# Re: Bijections (was OM = SIGMA1)

```Bruno Marchal skrev:
> 0) Bijections
>
> Definition: A and B have same cardinality (size, number of elements)
> when there is a bijection from A to B.
>
> Now, at first sight, we could think that all *infinite* sets have the
> same cardinality, indeed the "cardinality" of the infinite set N. By N,
> I mean of course the set {0, 1, 2,  3,  4,  ...}
>
What do you mean by "..."?
> By E, I mean the set of even number {0, 2, 4, 6, 8, ...}
>
> Galileo is the first, to my knowledge to realize that N and E have the
> "same number of elements", in Cantor's sense. By this I mean that
> Galileo realized that there is a bijection between N and E. For
> example, the function which sends x on 2*x, for each x in N is such a
> bijection.
>
What do you mean by "each x" here?```
```
How do you prove that each x in N has a corresponding number 2*x in E?

If m is the biggest number in N, then there will be no corresponding
number 2*m in E, because 2*m is not a number.
> Now, instead of taking this at face value like Cantor, Galileo will
> instead take this as a warning against the use of the infinite in math
> or calculus.
>
--
Torgny Tholerus

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