Re: Two results of set geometry

On 12 Sep., 12:55, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> writes:

The set of natural numbers, if it could exist, would be equivalent to
another natural number. Therefore it cannot exist. (Tip, hint: there
is a bijection between sets and naturals).

Where might one find this bijection described?


Here:

1 <--> {1}
2 <--> {1, 2}
3 <--> {1, 2, 3}
....

and even the points "..." in {1, 2, 3, ...} describe nothing but
natural numbers.

But a stronger result is that

lim{n --> oo} |{2, 4, 6, ..., 2n}| / 2n < 1.

So there is always, i.e. for any n, a greater natural number than
|{2, 4, 6, ..., 2n}|, and hence, there is a greater natural number
than
|{2, 4, 6, ...}|. As this cannot be true {2, 4, 6, ...} cannot exist.

Regards, WM


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