM 8/10/10

Real number between 0 and 1 versus Natural numbers 0 to Infinity--

Instead of pairing numbers the way the article suggests, why not simply map every natural number to it's 0 < x < 1 equivalent via adding a decimal point on the complete left side of the natural number, until we run into a case such as 10, 100, 200 etc, then simply add the zeroes on the right after the decimal then the number?

Example...

Naturals:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20...

Reals (between 0 and 1)
0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19, 0.02...

Assuming a natural number is X, you can represent an adjacent real by simply adding 0. to X as 0.X unless you hit a number with zeros at the end... simply take the zeroes and apply them after 0. and then append the rest of the natural number, so 213000 would be 0.000213

The mere fact that this method works for all natural numbers fitting within 0 and 1 shows that the real number set is significantly larger because this analysis did not include the real numbers between other natural numbers such as 1-2, 2-3... etc.

22. 22. arnolddsouza in reply to Rich_G 10:51 AM 1/25/12

No, they are the same size. To understand this, you have to understand how sizes of sets are compared - by a 1-to-1 mapping. As long as you can map each member of one set (A) to a member of the other set (B) such that: a. No member of set A is mapped to more than one member of B, and b. All members of A are mapped to at least one member of B, and the inverse of a. and b. are also true, then A and B are of the same (possibly infinite) size.

In this case, we have:
A = {1, 2, 3, ... }
B = {2, 3, 4, ... }

Consider the mapping 1<->2, 2<->3, 3<->4, .... Clearly every member of the first set is mapped with one and only one member of the second set and vice-versa. Hence the two sets are of the same size.

23. 23. mediawolf 02:33 AM 5/21/12

OK, so this entire argument hinges on the fact that one set gets to have members with an infinite number of digits, and the other set doesn't.

We're saying that what we mean by "natural numbers" is the infinite set of numbers-with-a-finite-number-of-digits.

(Whereas reals between 0 and 1 include infinite-digit values.)

Were we to lift the restriction, then:

a) The two sets could be put into one-to-one correspondence by taking each member of one set, reversing the digits and adding/removing the decimal point. Irrational numbers would map to infinite-digit whole numbers (we could call them super-naturals).

b) Cantor's argument would not invalidate the correspondence, since it could be just as easily applied to the natural side to create new ones that "aren't in the list".

c) The infinity-of-magnitude of the naturals would be exactly equal to the infinity-of-resolution of the reals between 0 and 1.

d) But the set of ALL reals—not just between 0 and 1—now this would be a "bigger" infinity, one the others can't map to. And (to oversimplify) this is because it's infinite in both magnitude and resolution.

khaley, SteveC, would you agree?

24. 24. kpontis 10:11 PM 6/4/12

The easiest way I can figure this is you can't match 123.456 with 123456 if 123456 is already matched with 12345.6

That is how you can have a larger infinity of real numbers than natural ones.

25. 25. SnowAngel in reply to RPost1 06:03 PM 6/27/12

There is, and it involves surreal numbers.
To explain it, I'll use your nice example.
Since it's generally accepted that 0.999... = 1, I won't try to prove it to you.
Each and every real number is surrounded by surreal numbers, which are much more closer to it than any other real number.
What it means, is that there can be an infinity of numbers between 0.999... repeating and 1, and each number will come closer and closer to the actual numerical value but never reach it.
This also applies downwards.
Some may argue that you cannot write 0.999...8 since the infinite 9 makes the 8 invalid.
However, surreal numbers come to the rescue, with an infinity of numbers below 0.999..., always coming closer yet never reaching 0.999..., while staying closer to its numerical value than any other real number can.

26. 26. Cartesiantheater in reply to RubenV 10:42 AM 12/4/12

Because. For every "largest" element in the first set, call it N, there is still a number in the "smaller" set called M = N+1 that corresponds to it (well, in this case, is equal to it).

For example, in the set 1,2,3... choose the number 156729987398723982732. Does the set 2,3,4... contain that number? Yes, and not only that, it also contains 156729987398723982732+1, 156729987398723982732. It also contains 156729987398723982732 + 156729987398723982732. So you see, for each and every number in 1,2,3,... there is a number from 2,3,4... that can be paired with it.

Your confusion stems from regarding the numbers as anything more than names for the elements of the sets. If the first set was orange, apple, spaceorange,... and the second set was apple, spaceorange, banana, you would have no problem with the notion. These two sets have the same number of elements, because for each element in one set there is ALWAYS going to be a number in the other set that corresponds to it, INCLUDING the number 1, which is in the first set but not the other. In order for the first set to be bigger than the second, it must be true that there is a largest natural number, because as long as there isn't, there can be found a number from the second set that corresponds to any number in the first. 1 corresponds to 2, 2 corresponds to 3,... 156729987398723982732 corresponds to 156729987398723982733, and so on, forever.

They do not have exactly the same NAMES to their elements, but they have exactly the same cardinality. In essence, they are the same set, but with different names.

27. 27. Cartesiantheater in reply to SnowAngel 10:47 AM 12/4/12

Still, there is no reason to resort to non-standard calculus and hyperreals (why do you call them surreals?). The several proofs that 0.999... = 1 are easy to do and understand.

28. 28. Cartesiantheater in reply to Nani77 10:51 AM 12/4/12

Two sets are "of the same size" if for each and every element in one set you can match it with exactly one element from the other set.

When you try to do this with the natural and real numbers, it turns out that you cannot find a match from the natural numbers for every real number. This means that the set of real numbers has more elements than the set of natural numbers.

