How did Cantor prove that there were more real numbers than integers? He set up a mapping between every single integer and a unique real number and then showed that there were still some real numbers not associated with an integer; this proved that the real numbers had a larger cardinality than the integers.
In the same way I can show you a mapping that associates every single real number with a unique clock face (all the clock faces a properly working clock can produce in this case) but I can also show you clock faces (an infinite number of them in fact) that are not involved in this mapping; I can show you clock faces not associated with a real number, thus the number of all possible clock faces must have a larger cardinality than the real numbers.
It’s incontrovertible that every number on the real number line is associated with unique clock face and it’s also incontrovertible that not every clock face is associated with a unique number on the real number line; this is the very method one uses to determine the cardinality of infinite sets, it worked for Cantor and the logic is ironclad.
In the same way I can show you a mapping that associates every single real number with a unique clock face (all the clock faces a properly working clock can produce in this case) but I can also show you clock faces (an infinite number of them in fact) that are not involved in this mapping; I can show you clock faces not associated with a real number, thus the number of all possible clock faces must have a larger cardinality than the real numbers.
This is not sufficient to show that you have a larger cardinality. This is essentially claiming that if I have sets A and B, and a bijection between A and a proper subset of B, then A and B must have different cardinality. This is wrong. To see a counterexample, take say the map from the positive integer to the positive integers from n → n+1. In this example, A and B are both the positive integers. Since A=B they must be the same cardinality. But we have a 1-1, onto map from A into a proper subset of B since the map only hits 2,3.4… and doesn’t hit 1.
What Cantor did is different. Cantor’s proof that the reals have a larger cardinality than the natural numbers works by showing that for any map between the positive integers and the reals, there will be some reals left over. This is a different claim than exhibiting a single map where this occurs.
...screw it, I’ll reply to this one just to point out what you should be looking up. That is not Cantor’s proof, Cantor used (invented) the diagonal argument[0]. Nor is that a correct proof; if it were, it would prove that Q would have a larger cardinality than Z. You may remember this surprise? Q has the same cardinality as Z but R has strictly larger cardinality? If this doesn’t sound familiar to you, you need to relearn basic set theory. If this does sound familiar to you but you don’t see why it applies, you need to better develop your ability to analyze arguments, and relearn basic set theory (going by your previous statements).
EDIT: JoshuaZ points out a clearer counterexample to your argument in a brother comment.
That should do for a start, though a more organized textbook may be preferable. Now you have at least something to read and I will spend no more time addressing your arguments myself as the linked pages do so plenty well.
[0]Yes, I know this was not his original proof. That is not the point.
it’s also incontrovertible that not every clock face is associated with a unique number on the real number line;
We can do a constructive counter-example for this one too, if you don’t like space-filling curves. Take any real number in [0, 1) and construct two real numbers each of which is also in [0,1) by concatenating the first, third, fifth, etc., digits to make one real number and the second, fourth, sixth, etc., digits to make the second real number. Treat those real numbers as specifying fractions of a revolution for the two clock hands, as in my previous comment. Now every clock face is associated with a unique number in a subset of the real number line and vice versa.
How did Cantor prove that there were more real numbers than integers? He set up a mapping between every single integer and a unique real number and then showed that there were still some real numbers not associated with an integer; this proved that the real numbers had a larger cardinality than the integers.
In the same way I can show you a mapping that associates every single real number with a unique clock face (all the clock faces a properly working clock can produce in this case) but I can also show you clock faces (an infinite number of them in fact) that are not involved in this mapping; I can show you clock faces not associated with a real number, thus the number of all possible clock faces must have a larger cardinality than the real numbers.
It’s incontrovertible that every number on the real number line is associated with unique clock face and it’s also incontrovertible that not every clock face is associated with a unique number on the real number line; this is the very method one uses to determine the cardinality of infinite sets, it worked for Cantor and the logic is ironclad.
John K Clark
This is not sufficient to show that you have a larger cardinality. This is essentially claiming that if I have sets A and B, and a bijection between A and a proper subset of B, then A and B must have different cardinality. This is wrong. To see a counterexample, take say the map from the positive integer to the positive integers from n → n+1. In this example, A and B are both the positive integers. Since A=B they must be the same cardinality. But we have a 1-1, onto map from A into a proper subset of B since the map only hits 2,3.4… and doesn’t hit 1.
What Cantor did is different. Cantor’s proof that the reals have a larger cardinality than the natural numbers works by showing that for any map between the positive integers and the reals, there will be some reals left over. This is a different claim than exhibiting a single map where this occurs.
...screw it, I’ll reply to this one just to point out what you should be looking up. That is not Cantor’s proof, Cantor used (invented) the diagonal argument[0]. Nor is that a correct proof; if it were, it would prove that Q would have a larger cardinality than Z. You may remember this surprise? Q has the same cardinality as Z but R has strictly larger cardinality? If this doesn’t sound familiar to you, you need to relearn basic set theory. If this does sound familiar to you but you don’t see why it applies, you need to better develop your ability to analyze arguments, and relearn basic set theory (going by your previous statements).
EDIT: JoshuaZ points out a clearer counterexample to your argument in a brother comment.
Here. Here are some Wikipedia links to get you started.
http://en.wikipedia.org/wiki/Hume%27s_principle
http://en.wikipedia.org/wiki/Galileo%27s_paradox
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
http://en.wikipedia.org/wiki/Equinumerosity
http://en.wikipedia.org/wiki/Bijection
http://en.wikipedia.org/wiki/Dedekind-infinite_set
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
http://en.wikipedia.org/wiki/Cantor%27s_theorem
http://en.wikipedia.org/wiki/Cardinality
http://en.wikipedia.org/wiki/Cardinal_number
http://en.wikipedia.org/wiki/Injective_function
http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem
That should do for a start, though a more organized textbook may be preferable. Now you have at least something to read and I will spend no more time addressing your arguments myself as the linked pages do so plenty well.
[0]Yes, I know this was not his original proof. That is not the point.
We can do a constructive counter-example for this one too, if you don’t like space-filling curves. Take any real number in [0, 1) and construct two real numbers each of which is also in [0,1) by concatenating the first, third, fifth, etc., digits to make one real number and the second, fourth, sixth, etc., digits to make the second real number. Treat those real numbers as specifying fractions of a revolution for the two clock hands, as in my previous comment. Now every clock face is associated with a unique number in a subset of the real number line and vice versa.