Re: Well Ordering the Reals


Daryl McCullough wrote:
> Tony Orlow says...
> >
> >MoeBlee said:
> >> Tony Orlow wrote:
> >>
> >> > Actually, most of the standard axioms would get scrapped
> >>
> >> Then please say exactly what you would scrap in this list:
> >>
> >> classical first order logic
> >>
> >> identity theory
> >>
> >> extensionality
> >> separation schema
> >> power set
> >> union
> >> pairing
> >> infinity
> >>
> >> regularity
> >> choice
> >>
> >> replacement schema
>
> >I am not sure about that. I see a system where the real line and
> >quantity form one kind of set and infinity, and discrete counting
> >systems form another, as in standard set theory, but where N=S^L
> >as a rule for symbolic systems is observed, and the inverse
> >function rule is applied for quantitative sets. The power set
> >relation is important, but given undue attention and importance.
> >Anyway, I think what I envision is simply a different starting point.
>
> It doesn't matter what your starting point is. If you are going to
> claim something that is a contradiction with existing set theory,
> you should be able to say which axiom of existing set theory is
> false.
>
> Moe's list is actually longer than it needs to be. You don't
> need regularity or choice for most of the results about reals
> and naturals.

Nor the replacement schema. I included those just for the purpose of
having a comprehensive list to see just what it is that Tony disagrees
with in set theory, not necessarily confined to questions about the
reals. For that matter, as you know, in some instances, we don't even
have to adopt certain of the axioms but can instead talk about the
theorems following from the axioms, irrespective of whether the axioms
themselves are accepted. In that sense, the question for Tony and
others is not what axioms they reject, but rather what do they claim
are the mistakes in the arguments showing that the theorems do follow
from the axioms.

MoeBlee

.

Relevant Pages

  • Re: Finite schema?
    ... no quantifiers over classes, ... Paul Cohen's "Set Theory and the Continuum Hypothesis", ... in a set theory that can be converted to a finite axioms in a class ... Schema of Separation in Z ...
    (sci.math)
  • Re: what is wrong with this theory.
    ... identity) by the following non logical axioms: ... Ayparadoxical then the left hand of the schema will be ... paradox (which is a *logical* paradox, prior to any theory, ... (which is, like the language of set theory, simply the first ...
    (sci.logic)
  • Re: Skolems Paradox and why is math the way it is?
    ... interpretation takes anything seriously at all. ... > believing the axioms are correct in some sense or something like that. ... Set theory was billed to me as the type-free be-all theory, ... It still doesn't mean that the reals ...
    (sci.math)
  • Re: Cantorian pseudomathematics
    ... > produces any infinite values. ... It's a theorem of set theory. ... Forget about axioms here. ... > naturals, the universe is the real number line which includes all quantities, ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... most of the standard axioms would get scrapped ... you claim that set theory is ... theory in which to express virtually all of mathematics. ... S (call this function 'omega pre S'). ...
    (sci.math)
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