Re: Well Ordering the Reals

Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx> writes:

> TO says that we sasy things we have never said. Omega, if
> representing the first non-finite ordinal, has no immediate
> predecessors, and no ordinal has omega in the chain of its immediate
> successors.

Not only "immediate". Iteration does not cross the realms.

>> Any finite natural you may consider as a predecessor to omega is
>> finite,
>
> In the obvious sense, while all finite ordinals precede omega,

They don't. They are smaller than omega, but by virtue of meeting
different axioms, not by virtue of forming some chain of succession
with omega.

> none of them have omega in the chain of their immediate successors.

Strike the "immediate". Claiming omega as a successor of any sort is
confusing language, and Tony is confused easily.

> TO conflates being linked in the successor/predecessor relation with
> the more general order relation.

Yes, you get it. Just use this language consistently, don't apply
"successor/predecessor" to omega, and you have one phrase less with
which to confuse Tony. There will be enough other terms remaining for
that purpose, don't worry.

--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... and no ordinal has omega in the chain of its immediate ... >> In the obvious sense, while all finite ordinals precede omega, ... >> none of them have omega in the chain of their immediate successors. ...
    (sci.math)
  • Re: Christmass theory.
    ... Of course, this begs the question, what would the order type of a ... so Omega is the order type of $, and thus Omega is a member of $. ... Burali-Forti paradox! ... von Neumann ordinals is that if the class of all ordinals were a ...
    (sci.logic)
  • Re: Axiom of choice
    ... like i said; i hate omega. ... But with ordinals, omega + 1 is distinct from omega. ... Notice that mathematicians ... so by writing the operands as cardinals or ordinals. ...
    (sci.math)
  • Re: Axiom of choice
    ... like i said; i hate omega. ... Hotel ideas of infinity) ... But with ordinals, omega + 1 is distinct from omega. ... pointing out that they agree for finite cardinals. ...
    (sci.math)
  • Re: Question
    ... appalling ignorance about usenet ettiquette. ... subtraction is really addition of additive ... writing "omega - n = ... The ONLY ordinals strictly less than omega are the natural numbers. ...
    (sci.math)