I think Cantor made a real mess of this field. In mapping the counting numbers (proportional to some finite limit N) to the rational numbers (proportional to the finite limite N²) he made infinity impossible to understand in logical terms. Because it is infinity, that doesn't make everything magically correct. You should be able to think of infinity as some value N which is CONSISTENTLY increased without restriction. He insisted on increasing N inconsistently and therefore came up with some rather foolish results.

Here is how I do it:

http://lesliegreen.byethost3.com/articles/new_maths.pdf

I sympathize with you; I myself have reservations about Cantor’s diagonal proof of the uncountability of the real numbers. For example, he seemed to assume that there was a bijection between each real number as its infinite decimal representation – this is not quite true, as seen by the fact that

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Here is how Jade Tan-Holmes does it on her YouTube channel Up and Atom: An Alternative Proof That The Real Numbers Are Uncountable.

]]>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 no set with a cardinality greater than ~~

EDIT: sorry, I got big confused with small or something. If the continuum hypothesis is true, then there is a set bigger than the set of naturals but smaller than the set of reals.]]>

Here is how I do it:

http://lesliegreen.byethost3.com/articles/new_maths.pdf

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.

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

]]>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.

]]>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.

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

]]>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.

]]>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.

]]>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.

]]>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

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.]]>