The cardinality of the number of points on a line is the same as the number of points in a square or the number of points in a cube or the number of clock faces a properly operating clock will display. However the position of the clock hands where the short hand is exactly at 12 and the long hand is exactly at 6 is NOT included in the set of all valid clock faces, or just turn a clock upside-down and you will see a clock face that a proper clock will never display when it is right-side up. Thus the number of all possible clock faces must be have a higher cardinality than the number of valid clock faces or the number of points on a line; it is the same larger cardinality as the set of all 2 dimensional curves, which is the highest cardinality I can give a simple example of. The boring cardinality of valid clock faces is just the same as the points on a line.
Here is another way of seeing this. Cantors theorem says that for any set X the set of all subsets of X (called the Power Set of X) has a greater cardinality than X. For example 1,2,3,4,5.… would be one way of arranging all the integers, 2,1,3,4,5.… would be another. The set of all possible ways of arranging the integers is the Power Set of the integers, and it would have a higher cardinality than the integers (in this case the cardinality of the real numbers), and thus cannot be put into a list.
If C is the set of all clock faces a working clock can produce, then it is equivalent to an infinite set where the elements are a pair of real numbers (one for each hand) between 0 and 12; so {12,12} {3,12} and {.5,6} would all be members of this set, but {12,6} and {3,9} and {.5,12} would not be. This set C may be infinite but it’s just ONE way the real numbers between 0 and 12 can be paired up. The Power Set of C would be the set of all subsets of paired numbers between 0 and 12, all the ways a pair of 2 real numbers can be arranged, all the ways 2 clock hands can be arranged not just the ways a properly operating clock will produce them.
Therefore the set of all possible clock faces has a higher cardinality than C the set of real numbers.
Firstly, thank you for stating what you meant by clock faces. You should really have stated that explicitly, though, as it’s not a standard term. Also I had to read that twice to notice you were making a distinction between “clock faces” and “valid clock faces”.
But this is simply wrong:
However the position of the clock hands where the short hand is exactly at 12 and the long hand is exactly at 6 is NOT included in the set of all valid clock faces, or just turn a clock upside-down and you will see a clock face that a proper clock will never display when it is right-side up. Thus the number of all possible clock faces must be have a higher cardinality than the number of valid clock faces or the number of points on a line; it is the same larger cardinality as the set of all 2 dimensional curves, which is the highest cardinality I can give a simple example of.
If S is strictly contained in T, and S is a finite set, then T necessarily has strictly larger cardinality than S. The same does not hold for infinite sets—this is just the old “Galileo’s paradox”; Z has the same cardinality as N despite strictly containing it.
The Power Set of C would be the set of all subsets of paired numbers between 0 and 12, all the ways a pair of 2 real numbers can be arranged, all the ways 2 clock hands can be arranged not just the ways a properly operating clock will produce them.
Therefore the set of all possible clock faces has a higher cardinality than C the set of real numbers.
EDIT: Sorry, I wrote something wrong here before due to misreading! Thanks to steven0461 for catching the real problem.
You seem to be equivocating between C and the power set of C. C is in bijection with R, its power set is not. (And since C is in bijection with R, its introduction was really unnecessary—you could have just used the power set of R.) (You also seem to be using unordered pairs when you want ordered pairs, but that’s a more minor issue.)
In short this has a number of errors (fortunately they seem to be discrete, specifically locatable errors) and I suggest you go back and reread your basic set theory.
You can set up a one to one correspondence between all the points on a line (or in a square or in a cube) and all the clock faces a working clock can produce but you cannot do the same with all possible clock faces. And I thought I was clear, I don’t know what you mean about me equivocating between a set and it’s power set; the faces a working clock can produce is just ONE way all real numbers can be paired together, the power set is ALL the ways 2 real numbers can be paired together, it has a larger cardinality than the points on a line and is the number of all possible clock faces.