29. 29. Cartesiantheater in reply to RPost1 10:54 AM 12/4/12

The problem is that 1 and 1.000000... and 0.999999... are all the same number. Each member of each set must be paired with one UNIQUE number for the two sets to have the same cardinality.

30. 30. happykat 03:12 AM 12/9/12

Here is my take on the differences in infinities.

Assume that a wire is carrying a current of 10 amps. Now lets create a graph that shows the voltage being applied to the wire on the horizontal axis. Since wattage is the result of multiplying volts times amps lets have the vertical axis represent the the wattage for each point on the horizontal, or voltage, axis for the fixed value of 10 amps.

Then let us designate the horizontal axis as a voltage from 1 to infinity. Obviously the vertical axis will eventually reach infinite wattage when the horizontal axis reaches infinite voltage.

Now let's redraw the graph with the horizontal axis representing voltages from 1000 volts to infinite volts. Both graphs reach the same conclusion at infinity but they start from different points so the slope of the graph will be will be different. So my claim is that the final infinity is always going to be the same but the route to it will differ with different starting points. And that is the real difference in different infinities.

31. 31. robtomorrow 02:56 AM 12/14/12

I understand that one set of infinite numbers is larger that the other, but does that mean that the value of infinity is different in each case.

32. 32. ThePillow in reply to RubenV 11:07 PM 12/16/12

Actually {1,2,3,4...} and {2,3,4...} are not sets of different sizes. Intuitively it seems obvious that they should be, however infinity leads to strange conclusions.

Clearly you can't pair every element in the 2nd set with its equivalent in the 1st set (because there's no 1 in the 2nd set). But to show they're the same size we only need to show that some 1 to 1 pairing can be made (not that a particular 1 to 1 pairing can be made).

So we could match them as follows:

1,2
2,3
3,4
4,5
etc.

There will never be a number in one of the sets that we can't pair with some number from the other set (it doesn't need to be the same exact number).

Here's an even stranger case of infinities of the same size: the set of all natural numbers and the set of all even natural numbers. It seems obvious that there are half as many even numbers but, again, infinity yields strange results. We can take each natural number and pair it with it's double:

1,2
2,4
3,6
etc.

And no matter how many numbers we consider, we can always make a 1 to 1 pairing of the naturals and the even naturals.

33. 33. anfarahat in reply to Rich_G 04:45 PM 4/19/13

use the one-to-one map: {1,2,3,...} --> {2,3,4, ...}, n --> n+1

34. 34. skytag in reply to Larry in Tampa 04:02 PM 7/25/13

The entire mathematical community recognizes this as a valid proof, but you suspect the conclusion isn't valid. *sigh*

35. 35. skytag in reply to anfarahat 04:04 PM 7/25/13

"use the one-to-one map: {1,2,3,...} --> {2,3,4, ...}, n --> n+1"

Use it for what?

36. 36. skytag in reply to robtomorrow 04:06 PM 7/25/13

Infinity doesn't have a "value." The term represents the "size" of all sets with infinitely many elements.

37. 37. skytag 04:09 PM 7/25/13

People never cease to amaze me. The information provided in the article is universally accepted by those in the mathematical community, yet reading over the comments I see several people who believe they know better, or believe they have a better proof (when in fact their proof contains a fatal flaw).

Yes, all the world's mathematicians are wrong and you're right. *sigh*

38. 38. skytag in reply to RPost1 04:22 PM 7/25/13

"IMHO that the reasoning presented is far more complex than necessary and that simpler is better."

You are wrong. Your "proof" may seem simpler to you, but it is fatally flawed and not a proof at all.

"Let each natural number, say 27, be paired with itâ€™s equivalent real-number partner which is 27.000 . . . (an infinite number of zeros)"

To prove the desired result you have to show that no one-to-one mapping exists, or stated another way, that every mapping fails to be one-to-one. The flaw in your proof is that you only show one specific mapping fails to be one-to-one.

Cantor's proof is function-agnostic, as in he makes no assumptions about it. As the article states, "Cantor thus assumes that the naturals and the reals from zero to one have been put into such a correspondence." It makes no assumptions about the correspondence other than to assume it is one-to-one.

39. 39. skytag in reply to RPost1 04:32 PM 7/25/13

"Consider the set of real numbers that have only zero to the left of the decimal point. This set of real numbers DOES have a one-to-one correspondence with the natural numbers."

People like you boggle my mind. You look at a mathematical result in Scientific American that is universally accepted as true by the mathematical community and even comes with a proof and you summarily declare it isn't true because you don't understand why it's true. All of the world's mathematicians are wrong and you're right.

The flaw in your proof is that you have only established a one-to-one correspondence between the natural numbers and real numbers with a finite number of digits to the right of the decimal point, and all of those real numbers are also rational numbers. In your mapping, none of the irrational numbers have a corresponding partner in the natural numbers.

For example, which natural number corresponds to the decimal portion of Pi?

Nice work, sport.

40. 40. Mat in reply to skytag 11:19 PM 8/2/13

A concensus, even by the best experts, does not make something more true.

41. 41. chaosdot in reply to skytag 09:18 PM 12/3/13

"For example, which natural number corresponds to the decimal portion of Pi?"
The set of natural numbers is infinite, so there exists an infinitely long number whose digits match one to one each digit of pi.

What number is this? the same number that pi is if you take the dot and the 3 from it. That is it is the same number you would you use to represent the decimal portion of pi.

The same applies to any number between 0 and 1. Adding a dot to the numberline shouldn't do anything special. Each and every number that you present as a decimal has a corresponding natural number which is the sequence of digits used to represent that decimal portion, so long as you use numbers(digits) you'll be unable to generate a number not present in the sequence of naturals.

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