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#1 2010-01-17 04:34:35

Fruityloop
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More real numbers that natural numbers?

I don't know how many people have seen Cantor's proof that the set of real numbers is larger than the set of natural numbers (1,2,3,4...) but he shows that you can't produce a 1-to-1 pairing of the real numbers and natural numbers like this....

You now create a new real number that is not on the list by selecting a number that differs from the first number in the first position after the decimal point and differs from the second number in the second position after the decimal point and so on.  For example, the number .2738..... won't be on the list because 2 differs from 8, 7 differs from 6, 3 differs from 9, 8 differs from 2 and so on.  This seems all logical and reasonable , but what if I use his same argument to show that the set of natural numbers can't be paired with itself.

If I select the new natural number 5832... I have a natural number that isn't on the list because 5 is different from 1, 8 is different from nothing, 3 is different from nothing and so on. So let's say I choose to pair the natural numbers with the set of real numbers like this...

There now seems to be a 1-to-1 correspondence between the natural numbers and the real numbers. Once you get past .9 the real numbers are simply the natural numbers reversed with a decimal point.  If you try and create a new real number that isn't on the list you would be following the above logic where you were trying to show that the natural numbers can't be paired with themselves.

Last edited by Fruityloop (2010-01-17 14:49:39)

The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

#2 2010-01-17 05:36:05

Ricky
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Re: More real numbers that natural numbers?

Remember the list is infinite.  The act of choosing which decimal place is which number is an infinite process.  For those who are a bit more advanced, remember that this does not involve the axiom of choice.

If I select the new natural number 5832...

This number appears on your list of integers in the 5832th spot, so there is no contradiction: it is on your list.

There now seems to be a 1-to-1 correspondence between the natural numbers and the real numbers.

The real (actually, rational) number:

0.211111111111111111111111...

Would not appear on that list.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

#3 2010-01-17 14:57:05

Fruityloop
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Re: More real numbers that natural numbers?

If I select the new natural number 5832...

Ricky wrote:

This number appears on your list of integers in the 5832th spot, so there is no contradiction: it is on your list.

Maybe I didn't make this clear enough, but the natural number 5832... is not the number 5832.  The three little dots means it continues forever.  So it is not listed in the 5832th spot.

There now seems to be a 1-to-1 correspondence between the natural numbers and the real numbers.

Ricky wrote:

The real (actually, rational) number:

0.211111111111111111111111...

Would not appear on that list.

Actually, that real number can be paired with a natural number it is
...111111111111111111111112

Every time you add a 1 to the end of your real number you create a new real number, so I add a 1 to the front of my number to match it.

Last edited by Fruityloop (2010-01-17 15:04:47)

The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

#4 2010-01-17 17:10:14

Ricky
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Re: More real numbers that natural numbers?

Maybe I didn't make this clear enough, but the natural number 5832... is not the number 5832.  The three little dots means it continues forever.  So it is not listed in the 5832th spot.

Any natural number can only have a finite number of digits.  Such is not true for a real number.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

#5 2010-01-18 00:32:08

Fruityloop
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Re: More real numbers that natural numbers?

OK.  I think you're right.  My argument doesn't work because any natural number I come up with can have only a finite number of digits so it must be on my list of natural numbers.  Any real number that doesn't end can't be on my list of pairing the natural numbers with the real numbers.  I guess this is why Cantor paired up the numbers the way he did.  It is very weird because we don't normally think of there being different 'sizes' of infinity.

The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

#6 2010-01-18 04:12:15

Ricky
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Re: More real numbers that natural numbers?

And to make sure your understanding is right, you should realize why Cantor's diagonal proof for the reals does not work for the rationals (which can have infinitely many digits as well).

Another good question: Is there a set which is larger than the integers (can't be paired with them), but smaller than the reals?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

#7 2010-01-18 07:24:36

soroban
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Re: More real numbers that natural numbers?

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#8 2012-02-07 21:31:00

Alex23
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Re: More real numbers that natural numbers?

The correct phrasing is more transcendental numbers than natural numbers. Real numbers are not "real" but the name historically came about to make the distinction between them and imaginary numbers.

Conceptually what Cantor spectacularly proved was that infinity extends not just outwards from the number line, towards +∞ and -∞, but also "inwards". The number line is more like an infinite fractal-like 1-dimensional grid.
This is also equivalent to points not having size. That's why no matter the line segment or area, the cardinality of the points is the same, C, the continuum.

However, in practice we have never used but computable rational numbers. Only mathematicians will leave a root of 2 at the end of an equation. An engineer will carry out that root of 2 from equation to equation but at the very end he will round of to any degree of accuracy he needs. Same goes for π ≈ 355/113.

Back to Cantor, his discovery makes explicit something many would have thought but not be able to prove. That there is no next real number. From this also follows, 0.999... = 1.

Last edited by Alex23 (2012-02-07 21:54:46)

#9 2012-06-25 12:00:54

anonimnystefy
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Re: More real numbers that natural numbers?

Ricky wrote:

And to make sure your understanding is right, you should realize why Cantor's diagonal proof for the reals does not work for the rationals (which can have infinitely many digits as well).

Another good question: Is there a set which is larger than the integers (can't be paired with them), but smaller than the reals?

There is of course P(N), the number of subsets of the set of naturals. It is the same as the number of reals, but I cannot remember the proof. We had a lecture on this stuff once. It was very interesting.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#10 2014-01-14 16:28:49

n00b
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Re: More real numbers that natural numbers?

Happened to come across this old thread via a google search. Anyways, we were asked to prove this for a homework assignment. Obviously Cantor's proof is elegant and so it is widely used. For my proof, I constructed a 1-to-1 mapping with the natural numbers mapping to their reciprocal. Then I merely pointed out a real number in that interval (I used 2/3).

Does this properly prove that there are more reals than natural numbers or am I missing something?

#11 2014-03-22 10:19:57

eigenguy
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Re: More real numbers that natural numbers?

n00b wrote:

Happened to come across this old thread via a google search. Anyways, we were asked to prove this for a homework assignment. Obviously Cantor's proof is elegant and so it is widely used. For my proof, I constructed a 1-to-1 mapping with the natural numbers mapping to their reciprocal. Then I merely pointed out a real number in that interval (I used 2/3).

Does this properly prove that there are more reals than natural numbers or am I missing something?

I don't know if n00b is still around, but yes, if I understand your description of your proof correctly, then you have missed the whole idea.

It is not sufficient to show that there is a 1-1 mapping from the natural numbers into the reals that misses some numbers. After all, your mapping, and your example, were actually in the rational numbers. So if your reasoning were correct, there would be more rational numbers than naturals, which is false. In fact, by the same idea, a simple mapping would show that there are more natural numbers than natural numbers! The 1-1 mapping

maps the natural numbers to themselves, but misses 1 (or if you prefer to define the Natural numbers to include 0, it misses 0).

What these mappings really show is that the natural numbers are infinite (the definition of "infinite set" is "a set which has a 1-1 mapping with a proper subset of itself"), and that the cardinality of the Reals and Rationals are both greater than or equal to the Naturals, which also follows from the simple fact that the Naturals are a subset of both.

By definition, two sets are the same size, or cardinality, if there is a 1-1 correspondence between them that includes every element of both sets. Cantor's proof shows that every mapping from Natural numbers into the Real numbers must miss at least one real number. Therefore the cardinality of the Real numbers cannot be equal to that of the Naturals. Combined with "greater than or equal to" already noted, we get that the cardinality of the Reals is strictly greater than that of the Naturals.

Last edited by eigenguy (2014-03-22 10:20:32)

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich