On Dec 22, 5:38 pm, Marshall <marshall.spi...@gmail.com> wrote: > On Dec 22, 5:27 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> > wrote: > > > On Dec 19, 5:20 pm, Virgil <Vir...@home.esc> wrote: > > > > Not at all! > > > > I merely note that every rational integer is included among the reals, > > > in both the Dedekind construction and the Cauchy construction, and 1 is > > > nicely both rational and integral. > > > Oh, so 1 = 1.000... now, eh. > > > Good luck with that. > > He won't need it. 1 the ratio is the same number as 1 > the natural, is the same number as 1 the member of > any other set. > > Unless you confuse numbers and their representations. > > Marshall
In set theory there are only sets.
Of course I think that 1 = 1.000..., but it makes a Platonist out of you.
Consider the general construction via ordinals of the naturals, vis-a- vis a standard construction of the reals. If you want to encode their representations then it's of products of sets, for example those.
So, implicitly, writing "1 = 1.000..." encodes both of them. There are simpler ways to model various real-valued processes.
There are more obvious consequences, more of them. Area is the sum.
Rationals have much simpler encodings than the general standard case, which finds itself "incomplete" in being the "complete" ordered field, in that each arithmetic operation would be definite, constructively then via completeness, Dedekind/Cauchy/Eudoxus constructions are insufficient to model the real, real numbers, number-theoretically.