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.

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.

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.

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.

Taken as a whole Everett's multiverse, where all things happen, probability is not a useful concept and everything is deterministic. However for observers like us trapped in a single branch of the multiverse, observers who do not have access to the entire wave function and all the information it contains but only a small sliver of it, probability is the best we can do.

I am unable to imagine an interpretation of this paragraph that makes it true.

Probability would be necessary for belief formation even if reality consisted of only a single world. More generally, the usefulness of probability to belief formation does not depend on any particular features or properties of the reality the belief-forming agent (or collection of agents, e.g., the people having this conversation) happens to find itself in (except for the trivial consideration that some realities cannot contain belief-forming agents).

Also, I am extremely skeptical that literally all things happen in the Everettian multiverse. For example, I would be extremely surprised if there exists or will ever exist a branch in which the law of the conservation of momentum is violated. The principle of charity demands that I assume that the OP (johnclark) knows that, but I have been in enough conversations on LW about many worlds to have strong evidence that some of the readers will take "all things happen" literally.

I enjoyed the review; but there's one comment I believe to be in error.

...Byrne does not make it clear if this means they are as numerous as the number of points on a line, or as numerous as an even larger infinite set like the set of all possible clock faces...

I'm pretty sure that the set of all possible clock faces has the same cardinality as the set of points in a line (see space-filling curve).

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