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.
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.