Re: Skolem's 'Paradox'

Aatu Koskensilta wrote:
> Ross A. Finlayson wrote:
> > Aatu, I derive humor from that. (Dr.) Koskensilta, what do you think
> > of the notion that a described well-ordering of the reals implies V =
> > L?
>
> It's not true, as far as I can see. I cancelled the post in which I
> (errorneously) claimed so. V = L does imply the existence of a definable
> well-ordering of the reals. Feferman's result is that one can, by
> forcing, given a formula R(x,y) modify any model of ZFC into one in
> which R(x,y) does not define a well-ordering of the reals.
>
> Also, despite my earth-shattering contributions to both philosophy and
> mathematics I have no academical degree of any sort. I am but a lowly
> student.
>
> > That leads me back to discussion of well-orderings of the real numbers.
> > One notion that has been recently addressed is the existence of an
> > ordinal for each cardinal, and each ordinal is well-ordered. Thus, for
> > any set, if it has multiple cardinals than via Cantor-Bernstein
> > infinite sets are equivalent, if it has multiple ordinals that is
> > satisfying to that there is the existence of at least one ordinal, in
> > ZF, and thus Choice is a theorem of ZF as any set is well-orderable.
>
> I have no idea what you're trying to say. What does a set "having
> multiple cardinals" mean?
>
> > If you research to where I mention Feferman's result, in "On
> > Well-Ordering(s) of Sets Dense in the Reals, Infinity", what do you
> > make of the quote from Friedman?
>
> The single sentence
>
> Even for countable sets of rationals, we know that, in various
> appropriate senses, we must use transfinite induction of arbitrary
> countable well ordered lengths.
>
> alone doesn't tell us what Friedman is talking about. More context is
> needed.
>
> --
> Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)
>
> "Wovon man nicht sprechen kann, daruber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Ah, you are an astute researcher. Some of the context is discussed in
the reference.

Feferman's result in the words of Herb "provides a model for V=/=L, but
not V = L." Thus, if other reasoning leads to a well-ordering of the
reals, then V = L.

This talk has drifted away from that of Skolem, and Skolem's paradox,
in this thread here opened by William of Ockham in his request for
"Filosofical Refutations of the Cantorian Hypothesis."

What has been discussed is the ramifications of the well-ordering of
the reals with regards to Cantor's first that there are adjacent points
in the normal ordering of the reals, or the rationals are uncountable
or the reals not a set. In corolarry, Choice, or the well-ordering
principle, would be a theory of ZF because of the existence of an
ordinal for each cardinal. ZF being inconsistent due to quantification
over sets with the predicate true leading to a universal set, then in
the sense of reverse mathematics non-logical axioms are pared from that
system. A set of zero axioms, or rather, the null axiom system, offers
the additional benefit, in the sense of those valuing completeness in a
logical theory, of not being incomplete, towards consistency and
completeness in a logical system.

We're students, of ourselves. The student is the teacher.

Well-order the reals. Iota means Vitali does not say there are
non-measurable sets, and Banach-Tarski is different, about points on a
line, as members of continua, or not, points, polydimensional points.

Skolemize, the naturals are unchanged.

Ross

.

Relevant Pages

  • Re: Rational numbers, irrational numbers: each dense in real numbers
    ... of the well-ordering X_i setminus i, that is defined to equal X_{i ... X (whether or not it's the ordinary successor relation of the ordinals ... In ZFC, if the reals are a set, each subset of the reals can be well- ... reals, as are the irrationals. ...
    (sci.math)
  • Re: RAF: Rational numbers, irrational numbers: each dense in real numbers
    ... the reals, give an explicit well-ordering of the reals. ... It seems you accept that a well-ordering of the reals, ... there exist as many ordinals beta less than ... if there are uncountably many strictly decreasing ...
    (sci.math)
  • Re: Skolems Paradox
    ... don't recall an explanation that V = L and a well-ordering of the reals ... if it has multiple ordinals that is ... I think infinite sets are equivalent. ...
    (sci.logic)
  • Re: Skolems Paradox
    ... V = L does imply the existence of a definable well-ordering of the reals. ... One notion that has been recently addressed is the existence of an ordinal for each cardinal, ... Thus, for any set, if it has multiple cardinals than via Cantor-Bernstein infinite sets are equivalent, if it has multiple ordinals that is satisfying to that there is the existence of at least one ordinal, in ZF, and thus Choice is a theorem of ZF as any set is well-orderable. ...
    (sci.logic)
  • Re: Well Ordering the Reals
    ... >>In a well-ordering of the reals, ... >>an initial segment indexed by the natural integers. ... For each countable initial segment is generated a unique convergent ...
    (sci.math)
Quantcast