How big is my set?

  • #1

Main Question or Discussion Point

I am not a mathematician, I am a physicist. Out of curiosity, how big is the following set?

Let me first define:
A_0 = a point
A_1 = the reals
A_2 = the reals^2 (the plane)
A_3 = the reals^3
.
.
.

The set I want to know the size of is the sum (union) of all the A_n sets, where n goes to infinity. Also, can i assign a cardinality?
 

Answers and Replies

  • #2
disregardthat
Science Advisor
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Let f_n be an injection from R^n to [n-1,n), n>0, then we can define a function F on U = [tex]\bigcup_{n=1} R^n[/tex] to [tex][0,\infty)[/tex] by F(x) = f_n(x) whenever x is an element of R^n. This is a bijection, so U + your point (R^0) has the same cardinality as R.

I have explained the fact that R^n has the same cardinality as R (and hence [n-1,n) ) in an earlier post of mine if you are not familiar with this fact:

https://www.physicsforums.com/showpost.php?p=3021974&postcount=11

As I have used above, U + a point, or more formally [tex]U \cup \{x\}[/tex] where x is not in U has the same cardinality as U whenever U is an infinite set. We can prove that as such: take a countable sequence a_n, n>0, of distinct elements from U, and define [tex]f : U \to U \cup \{x\}[/tex] by [tex]f(a_n) = a_{n-1}[/tex] for n>1, f(a_1) = x, and f(y) = y whenever y is not in the sequence. This is a bijection.
 
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