Subject: 1-1 Correspondence & Countability
From: "William J. Rapaport" <rapaport@buffalo.edu>
Date: Tue, 1 Sep 2009 21:42:49 -0400 (EDT)

A student writes:
"Do all sets have the property of being countable in base 1, since they
will necessarily have a one to one correspondence with the positive
natural numbers?"

Interesting question!

Base 10, of course, is the usual way we write the natural numbers:
0,1,2,...,9,10,11,...

Base 2, as I hope you all know, limits itself to 2 "digits" or "bits" 0
and 1: 0,1,10,11,100,...

Base 1 limits itself to 1 digit: /: /,//,///,////,/////...  (Is there a
way to write 0 in base 1? I'll let you think about that :-)

But the base that is used is irrelevant to how many objects are in a
set or how you count them. The base just determines how you write the
number, not how big the number is.

Not all sets are countable: The reals are not, nor are the complex
numbers, nor is the set of all points on a line, etc. And it doesn't
matter which base you try to count with.