Re: how to list all of the real numbers

In article <1188090017.911925.188730@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx> wrote:

On Aug 25, 1:55 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1188070175.377352.165...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:



On Aug 24, 6:17 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1187994742.323515.123...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

If both of O < c and O > c hold, cardinals are non-trichotomous or the
reals aren't a set.

If 0 < c and c < 0, cardinals are trichotomous and the reals are a set,
and anything else anyone wants to claim is also true.

No, if that is so then where the reals are a set cardinals wouldn't be
trichotomous, because the ordering properties don't hold, not directly
via violation of excluded middle, to inconsistency and Russell is the
pope.

In every order relation of my acquaintance, 0 < c AND c < 0 is a
contradiction, which implies everything and anything.



That the cardinality of the continuum would be consistently equivalent
to a wide variety of cardinals, but not any particular one, is well-
known and swept under the rug.

Are you claiming that merely knowing which aleph or beth fits the
cardinality of the continuum causes inconsistencies?

No, I generally don't.

But you particularly do above.


You appear to be committing a fallacy because
the point is that there is not an ordering relation yielding
trichotomy, in the cardinals, for various reasons to do with the large
number of real numbers..

Is it that there is no order relation or merely that some details of
that relation are unknown?

Where Beth_0 is defined to equal c the cardinality of the continuum,
that's its definition, questions as to whether Aleph_1 = Beth_0


Beth_ 0 = aleph_0, dimwit, and is smaller than c, as can be seen in all
reasonable references, e.g.,
http://en.wikipedia.org/wiki/Beth_number
http://planetmath.org/?op=getobj&from=objects&name=BethNumbers


involve what is called the Continuum Hypothesis, where the hypothesis
is that Aleph_0 < Aleph_1 = Beth_0, aleph, beth, gimel being the first
three letters of the Hebraic alphabet in a convenient expansion of the
Latin mathematical symbol set, where there exist no cardinals between
Aleph_0 and Aleph_1. There are bijections between c and Aleph_1, e.g.
in binary digital coding of the unit interval, and according to
Cohen's analysis of the continuum hypothesis it is undecideable of
whether CH holds or not in ZFC, it is at once a) decided and b)
undecided, decided as there exists an exemplar, undecided via Cohen's
accepted demonstration of the independence of the continuum hypothesis
via forcing in model theory.

(There is no model of ZF in ZF.)

So, nobody claims to know which Aleph Beth_0 is, which is by
definition c.

Wrong, as usual!

I noticed that where there are non-measurable sets, then, that would
seem to indicate a negation of the continuum hypothesis

Measurability and non-measurability has nothing to do with the continuum
hypothesis, so Ross is still wrong, as usual.
.

Relevant Pages

  • Re: how to list all of the real numbers
    ... trichotomous, because the ordering properties don't hold, not directly ... trichotomy, in the cardinals, for various reasons to do with the large ... involve what is called the Continuum Hypothesis, ... degenerate interval of the reals, ...
    (sci.math)
  • Re: how to list all of the real numbers
    ... reals aren't a set. ... trichotomy, in the cardinals, for various reasons to do with the large ... accepted demonstration of the independence of the continuum hypothesis ... greater than any given ordinal, with the ordinals mapping to, say, ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... The continuum hypothesis is ... The situation is quite similar to geometry where the parallel axiom ... also consistent with the other axioms. ... In some models you have that there are no cardinals ...
    (sci.math)
  • Re: Is zero even or odd?
    ... possibility exists that there may be cardinals that are not comparable ... The assertion that c = aleph_1 is called the Continuum Hypothesis. ... will never be proved or disproved in ZFC (Zermelo-Frankel set theory plus ... Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ...
    (sci.math)
  • Re: Is zero even or odd?
    ... possibility exists that there may be cardinals that are not comparable ... The assertion that c = aleph_1 is called the Continuum Hypothesis. ... will never be proved or disproved in ZFC (Zermelo-Frankel set theory plus ... Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ...
    (sci.electronics.design)
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