On 26 Mai, 21:49, Virgil <virg...@nowhere.com> wrote:
> > If we define: > > > 1 is a natural number > > and > > with n also n+1 is a natural number > > and > > N is the smallest set that satisfies both conditions > > > then N is uniquely specified. > > Of course there can be different models for N and there can be > > different names for the elements of N. But that does not matter. The > > natural numbers do not enter mathematics because someone "defines" > > them, names them, or makes models of them, but because the natural > > numbers are simply existing and mathematics is built upon them.
> But according to WM, no such thing as N can exist. > So WM wold throw out the naturals on which so much is built.
The question is not whether the complete set of all naturals exists. That question alrady is nearly as ridiculous as any affirmative answer.
The question is whether we could inform someone who does not yet know, what we understand by the sequence of natural numbers.
In order to answer this question, we need not wait until SETI gets contact. We can answer it in every first class of every elementary school. Of course we can inform any child with average intelligence what we understand by this sequence 1, 2, 3, ...
Only logicians seem to see problems where no problems are. (As some kind of compensation they see no problems where problems are.)
Of course this sequence 1, 2, 3, ... is uniquely defined, because it is possible to inform any intelligent being about it. There is no the slightest difference between the idea of N and the idea of a sonata or pi.
On 26 Mai, 23:09, Virgil <virg...@nowhere.com> wrote: > In article > <b54a16b3-914d-47eb-bd89-967cc10d2...@u10g2000vbd.googlegroups.com>, > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 25 Mai, 17:40, WM <mueck...@rz.fh-augsburg.de> wrote: > > > ... classical logic was abstracted from the mathematics of finite sets > > > and their subsets .... Forgetful of this limited origin, one > > > afterwards mistook that logic for something above and prior to all > > > mathematics, and finally applied it, without justification, to the > > > mathematics of infinite sets. [H. Weyl]. > > > > One example is this. For infinite sets with a linear ordering ³=<² the > > > implication > > > En Am: m =< n ==> Am En m =< n (1) > > > is held correct, but not the equivalence > > > En Am: m =< n <==> Am En m =< n (2) > > > For any finite set with linear ordering however, (2) is true. The > > > reason is that all elements of a finite set are subject to > > > investigation. Therefore, if in ³completed², i.e., ³actual² infinity, > > > as used in set theory, all elements are available, then also (2) > > > should be used. (2) can only be false, if potentially infinite sets > > > are considered, i.e., non-static sets which are never complete but > > > allow that elements can be added. > > But non-static sets do not exist in any mathematical set theory.
They do not exist in what is commonly called ste theory and what is eternally false mathematics.