# Cardinality of the set of binary-expressed real numbers

#### Pengkuan

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#### zylo

So all natural numbers are finite?

#### Pengkuan

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#### zylo

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.

#### [email protected]

Damn, two Cantor disbelievers arguing with eachother. This is great stuff...

1 person

#### idontknow

$$\displaystyle y=1/n$$ is simply more dense on $$\displaystyle axis$$ than $$\displaystyle y=n$$

#### Pengkuan

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
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.

#### zylo

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......
.
.

#### [email protected]

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)