There are 2 probability puzzles that I like: 1) Suppose I tell you that I have 2 children and one of them is a boy, what is the probability that I have 2 boys? The correct answer is not 1/2 but 1/3. How can that be? Well there are 4...
I've just finished this book and its one of the most enjoyable things I've read in a long time. Being a staple of science fiction and the only interpretation of quantum mechanics to enter the popular imagination it's a little surprising that "The Many Worlds of Hugh Everett" by Peter...
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.... (read 412 more words →)
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.... (read more)
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... (read more)
There are 2 probability puzzles that I like:
1) Suppose I tell you that I have 2 children and one of them is a boy, what is the probability that I have 2 boys?
The correct answer is not 1/2 but 1/3. How can that be? Well there are 4 possible combinations, BB,GG,BG and GB but but at least one is a boy so you can get rid of GG. So all that's left is BB,BG and GB; and in only one of those 3 possibilities do I have two boys.
2) Now I tell you that I have 2 children and one of them is a boy born on a Tuesday. What is... (read 182 more words →)
I've just finished this book and its one of the most enjoyable things I've read in a long time. Being a staple of science fiction and the only interpretation of quantum mechanics to enter the popular imagination it's a little surprising that "The Many Worlds of Hugh Everett" by Peter Byrne is the first biography of the originator of that amazing idea. Everett certainly had an interesting life, he was a libertarian and a libertine, became a cold warrior who with his top secret clearance was comfortable with the idea of megadeath, became wealthy by started one of the first successful software companies until alcoholism drove him and his company into the... (read 1614 more words →)
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... (read more)