Re: Well Ordering the Reals

Daryl McCullough said:
> 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.
>
> --
> Daryl McCullough
> Ithaca, NY
>
>
>
If you want to know where I think the root problems lie, I can reiterate
something I think modern mathematicians need to consider. One of the very root
problems in set theory as I see it is a misunderstanding of the nature of
inductive proof. In a traditional proof, one starts with some set of premises
assumed true, and given logical rules of deduction, creates a chain or network
of implications that finally imply the result. In a recursive proof such we
have in induction, we have an implication that always implies another
implication, and so we have an infinite chain of implications, thus covering
the infinite linear set, generally considered the naturals. When induction is
used to prove that every natural is finite it's a mistake. While it is true
that adding 1 to a finite number yields a finite number, that is a finite
operation. If you add 1 n times, then you have added n to the value of the
largest element. If n is infinite, then you have added an infinite value. In
order for induction to prove anything in the infinite case, the property being
proven must be in the form of an equality, and any inequality must be shown not
to converge to zero at infinity. When one says "x is finite", what one means is
"x<oo", where oo is any infinite value. The limit of x, as x apporaches oo, is
oo, in which case x<oo is false. Similarly, induction can be used to prove that
1/x>0, but since the limit of 1/x at oo is 0, this proof does not hold for the
infinite case. Once this issue is noted, some other problems can be addressed.
--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

Relevant Pages

  • Re: infinity
    ... If you're modifying induction at all, then you've modified Peano arithmetic. ... are infinite natural numbers. ... nonstandard models of PA: there, ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... One of the very root ... >> have in induction, we have an implication that always implies another ... >> the infinite linear set, generally considered the naturals. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... One of the very root ... > have in induction, we have an implication that always implies another ... > the infinite linear set, generally considered the naturals. ...
    (sci.math)
  • Re: Cantor Confusion
    ... What numbers are subject to complete induction? ... A is not infinite. ... potentially infinite sets, ... Calling it infinite does not change its properteis either. ...
    (sci.math)
  • Re: infinity
    ... >>>system of axioms, the axioms are absolute. ... But, if you divide both sides by zero, you can get ... >> induction to prove an inequality which is decreasing with each iteration. ... > to establish its initial case, not the fact that the infinite sum ...
    (sci.math)
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