"Ralf Bader" <email@example.com> wrote in message news:firstname.lastname@example.org... > Julio Di Egidio wrote: >> "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.
Thank you and Zeit Geist for pointing my in the right direction. The whole thing is actually quite interesting, although for many of you this may be basic stuff.
Following links from non-well-founded set theory, I get to the axiom of superuniversality and axiomatic nonstandard analysis, a connection that is pretty valuable for me. OTOH, I also get a connection to computer science, namely to bisimulation, corecursion and coinduction.
So, I'll have to study to eventually get back to a definition of ordinals (and cardinals, etc.), but I already don't see why it should be a trainwreck...