Re: how to list all of the real numbers

On Aug 26, 2:12 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1188161306.949947.21...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

On Aug 25, 8:10 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
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.

No, I don't.

You don't what? I don't recall saying you did anything.



Beth_1 is a synonym for c

Says who? Beth_1 is 2^ Beth_0 according to my sources, and nobody knows
whether that is c or not.

I happily follow the convention, luckily
the above statements I make do still hold true given a simple
substitution of terms following the initial definition as above, i.e.,
I miswrote Beth_0 for Beth_1, but in a defensive way. Few care.

How is writing Beth_0 for Beth_1 "defensive"?
Offensive maybe, but defensive?



While it irks me to find myself wrong, I can see that my arguments
about the inapplicability of a cardinal to the reals should see some
reanalysis if I am to convince others via this particular avenue that
there are inconsistencies in the treatment of infinite cardinals.

As the only such inconsistencies Ross has displayed are his own, perhaps
he should go back to the books for a wile.

On the one hand, in well-ordering the reals

Have you actually done that explicitly, Ross?

If so, your name is made, but please post how.

[snipped the rest as useless.]

The cardinal Beth_1 is defined to be the cardinality of the powerset
of N which is equal to the cardinality of the reals, it's defined to
be that value which notation aside are the same thing. Above here, I
defined Beth_0 to be that, which while incorrect, does read correctly,
in the senses that defensively I could call myself technically
correct, and defensively I can brush off your insults.

Browsing Penrose's "The Road to Reality": "The notion of cardinality
does not seem to be sufficiently refined to capture the appropriate
concept of _size_ for the spaces that are encountered in physics."
Also, "It is perhaps remarkable, in view of the close relationship
between mathematics and physics, that issues of such basic importance
in mathematics as transfinite set theory and computability have as yet
had a very limited impact on our description of the physical world."

In a physical universe where physical objects are mathematical
objects, and functions between them physical objects, the universe is
infinite and existence is a counterexample to the notion that, for
example, infinite mathematical/physical universe and powerset of the
universe, identical, have that identity fails, which would be
inconsistent, in, for example, the ontological sense. (The universe
is infinite and I hear that's possible even in a theory with a Big
Bang.)

Another notion to consider is that of the universe of regular sets, of
ZF(C), and how as a collection, and basically as a set, with the
ability to quantify over elements of the collection where membership
as in set membership (E) and partition as in set subsets (<), that via
Rusell's argument that collection contains as a member itself, so that
ZF's universe of regular sets contains an irregular set.

Then as above I'd like the readers here to consider the notion of the
function as described, and how it seems to fulfill the properties of
being a CDF as illustrated, and if so that it describes a uniform
probability distribution over the natural integers, then with regards
to number-theoretic densities of various subsets of the naturals in
the naturals, how then the sizes of those subsets may be quantified
and preserved as invariant quantities in various transitive operations
on those quantities.

After some discussion of as indicated Cartan-guided notions of
infinities and tacitly their preservation under inverse as
infinitesimals, Cantor's flu bugs, "These matters will have
considerable importance for us later."

So, there are some examples of perceived inconsistencies in ZF, and
then there is the notion of the existence of such a function that maps
in the natural order of the reals and naturals the naturals into a
dense subset of the reals in the reals, which would seem to be a
contradiction to standard analysis and the existence proof of a
successor to a least element in a dense subset of the unit interval:
well-ordering them.

Ross

--
Finlayson Consulting

.

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