You also say an entire paragraph is “simply wrong” but you don’t say what you object to other than to note that the laws concerning finite sets are different than those concerning infinite sets and obviously I agree. But what don’t you like in my statement?
You can set up a one to one correspondence between all the points on a line (or in a square or in a cube) and all the clock faces a working clock can produce but you cannot do the same with all possible clock faces.
The set of all clock faces a working clock can produce—call this the set of all valid clock faces—has the same topology (and cardinality) as a circle. The set of all possible clock faces has the same topology (and cardinality) as a 2-dimensional torus.
However, the cardinality of a 2-dimensional torus is the same as the cardinality of a square, which is the same as the cardinality of a line (as you yourself recognize), which is the same as the cardinality of a circle.
Therefore the set of all valid clock faces has the same cardinality as the set of all possible clock faces.
the faces a working clock can produce is just ONE way all real numbers can be paired together, the power set is ALL the ways 2 real numbers can be paired together, it has a larger cardinality than the points on a line and is the number of all possible clock faces.
A power set indeed has a larger cardinality than the set it is a power set of. However, the set of all possible clock faces is not the power set of the set of all valid clock faces.
There are two hands, an hour hand and a minute hand. The set of all possible positions that the hour hand can take describes a circle. The same is true of the minute hand: its set of all possible positions describes a circle. Consequently, the set of all ordered pairs of possible positions (h,m), where h is the position of the hour hand and m is the position of the minute hand, is the Cartesian product of the two individual sets, and thus the Cartesian product of two circles. This is a two-dimensional torus.
Did you take into account that the positions of the two hands are not independent? When the hour hand of a given clock is at its 12:00:00 position, there’s only one possible location of the minute hand for that clock, and this is true for any position of the hour hand.
If you read elsewhere in the thread, you’ll see that johnclark draws a distinction between all possible clock faces and “valid clock faces”, i.e., those that obey the constraint you describe. Constant is addressing the former, not the latter.
You’re talking about what I’ve been calling valid clock faces, the faces of a working clock. That set forms a circle. Right here I’m talking about what we’ve been calling the set of possible clock faces, where we no longer assume the gears of the clock are constraining the positions of the hands. This set forms a two-dimensional torus, the surface of a donut.
You can set up a one to one correspondence between all the points on a line (or in a square or in a cube) and all the clock faces a working clock can produce but you cannot do the same with all possible clock faces. And I thought I was clear, I don’t know what you mean about me equivocating between a set and it’s power set; the faces a working clock can produce is just ONE way all real numbers can be paired together, the power set is ALL the ways 2 real numbers can be paired together, it has a larger cardinality than the points on a line and is the number of all possible clock faces.
Evidently you are more confused than I realized. OK, last attempt at explaining this. The power set of R would be the set of all subsets of R, not just the set of all size-2 subsets of R. (I will ignore for now that you are talking about pairs and what you want here is ordered pairs.) The set of pairs of reals is in bijection with R. And any clock face, valid or not, can be described by an ordered pair of reals; there is no such clock face, valid or not, as {1, 3, 5} or {n in N | 2n+1 is prime}. Your conclusion that the set of clock faces has a higher cardinality than R does not follow, and in fact is false—as I pointed out in a cousin comment, R x R is in bijection with R, and as the set of pairs of reals injects into this, the result follows by Schroeder-Bernstein.
You seem to be equivocating between “the set of all clock faces (valid or not)” and “the set of all ways of pairing up members of R” (which could mean any of several different things, but for now we’ll leave it unspecified as the distinction is irrelevant—they’d have the same cardinality). The latter does indeed have cardinality greater than that of R, but this is an entirely different set than the former.
You really need to be more precise with your language. “The set of all ways 2 real numbers can be paired together” would usually be interpreted to mean “the set of all 2-element subsets of R”, not the much larger set above.
Earlier you wrote:
The Power Set of C would be the set of all subsets of paired numbers between 0 and 12, all the ways a pair of 2 real numbers can be arranged, all the ways 2 clock hands can be arranged not just the ways a properly operating clock will produce them.
