In article <email@example.com> WM <firstname.lastname@example.org> writes: > On 16 Dez., 03:15, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: ... > > > > > > Without the axiom of infinity omega would not be immediatelry > > > > > > existinging. > > > > > > So apparently there is a definition of omega without the axiom > > > > > > of infinity. > > > > > > Can you state that definition? > > > > > > > > > > Look into Cantor's papers. Look into my book. > > > > > > > > I have never seen there a proper definition of omega. > > > Yes, so the definition uses the axiom of infinity. Without that axiom > > omega would not be immediately existing, as I stated. > > No, you stated that you had never seen there a proper definition of > omega.
Look at the context. It started with my asking for a definition of omega without the axiom of infinity. You state that there is such a definition in your book. I find such a definition nowhere in your book. And the 'proper' should be read in the context of 'without the axiom of infinity'.
> > > > > > There are no concepts of mathematics without definitions. > > > > > > > > > > So? What is a set? > > > > > > > > Something that satisfies the axioms of ZF for instance. > > > > > > Is that a definition? > > > > That is not something unheard of. In mathematics a ring is something that > > satisfies the ring axioms, and that is pretty standard. > > And omega is something that does never end.
Whatever that may mean.
> > > But in case you shouldn't have been able to find a definition of > > > actual infinity, here is more than that: omega + 1. > > > > Ok, so actual infinity now is omega + 1. > > No, that's more than that. Omega already is actual infinity. And here > are some other statements about actual infinity:
So actual infinity is omega?
> Let us distinguish between the genetic, in the dictionary sense of > pertaining to origins, and the formal. Numerals (terms containing > only > the unary function symbol S and the constant 0) are genetic; they are > formed by human activity. All of mathematical activity is genetic, > though the subject matter is formal. > Numerals constitute a potential infinity. Given any numeraal, we can > construct a new numeral by prefixing it with S. > Now imagine this potential infinity to be completed. Imagine the > inexhaustible process of constructing numerals somehow to have been > finished, and call the result the set of all numbers, denoted by |N.
But N is *not* defined as an inexhaustable process of constructing numerals that somehow has been finished. The axiom of infinity state (together with the actual definition) state that it does exist, not how it is created. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/