Re: Ultimate debunking of Cantor's Theory


"Calvin" <crice5@xxxxxxxxxxxxxx> wrote in message news:1184268207.386552.179480@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Jul 12, 2:55 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
Calvin <cri...@xxxxxxxxxxxxxx> wrote:
> Are there other noteworthy forms of the diagonal argument?


There are many. It is the underlying tool used for many of the most important results in set theory and logic. In addition to the two results already described, it is the basis of:

* Goedel's first incompleteness thereom;
* Turing's proof of the unsolvability of the halting problem
* Church's proof that the set of theorems in ZF is not recursively enumerable

Asking if there are any other noteworthy forms of the diagonal construction in set theory is a bit light asking if there are any other noteworthy uses of limits in calculus.


A related theorem and proof is the theorem that for any set S, there is
no surjective function from it to its power set, P(S) (the set of all
subsets of S).

Suppose f is a function from some S to its power set P(S), f: S --> P(S)
consider the subset of S of all members which do not map into their
images under f, i.e., M_f = {s in S: not s in f(s)}.

Given any f:S -->P(S), such a set is always well defined, but a moment's
thought shows that one cannot have f(x) in M_f for any x in S.

You're presenting half of the continuum hypothesis, right?


No, he is showing another use of the diagonal argument, which shows that you are given a set of any cardinality, you can construct a set with strictly larger cardinality.

You're showing that the power aleph1 is greater than
the power aleph0, but you're not really showing that
it is the same as the power of the reals (called c),
the continuum hypothesis being that aleph1 = c.

No. Aleph1 > Aleph0, by definition.
The diagonal argument shows c > Aleph0

It doesn't prove c=Aleph1, and indeed that cannot be proven in ZFC. You are however free to add it as an axiom if you which, in which case it obviously is provable.


My limited understanding is that the continuum
hypothesis has been shown to be neither provable nor
disprovable.


That is correct.


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