It’s really not clear what sets you’re referring to here. “All subsets of paired numbers between 0 and 12?” “All the ways a pair of 2 real numbers can be arranged?” I can guess at what you mean but I can’t be certain I’m right—especially because you are using these as if they are self-evidently the same, while my best guesses for what you mean by each of them, if they were taken in isolation, would be very different sets! Please go back and learn the standard terminology so people have some idea what you’re saying.
You also say an entire paragraph is “simply wrong” but you don’t say what you object to other than to note that the laws concerning finite sets are different than those concerning infinite sets and obviously I agree. But what don’t you like in my statement?
I did not just say “the laws concerning finite sets and infinite sets are different”; I pointed out specifically which principle you appeared to be attempting to use that is not valid. Downvoted.
The power set of R would be the set of all subsets of R, not just the set of all size-2 >subsets of R.
I know that, but I’m not talking about R, I’m talking about the set a working clock could produce, call it VC for valid clock, the elements of this set consist of 2 real numbers. VC has the same number of points as there is on a line or in a square or in a cube. VC is one way all the real numbers can be put into pairs to form a set, but it is not the only way, there are infinitely many other ways and other sets. It’s easy to find a mapping between the points on a line and all the clock faces a working clock can produce:
Every single point on the circular rim a clock is associated, without exception, to the face a working clock could display. Every single point. There is no room for a single extra association, much less the infinite number of them that would be needed. You could pick a point on the rim and say it is associated with the small hand being exactly at 12 and the large hand exactly at 6 but that would be untrue, that point has already been associated with a working clock face as can be seen just by moving the hour hand to point to that point, so now the same point is associated with 2 very different clock faces and that is a invalid mapping.
It’s impossible to find a mapping between the points on a line (or on a circular rim) and all possible clock faces, so it must have a higher cardinality
The set of pairs of reals is in bijection with R.
No, one (not “the”) infinite set whose elements are pairs of real numbers is in bijection with R, the set VC; but there are an infinite number of other infinite sets whose elements are pairs of real numbers, the set of all possible clock faces. This has a larger cardinality than R just like the set of all curves.
You seem to be equivocating between “the set of all clock faces (valid or not)” and >”the set of all ways of pairing up members of R”
I’m confused that you’re confused.
The latter does indeed have cardinality greater than that of R
Thank you.
but this is an entirely different set than the former. You really need to be more precise >with your language. “The set of all ways 2 real numbers can be paired together” would >usually be interpreted to mean “the set of all 2-element subsets of R”, not the much >larger set above.
How is “the set of all ways of pairing up members of R” different from “the set of all 2-element subsets of R” different from “the set of all ways 2 real numbers can be paired together”??
You say in the above that you agree with me that “the set of all ways of pairing up members of R” has a higher cardinality than the real numbers, and you certainly must agree that some of those number pairings a working clock would never produce, and you must agree that it would be easy to find a mapping between that set and the set of all possible clock faces. So what are we arguing about?
Every single point on the circular rim a clock is associated, without exception, to the face a working clock could display. Every single point. There is no room for a single extra association, much less the infinite number of them that would be needed. You could pick a point on the rim and say it is associated with the small hand being exactly at 12 and the large hand exactly at 6 but that would be untrue, that point has already been associated with a working clock face as can be seen just by moving the hour hand to point to that point, so now the same point is associated with 2 very different clock faces and that is a invalid mapping.
This sort of reasoning only works with finite sets. I’m not going to bother to address the rest of your comment, because it’s full of confusion and it’s clear you really need to go back and relearn basic set theory. It would be a waste of all our time to continue this argument further.
Every single point on the circular rim [of] a clock is associated, without exception, to the face a working clock could display.
And also, via a different association, to a face any clock (working or not) could display.
You are the victim of a very common misunderstanding, which is to forget that mappings between sets are allowed to vary when we use them for the purpose of comparing cardinalities.
