FISONs

[Hippasos Metapont]

Since from every FISON F(n) = {1, 2, 3, ..., n} we know by definition that it is neither sufficient nor necessary to make the union of FISONs ℕ, we can remove it from the union and find

            ∪{F(1), F(2), F(3), ...} = ℕ   ==>   ∪{ } = ℕ .

Among the FISONs of ℕ there is not, in any enumeration, a first one that is required to yield the union ℕ.

 Usually this is apologized by the fact, that even in

           {1, 2} ∪ {2, 3} ∪ {3, 1} = {1, 2, 3}                                                                      (*)

 it is impossible to find a first set which cannot be omitted from the union to yield {1, 2, 3}. But this argument fails. It is not a set of sets which is subject to Cantor's theorem B (every embodiment of different numbers of the first and the second number class has a smallest number) but only every set of ordinal numbers. Therefore we always have to enumerate the sets. In case of FISONs this is simple. We apply the natural order: {1, 2, 3, ..., n} --> n. Of course every other enumeration would also do. In case of the sets (*) we can use the written order from left to right. Then the first set not to be omitted is {2, 3} because after having omitted {1, 2} already, 2 would then be missing in the union. Every other order is possible and has a first set which cannot be omitted.

Regards, WM