> > For every n: FIS(n) of the first column is a subset of line n. > > > Claiming that the first column contains more than the lines is false > > in mathematics. None of the lines contains an actually infinite set. > > So, you have made yourself clear on how a majorant is different > from an upper bound (???).
Haha, hope that you have got it.
> The fact is that for any *given* natural number one can find > another different natural number whose existence derives > from the axiomatically *given* successor function
Given or not: For every element n of the table
1 2,1 3,2,1 ...
we find that the first n elements of the first column are collected in a line too. Since this holds for all n, there can never be more in the first column than in a line. Every line contains only a finite number of elements. That's mathematics! Not pseudo-logical blathering as you like it.
We (i.e. proper mathematicians) see that aleph_0 may be use as a name for the first line, but the number of its elements cannot be larger than every natural number. This is impossible. (Were it possible, we had a line in the table with aleph_0 elements. Try to find out why that is not the case!)
|N is a limit, 0.111... is a limit, but no limit can be written with aleph_0 symbols. TSame holds for for the limit paths in the Binary Tree. hTerefore, the Binary Tree cannot contain uncountably many paths.