On Friday, December 6, 2013 1:17:30 PM UTC+1, WM wrote: > Am Freitag, 6. Dezember 2013 12:33:19 UTC+1 schrieb Albrecht: > > > > > > > The diagonal argument from Georg Cantor is so silly and easy to destroy. Let's have a "antidiagonal" of all natural numbers. No problem: For any list of naturals we can take the first number and change it by adding 1 to gain d_1, take the second number and, if different from d_1, change it by adding 1 to gain d_2, else d_2:=d_1, and so on. > > > > > > For all numbers of the list, the (equivalent to Cantor's antidiagonal) gained antidiagonal d_list is natural (proof: a natural +1 is natural) and different from any number of the list (proof: it is at least 1 more than every number of the list by construction) and thus the list isn't complete. Since this holds for _any_ list of naturals --> the naturals are uncountable. > > > > But first, the list > > > > 1 > > 2 > > 3 > > ... > > has no diagonal, and second: d is infinite. This is the trick that always works. For instance, we can connect every node of the Binary Tree with the root node. Every connection is finite. All connections are in a countable set. And there are no further nodes to construct further connections. > > > > But it is the business model of matheology to cheat innocent newbees and mathematicians who are not so familiar with that matter by asserting that there are further connections (paths). From the fact that every connection can be extended to a longer one, they claim that there are connections that are longer than all finite ones. Simply a sleight of hand, an intentional fraud. Therefore in your example there is also a diagonal which stretches beyond all finite numbers of digits although every line has only a finite number of digits. And abracadabra ZFC is free of contradictions. > > > > Regards, wM
My given proof has only one flaw: in the case of an infinite list of pairwise different naturals, there must happen an infinitude of additions of 1's to gain the "antidiagonal" d. And an infinitude of additions isn't defined. So the proof fails. Cantor's second diagonal argument fails not because reals are defined as an infinitude of digits after the decimal point (in the case of decimal system).
I think there must something wrong with this definition. Only because all the reals we know are describable with an infinitude of digits after the decimal point does not lead necessary to the reverse conclusion that all numbers with an infinitude of digits after the decimal point would be real numbers (and especially would be senseful numbers in any concern).
One hint that this reverse conclusion is wrong is the by you well stated fact that there are only countable many names or numbers or definitions of numbers or constructions or whatever.
The platonic view of the mathematicians at this fact is nothing more than a confession of failure.