Re: 1-1/2+1/3-1/4+1/5-1/6+1/7

Han de Bruijn wrote:

Consequently, the empty set has the cardinality of the natural numbers?

What goes wrong here?

Let's try again.

An infinite composition of permutations need not to be a permutation.
Quite right. Ullrich's example:

1 2 3 4 5 6 7 8 9 ...
2 1 3 4 5 6 7 8 9 ...
2 3 1 4 5 6 7 8 9 ...
2 3 4 1 5 6 7 8 9 ...
2 3 4 5 1 6 7 8 9 ...
2 3 4 5 6 1 7 8 9 ...
..
..
2 3 4 5 6 7 8 9 10 ...

Which is the following bijection N -> N : f(n) = n + 1 , n in N .

Suppose we make a composite function F(n) = f(f(f(f( .. (n))) .. ))
of infinitely many of such bijections (successor functions).

Theorem: F(n) removes every natural (n) from the set of naturals N.
Proof by contradiction: name _one_ number (n) which is not removed.

Conclusion: F(N) is the empty set, F(N) = {} . And:
An infinite composition of bijections is not nececcarily a bijection.

Any bijections, uhm .. OBjections to the above?

Han de Bruijn

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