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

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

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 .

That f is not a bijection N -> N. It *is* a bijection N -> N \ {0}.
So, are you agreeing that there are bijections between N and at least
some proper subsets of N? Fascinating.

In an cast, Ullrich's example is *not* the bijection f. What you say
here seems utterly confused. I guess you mean to point out that each
permutation in Ullrich's list moves 1 to the next position, but the
claim that this sequence "is" the successor function is just wrong.

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

Good luck finding someone here who claims that you can define F thus.

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.

An infinite composition of functions is not defined at all.

--
Jesse F. Hughes

"[Iota]'s the smallest infinitesimal, Russell, there are smaller
infinitesimals." -- Ross Finlayson
.

Relevant Pages

  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... Let poo be the infinite composition of p with itself. ... You claim poo ... according to HdB, necessarily a permutation, an infinite concatenation ...
    (sci.math)
  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... An infinite composition of permutations need not to ... Ullrich's example is *not* the bijection ... permutation in Ullrich's list moves 1 to the next ... Jesse F. Hughes ...
    (sci.math)
  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... Let poo be the infinite composition of p with itself. ... But note that while an infinite composition of permutations is, according to HdB, necessarily a permutation, an infinite concatenation of naturals cannot even exist. ... a sequence of permutations can be non-existing, ...
    (sci.math)
  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... Han de Bruijn wrote: ... An infinite composition of permutations need not to be a permutation. ... An infinite composition of bijections is not nececcarily a bijection. ...
    (sci.math)
  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... The locution "set closed under the successor function" has a perfectly ... clear meaning while "applying the successor function an infinite ... But in no such case is an infinite composition defined or even definable. ...
    (sci.math)