On 11 Jun., 21:32, Virgil <virg...@nowhere.com> wrote:
> > The idea that the infinite union of finite segments1 > > 1,2 > > 1,2,3 > > ... > > results in an infinite segment is false. > > The idea that an INFINITE union of finite "segments", no two of which > are identical, must result in a finite segment is not even false. It is > ridiculous.
Nearly correct. Absolutely correct is this statement: The idea of an INFINITE union of finite "segments", no two of which are identical, is ridiculous. >
> At least not until WM, or someone, comes up with an axiom system in > which those properties WM posttulates are less self-contradictory than > they are at present.
There is no axiom system required. We accept the hierarchy of types; but we assume only one category of primary objects, the numbers; and one basic binary relation between numbers, namely "x is followed by y." All other relations of the various types are explicitly constructed, the quantifiers (Ex) and (Ax) being applied only to numbers and not to arguments of higher type. No axioms are postulated.