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.
Yes.
The set of all possible clock faces has the same topology (and cardinality) as a 2-dimensional torus.
Show me.
John K Clark
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?
John K Clark
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
Some of the laws of physics could change from universe to universe, but there must be some laws that remain invariant across the entire multiverse because without rules it would behave chaotically and if the multiverse behaved that way so would all the universes in it, including ours. However there is order in our universe, but what is fundamental and what is not? I think we probably all agree that purely mathematical things like pi or e would remain constant in all universes, but consider some of the physical things that might change:
The Planck constant. The speed of light. The gravitational (big G) constant. The mass of the electron, proton, and neutron. The electrical charge on the proton and electron. The inverse square law of gravity and electromagnetism. The conservation of Mass-energy, momentum, angular momentum, spin and electrical charge.The relative strength of the 4 forces of nature. The number of large dimensions in a universe. The Hubble constant. The ratio of baryonic matter to dark matter and dark energy.
It seems to me that the speed of light and Planck’s constant may be more fundamental than other “constants” and the basic structure of the laws of physics may be more fundamental than the constants they use. But I could be wrong, perhaps the things that always remain the same are none of the above and we haven’t even discovered them yet.
John K Clark
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
Are atoms real? Whatever the answer to that question is imagine if it were exchanged, that is suppose that magically the reality of atoms became unreal or the reality of atoms became real, would the world be in any way different as a result? I think the clear answer is no, therefore regardless of what the status of atoms may ultimately be, the question “Are atoms real?” is not real because real things make a difference and unreal things do not.
John K Clark
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