muzhiqingfeng said:
It's good to see my posts when not login,sometimes I'd like to login when some new posts replid to my post,
question1:
1-there only exist one.
2-there exist two,or more,but they are the same.
are they different?
It seems you speak a different native language, but I'll interpret.
There are two sets of objects, I'll call them sets A and B. Set A contains only one 'thing' whereas set B contains more than 1 but they are the "same." This is a difficult question to answer, partly because it might be wrongly worded.
Abstractly, any mathematician might say they are the same. If one takes the word "same" literally to mean
identical then the answer is yes; there cannot be two different things that are the same... by definition. The problem with this then, is that the statement is vacuously true. Simple logic: "
if P is true
then Q is true" is, by convention, always
true when P is
false. Here, P is "sets [1] and [2] exist" and Q is "then they are the same." The P here is always false since set [2] cannot exist -- it defies the notion of what a set is. Hence the statement is true. In layman's terms "is a trick question."
If the word "same" is open to interpretation then no. {3/4,12/16} is a set containing two different rational numbers that might be regarded as the same in some contexts and different in others. What if they were the sides of your TV screen and you ordered the 12/16 model (20 inches diagonal) and were sent the 3/4 model (5 inches diagonal)?
But then, what of two sets each having one (or the same number of) element(s)? {a} amd {1} are the same in size, but as symbols one would say the elements are different. It all depends on what meaning you attatch to them. I can count in letters to
five, "a, b, c, d, ... e." I can do it in number also. In this manner I would say the sets {1,2,3,4,5} and {a,b,c,d,e} are equivilant.
muzhiqingfeng said:
question2:
infinite point is one point,right?
or infinite points are infinitely many,but they are the same?
Are you referring to the point \(\displaystyle \infty\) in the extended reals or different magnitudes of Infinity?
