> "FredJeffries" <email@example.com> wrote in message > news:firstname.lastname@example.org... >> On Jun 20, 10:46 am, "Julio Di Egidio" <ju...@diegidio.name> wrote: >>> "WM" <mueck...@rz.fh-augsburg.de> wrote in message >>> news:email@example.com... >>> > On 20 Jun., 15:33, "Julio Di Egidio" <ju...@diegidio.name> wrote: >>> >> >>> >> (E.g. did anybody study the latter construction [the L=[0,L]]? Does >>> >> the "issue" exists at all, >>> >>> > This issue does not exist in any forum or journal where matheologians >>> > are the dominating fraction. >>> >>> If that distinction has any merit, then it's hard to believe that nobody >>> has >>> looked into it yet. Anyway, if that is so, this "issue" might very well >>> become the basis for my PhD thesis... so to speak. >> >> I cannot find your original question. From the few comments I have read >> it seems that you wish to abolish limit ordinals and relabel the >> successor of a limit ordinal to the limit ordinal's label, etc. Thus, >> your omega is what >> is currently called omega + 1, your omega + 1 is what is now referred to >> as omega + 1, omega + omega for you would be (omega + omega + 1)... >> >> Is that close? >> >> Is there some problem you believe you can solve with such a system? > > These are the two core posts: > > <https://groups.google.com/d/msg/sci.math/nIjyGmRkzCU/nh63p4qtXF8J> > <https://groups.google.com/d/msg/sci.math/nIjyGmRkzCU/ykYC-6KYFtwJ> > > The idea was not just that of a shift of labels: > > Consider von Neumann definition of ordinals: "each ordinal is the > well-ordered set of all smaller ordinals. In symbols, lambda = [0, > lambda)." > <http://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals> > > Now consider this alternative definition: each ordinal is the well-ordered > set of all non-strictly smaller ordinals. In symbols, lambda = [0, > ambda]. -- I.e. the set includes the ordinal to which it corresponds. > > That's what I am asking about, the second definition: if it works, if it > exists in the literature, etc. > > By the way, thanks very much for the references you give in the other > post. > > Julio
By foundation, no set in ZFC can contain itself as an element. Moreover, in the development of ZFC, an ordinal is by definition a hereditarily transitive set. A set S is transitive if any one of its members is also a subset; that is, x e X c S entails x e S. And "hereditarily" means that all elements of S share the property referred to, so here are themselves transitive. The von Neumann ordinals are the ordinals according to that definition. A von Neumann ordinal is well-ordered by the e-relation, i.e., x is less than y in that order if x is an element of y. And then one proves that any well-ordered set is order-isomorphic to an ordinal. Then comes transfinite induction, cardinals are by definition ordinals of a special kind, and so on. This is basic stuff in any textbook or lecture on set theory. Take any one and see what happens with your proposal - I'd say it will be a trainwreck.