Cardinality of the set of binary-expressed real numbers

Mar 2015
1,720
126
New Jersey
So all natural numbers are finite?
 
Mar 2015
1,720
126
New Jersey
Clearly we have to define finite.
Finite sequence: one that ends

I note that the sequence of natural numbers (0,1,2,...) does not end.
 
Dec 2015
973
128
Earth
\(\displaystyle y=1/n \) is simply more dense on \(\displaystyle axis\) than \(\displaystyle y=n\)
 
Dec 2015
103
1
France
Clearly we have to define finite.
Finite sequence: one that ends

I note that the sequence of natural numbers (0,1,2,...) does not end.
Yes. Infinity means no end in latin.
 

skipjack

Forum Staff
Dec 2006
21,388
2,411
Clearly we have to define finite.
It suffices to understand "enumeration", "infinitely many" and "infinitely long", as what's being referred to is that any enumeration of infinitely many infinitely long binary sequences allows an infinitely long binary sequence to be specified (by "diagonalization") that isn't in the enumeration.
 
Mar 2015
1,720
126
New Jersey
It's easy to set up a 1-1 correspondence between infinite binary digits and the natural numbers

0: 00000.......
1: 10000......
2: 01000......
3: 11000......
4: 00100.......
5: 10100.......
6: 01100......
.
.
 
Oct 2009
937
364
It's easy to set up a 1-1 correspondence between infinite binary digits and the natural numbers

0: 00000.......
1: 10000......
2: 01000......
3: 11000......
4: 00100.......
5: 10100.......
6: 01100......
.
.
So what is the natural number associated with 1111111....


(Why am I bothering)