1

I am starting a new class of theory of computation (you know, where we talk about the turing machine, etc etc) and I am having some difficulties to understand that $|\mathbb N| = |\mathbb Z|$ (they are the same infinite). Before I started this class my intuition always said that $|\mathbb N| < |\mathbb Z|$. Can some one here with your own words explain to me why are $\mathbb N$ and $\mathbb Z$ the same infinite?

Thanks!

Bram28
  • 100,612
  • 6
  • 70
  • 118
Daniel Oliveira
  • 159

1 Answers1

2

The smallest cardinality is called countable or listable. These are the sets which you can write out as a list. For example

$$\mathbb{N} = \{0, 1, 2, 3, 4, \ldots \}$$

and

$$\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \}$$

are both lists. One of them happens to be bidirectional.

It turns out that if you can list a set with a bidirectional list, then you can list it with a monodirectional list. The way to do this is essentially to fold the list in half and zipper the two halves together. That is,

$$ \mathbb{Z} = \{ 0, 1, -1, 2, -2, 3, -3, \ldots \}. $$

If you can write a a set $S$ as a monodirectional list $S = \{s_0, s_1, s_2, \dots\}$ in this way, you get a bijection between that set and the natural numbers by sending $s_n$ to $n$.

Trevor Gunn
  • 27,041
  • Thanks for the answer. But one thing I can´t understand is... As you can see here https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument Cantor said that infinite sequences of binary numbers are uncountable but at the same time I can think of a binary number as a decimal number and create a bijection f: N -> BinaryNumbers. Am I thinking in a wrong way here? Because I does not make sense... – Daniel Oliveira Apr 18 '17 at 01:10
  • 1
    @Daniel You can think of a finite binary sequence as a decimal number but not an infinite one. – Trevor Gunn Apr 18 '17 at 01:12