You can set up a one to one correspondence between all the points on a line (or in a square or in a cube) and all the clock faces a working clock can produce but you cannot do the same with all possible clock faces.
Perhaps I can help explain why this is wrong by giving a constructive counter-example. Correct me if I’m wrong, but by “the set of all clock faces” you mean the set of all positions the two hands of a clock could take. You can specify the position of the clock hands by stating the angles they make relative to any fixed position—say the 12 position for concreteness. Suppose the angles are expressed in radians and take values in the set [0, 2*pi). Multiply each angle by the conversion factor “1 rotation per 2*pi radians” to map the angles into the set [0, 1). Now you can express any specific “clock face” by a point in the unit square. Then you can return to my original point about space-filling curves showing that this has the same cardinality as a line segment.
You also say an entire paragraph is “simply wrong” but you don’t say what you object to other than to note that the laws concerning finite sets are different than those concerning infinite sets and obviously I agree. But what don’t you like in my statement?
Sniffnoy’s next remark was relevant. What he was saying was simply wrong was the idea that if one set A contains another set B then A must have higher cardinality than B. It seems that you have some confusion about how cardinality of infinite sets behaves. It might help to read the relevant Wikipedia entries starting with the basic one on cardinality or look at a standard textbook on set theory. Some of these issues will also be handled by a real analysis textbook.
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.
But what you’re talking about at the end is not the set of all possible clock faces. It’s the set of all possible ways you could divide that set into “valid” and “invalid” clock faces, that is, the set of all possible sets of clock faces.
The cardinality of the number of points on a line is the same as the number of points in a square or the number of points in a cube or the number of clock faces a properly operating clock will display. However the position of the clock hands where the short hand is exactly at 12 and the long hand is exactly at 6 is NOT included in the set of all valid clock faces, or just turn a clock upside-down and you will see a clock face that a proper clock will never display when it is right-side up. Thus the number of all possible clock faces must be have a higher cardinality than the number of valid clock faces or the number of points on a line; it is the same larger cardinality as the set of all 2 dimensional curves, which is the highest cardinality I can give a simple example of. The boring cardinality of valid clock faces is just the same as the points on a line.
Here is another way of seeing this. Cantors theorem says that for any set X the set of all subsets of X (called the Power Set of X) has a greater cardinality than X. For example 1,2,3,4,5.… would be one way of arranging all the integers, 2,1,3,4,5.… would be another. The set of all possible ways of arranging the integers is the Power Set of the integers, and it would have a higher cardinality than the integers (in this case the cardinality of the real numbers), and thus cannot be put into a list.
If C is the set of all clock faces a working clock can produce, then it is equivalent to an infinite set where the elements are a pair of real numbers (one for each hand) between 0 and 12; so {12,12} {3,12} and {.5,6} would all be members of this set, but {12,6} and {3,9} and {.5,12} would not be. This set C may be infinite but it’s just ONE way the real numbers between 0 and 12 can be paired up. The Power Set of C would be the set of all subsets of paired numbers between 0 and 12, all the ways a pair of 2 real numbers can be arranged, all the ways 2 clock hands can be arranged not just the ways a properly operating clock will produce them.
Therefore the set of all possible clock faces has a higher cardinality than C the set of real numbers.
John K Clark
Firstly, thank you for stating what you meant by clock faces. You should really have stated that explicitly, though, as it’s not a standard term. Also I had to read that twice to notice you were making a distinction between “clock faces” and “valid clock faces”.
But this is simply wrong:
If S is strictly contained in T, and S is a finite set, then T necessarily has strictly larger cardinality than S. The same does not hold for infinite sets—this is just the old “Galileo’s paradox”; Z has the same cardinality as N despite strictly containing it.
EDIT: Sorry, I wrote something wrong here before due to misreading! Thanks to steven0461 for catching the real problem.
