Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/14/05 

Date: 14 Jan 2005 13:06:51 -0800

briggs@encompasserve.org wrote:
> In article <1105688163.538902.182680@f14g2000cwb.googlegroups.com>,
"Ross A. Finlayson" <raf@tiki-lounge.com> writes:
> > The reals and any subset of the reals are totally ordered (by their
> > numeric magnitude). I guess I think a proper well-ordering of the
> > reals is its natural total order.
>
> Sure. If you believe that every subset of any totally ordered set
has
> both a smallest and a largest element then it follows that every
> totally orderd set is well ordered.
>
> You're a fellow who believes that N contains oo.
>
> You're a fellow who believes in two real numbers, infinitesimally
> close to one another so that there are no other numbers in between.
>
> So one would expect you to fall for this fallacy, hook, line and
sinker.
>
> John Briggs

The set N is the set of finite natural numbers. If there are
ubiquitous naturals then N is the set of all sets. The set of sets
that are rootsets of the empty set is the empty set, the contents of
which is the empty set. That doesn't change that the powerset of the
empty set is not the empty set, because the empty set is deemed a
subset of every set. That's where the powerset is order type is
successor except for the special case of the the ur-element, the only
set in the set theory that is also not a set, the ur-paradox that is
not. The set of all sets is its own powerset.

So, I don't necessarily believe that N contains oo: N = oo, and N E N+1
( N E P(N) ), and oo+1=oo.

There are only real numbers on the real number "line", that is to say
the set of real numbers. In a total well-ordering of the reals (or
rationals or irrationals), it's fair to say that for a given definite
real there is a previous and next real number, because the set contains
only real numbers. You can say that there exists a well-ordering of
the reals and I showed above in a simple and direct manner how there
could thus be a total well-ordering.

So, you say there is a fallacy and imply that "a total ordering is a
well ordering" is false. A total ordering on the reals has no greatest
and least element, it diverges towards infinity towards the negative
and positive. Neither does the set of natural numbers, it diverges
towards infinity in its total and well-ordering, nor does the set of
all integers. So that's one reason why that the least or greatest
would apply to a set of the reals with the infimum or supremum, the
greatest lower bound or least upper bound. Alternatively any subset of
the reals has a total ordering and thus a well-ordering.

The reasons I say that is not blind refutation and denial of
"orthodoxy" but informed logical progression using common vocabulary
and first order logic, coincidentally to work with the axiom free
theory that is strong enough to represent the natural integers and
having no non-logical axioms free from incompleteness, in these
modular, logical components.

That is to say, I think you understand what I say and why about that if
you can well-order the reals then its total ordering is a well
ordering. I don't have a hidden agenda, I just want to establish among
us that the truth value of that statement is true, that it is a theorem
of the logic (logical theory) in use, and that is thus usable as a
mathematical fact.

So, via well-ordering with the ability to enumerate the total ordering,
then there is determined a sequence of real numbers, in order, with
none missing between them. They're real numbers because there are only
real numbers in the real numbers. They're enumerable because of their
total well-ordering. So there is a sequence of real numbers and to
give names to one of them, iota, is a way to impart structure to the
sequence, in defining names and rules of operations upon these sequence
elements, representing contiguous elements, for their consideration and
discussion at leisure.

Deduction ensues and it can be seen that any finite sequence contains
at most one "definite" real number, the rest being "indefinite". Iota
might be non-constant, it's the least infinitesimal. There are smaller
infinitesimals, or perhaps not. Infinity happens to be larger than any
finite number.

John, what do you think about EF? Orthodoxy says the domain of EF is
countable, thus with measure zero. I disagree, because x/x = 1. Is
the impulse function a function? I guess I hope you would elaborate
and go into detail on your opinion on infinite sets in a set theory.
Regards,

Ross Finlayson 

--
"I know... I just love hearing it."

Relevant Pages

  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... a well-ordering, where the well-ordering is a total ordering, or are ... Dedekind cuts or Cauchy sequences insufficient to represent all reals? ... I really think the primary objective of calling infinite sets ... ordering, for example the complex numbers, that that total ordering is ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... a well-ordering, where the well-ordering is a total ordering, or are ... Dedekind cuts or Cauchy sequences insufficient to represent all reals? ... I really think the primary objective of calling infinite sets ... ordering, for example the complex numbers, that that total ordering is ...
    (sci.logic)
  • Re: abundance of irrationals!)
    ... All I know is that what we know about infinite ... > the sets I call finite have larges members. ... The set of all finite naturals is not infinite, ... Sets defined by mapping functions from the naturals to the reals which have ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... at least one of those is a well-ordering. ... > At least one total ordering is, ... well-ordering of the reals. ... magnitude, or scalar value, due to trichotomy of the reals. ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... at least one of those is a well-ordering. ... > At least one total ordering is, ... well-ordering of the reals. ... magnitude, or scalar value, due to trichotomy of the reals. ...
    (sci.logic)