On 19 Jun., 16:00, Sylvia Else <syl...@not.here.invalid> wrote:
> ZFC makes claims in the context of ZFC. You can't take it down using a > different set of axioms, because ZFC doesn't make statements under those > other axioms. If you want to attack ZFC, as distinct from inventing > competing sets of axioms, your only viable course is to seek to show > that it is inconsistent.
There are things more elementary than ZFC. Induction for instance: If a list is constructed as follows:
0.0 0.1 0.11 0.111 ...
then the anti-diagonal p = 0.111... does not exist or it is in one and the same line of the list.
Proof: All digits of p are in the list within their appropriate columns. Otherwise they could not constitute p.
So either all digits are within one line (containg p) or not. If not, then there must be at least two lines (or more) containing digits of p but not all ofd them.
Now we prove by induction: If n lines contain digits of p then one of these lines contain all digits that are contained by those n lines. Therefore n lines can be substituted by one line. This can be proven for all finite n. But there are no other than finite line nunbers. Hence the proof is complete.
The usual "refutation" of this argument is: There cannot be shown two lines with different digits of p because an infinity of lines is required. But it is obviously not sensible to claim an infinitude of lines while not even being able to show two lines which are required. No, for a set in linear order it is not possible to get the digits of p distributed over more than one line. So, if all are there, then all are within one line. This proves that "not all are there". Finished infinity is nonsense. Therefore aleph_0 und uncountability should be recognized as what they are: pure selfcontradictions. In particular because it is clear that set theory has not any single application in science including mathematics. (if it had, one could easily see it fail. But it cannot fail because it does yield any concrete result apart from the "hierarchy of infinities" that has the same relevance as Russell's tea pot).