You seem to be equivocating between C and the power set of C. C is in bijection with R, its power set is not. (And since C is in bijection with R, its introduction was really unnecessary—you could have just used the power set of R.) (You also seem to be using unordered pairs when you want ordered pairs, but that’s a more minor issue.)
In short this has a number of errors (fortunately they seem to be discrete, specifically locatable errors) and I suggest you go back and reread your basic set theory.
You can set up a one to one correspondence between all the points on a line (or in a square or in a cube) and all the clock faces a working clock can produce but you cannot do the same with all possible clock faces. And I thought I was clear, I don’t know what you mean about me equivocating between a set and it’s power set; the faces a working clock can produce is just ONE way all real numbers can be paired together, the power set is ALL the ways 2 real numbers can be paired together, it has a larger cardinality than the points on a line and is the number of all possible clock faces.
You also say an entire paragraph is “simply wrong” but you don’t say what you object to other than to note that the laws concerning finite sets are different than those concerning infinite sets and obviously I agree. But what don’t you like in my statement?
John K Clark
The set of all clock faces a working clock can produce—call this the set of all valid clock faces—has the same topology (and cardinality) as a circle. The set of all possible clock faces has the same topology (and cardinality) as a 2-dimensional torus.
However, the cardinality of a 2-dimensional torus is the same as the cardinality of a square, which is the same as the cardinality of a line (as you yourself recognize), which is the same as the cardinality of a circle.
Therefore the set of all valid clock faces has the same cardinality as the set of all possible clock faces.
A power set indeed has a larger cardinality than the set it is a power set of. However, the set of all possible clock faces is not the power set of the set of all valid clock faces.
Yes.
Show me.
John K Clark
There are two hands, an hour hand and a minute hand. The set of all possible positions that the hour hand can take describes a circle. The same is true of the minute hand: its set of all possible positions describes a circle. Consequently, the set of all ordered pairs of possible positions (h,m), where h is the position of the hour hand and m is the position of the minute hand, is the Cartesian product of the two individual sets, and thus the Cartesian product of two circles. This is a two-dimensional torus.
Did you take into account that the positions of the two hands are not independent? When the hour hand of a given clock is at its 12:00:00 position, there’s only one possible location of the minute hand for that clock, and this is true for any position of the hour hand.
If you read elsewhere in the thread, you’ll see that johnclark draws a distinction between all possible clock faces and “valid clock faces”, i.e., those that obey the constraint you describe. Constant is addressing the former, not the latter.
Yes. Thanks.
Ah. Okay.
Good point. The minute hand is entirely redundant.
You’re talking about what I’ve been calling valid clock faces, the faces of a working clock. That set forms a circle. Right here I’m talking about what we’ve been calling the set of possible clock faces, where we no longer assume the gears of the clock are constraining the positions of the hands. This set forms a two-dimensional torus, the surface of a donut.
Evidently you are more confused than I realized. OK, last attempt at explaining this. The power set of R would be the set of all subsets of R, not just the set of all size-2 subsets of R. (I will ignore for now that you are talking about pairs and what you want here is ordered pairs.) The set of pairs of reals is in bijection with R. And any clock face, valid or not, can be described by an ordered pair of reals; there is no such clock face, valid or not, as {1, 3, 5} or {n in N | 2n+1 is prime}. Your conclusion that the set of clock faces has a higher cardinality than R does not follow, and in fact is false—as I pointed out in a cousin comment, R x R is in bijection with R, and as the set of pairs of reals injects into this, the result follows by Schroeder-Bernstein.
You seem to be equivocating between “the set of all clock faces (valid or not)” and “the set of all ways of pairing up members of R” (which could mean any of several different things, but for now we’ll leave it unspecified as the distinction is irrelevant—they’d have the same cardinality). The latter does indeed have cardinality greater than that of R, but this is an entirely different set than the former.
You really need to be more precise with your language. “The set of all ways 2 real numbers can be paired together” would usually be interpreted to mean “the set of all 2-element subsets of R”, not the much larger set above.
