In Reply to: Re: Discussion question posted by Daniel Goroff on October 21, 1999 at 01:00:29:
>>Good answer in this context. How could we say more generally when something like a group or ring or field should count as just a copy of something else?
How about starting by saying that two sets which are "copies" of each other should have the same cardinality ? Matt argued that
we can't consider Z_7 a copy of the rationals because no 1-1 map
exists between Q and Z_7; that's because Z_7 isn't "big enough."
Similarly, the rationals probably should not be considered
a "copy" of the reals, since the reals are uncountable and Q
is countable -- so again no 1-1 map exists.
I don't know if every set of the same size should be considered a "copy" of every other set of the same size, but it seems like
someplace to start.