This full set [of all possible laws of physics under which sentient life can exist] is infinite,
How can you be so sure of this?
Presumably there is some level of resolution at which changes to the fundamental constants no longer have appreciable effects. Maybe it’s the thousandth decimal place; maybe it’s the googolth decimal place; but it seems extremely unlikely to me that there isn’t such a level. Given this assumption, the set of possible laws is clearly infinite.
It doesn’t, because the reals are infinite in two ways: any given interval is infinite, but the number of intervals is also infinite.
Also this only applies to changing the constants but keeping the general structure the same; you can also create further laws by changing the structure of the laws itself. There are a lot of degrees of freedom.
Presumably there is some level of resolution at which changes to the fundamental constants no longer have appreciable effects. Maybe it’s the thousandth decimal place; maybe it’s the googolth decimal place; but it seems extremely unlikely to me that there isn’t such a level. Given this assumption, the set of possible laws is clearly infinite.
Doesn’t this all imply that the set of meaningfully different laws is finite? Also, what if there is a smallest possible level of resolution?
You are aiming at meaningfully distinct. nbouscal is aiming at the functionally equivalent.
It doesn’t, because the reals are infinite in two ways: any given interval is infinite, but the number of intervals is also infinite.
Also this only applies to changing the constants but keeping the general structure the same; you can also create further laws by changing the structure of the laws itself. There are a lot of degrees of freedom.