Earlier you wrote:
It’s really not clear what sets you’re referring to here. “All subsets of paired numbers between 0 and 12?” “All the ways a pair of 2 real numbers can be arranged?” I can guess at what you mean but I can’t be certain I’m right—especially because you are using these as if they are self-evidently the same, while my best guesses for what you mean by each of them, if they were taken in isolation, would be very different sets! Please go back and learn the standard terminology so people have some idea what you’re saying.
I did not just say “the laws concerning finite sets and infinite sets are different”; I pointed out specifically which principle you appeared to be attempting to use that is not valid. Downvoted.
I know that, but I’m not talking about R, I’m talking about the set a working clock could produce, call it VC for valid clock, the elements of this set consist of 2 real numbers. VC has the same number of points as there is on a line or in a square or in a cube. VC is one way all the real numbers can be put into pairs to form a set, but it is not the only way, there are infinitely many other ways and other sets. It’s easy to find a mapping between the points on a line and all the clock faces a working clock can produce:
Every single point on the circular rim a clock is associated, without exception, to the face a working clock could display. Every single point. There is no room for a single extra association, much less the infinite number of them that would be needed. You could pick a point on the rim and say it is associated with the small hand being exactly at 12 and the large hand exactly at 6 but that would be untrue, that point has already been associated with a working clock face as can be seen just by moving the hour hand to point to that point, so now the same point is associated with 2 very different clock faces and that is a invalid mapping.
It’s impossible to find a mapping between the points on a line (or on a circular rim) and all possible clock faces, so it must have a higher cardinality
No, one (not “the”) infinite set whose elements are pairs of real numbers is in bijection with R, the set VC; but there are an infinite number of other infinite sets whose elements are pairs of real numbers, the set of all possible clock faces. This has a larger cardinality than R just like the set of all curves.
I’m confused that you’re confused.
Thank you.
How is “the set of all ways of pairing up members of R” different from “the set of all 2-element subsets of R” different from “the set of all ways 2 real numbers can be paired together”??
You say in the above that you agree with me that “the set of all ways of pairing up members of R” has a higher cardinality than the real numbers, and you certainly must agree that some of those number pairings a working clock would never produce, and you must agree that it would be easy to find a mapping between that set and the set of all possible clock faces. So what are we arguing about?
John K Clark
This sort of reasoning only works with finite sets. I’m not going to bother to address the rest of your comment, because it’s full of confusion and it’s clear you really need to go back and relearn basic set theory. It would be a waste of all our time to continue this argument further.
And also, via a different association, to a face any clock (working or not) could display.
You are the victim of a very common misunderstanding, which is to forget that mappings between sets are allowed to vary when we use them for the purpose of comparing cardinalities.
Perhaps I can help explain why this is wrong by giving a constructive counter-example. Correct me if I’m wrong, but by “the set of all clock faces” you mean the set of all positions the two hands of a clock could take. You can specify the position of the clock hands by stating the angles they make relative to any fixed position—say the 12 position for concreteness. Suppose the angles are expressed in radians and take values in the set [0, 2*pi). Multiply each angle by the conversion factor “1 rotation per 2*pi radians” to map the angles into the set [0, 1). Now you can express any specific “clock face” by a point in the unit square. Then you can return to my original point about space-filling curves showing that this has the same cardinality as a line segment.
Sniffnoy’s next remark was relevant. What he was saying was simply wrong was the idea that if one set A contains another set B then A must have higher cardinality than B. It seems that you have some confusion about how cardinality of infinite sets behaves. It might help to read the relevant Wikipedia entries starting with the basic one on cardinality or look at a standard textbook on set theory. Some of these issues will also be handled by a real analysis textbook.
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.
But what you’re talking about at the end is not the set of all possible clock faces. It’s the set of all possible ways you could divide that set into “valid” and “invalid” clock faces, that is, the set of all possible sets of clock faces.