On 16 Dez., 14:21, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > In article <a586deed-42c7-4523-acb2-1567183f0...@g12g2000vbl.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> 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'.
The axiom has stolen the correct definition from Peano, (Zermelo says from Dedekind, Dedekind says from Bolzano) and has reversed its meaning from infinite to finished. That's all. > > > > > > > > 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.
The same as the axiom says. The axiom says, in addition, that this thing is a set, but as it is not said what a set is, this addition is idle. > > > > > 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?
In fact, it is said so. Sometimes it is carelessly used also for potential infinity. > > > 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.
Of course it is. If with n you can do n + 1, and if 0 is there, then you have omega. The axiom states that such a thing (it says set, but does not say what a set is) does exist.