Re: how to list all of the real numbers

On Aug 27, 9:45 pm, lwal...@xxxxxxxxx wrote:
On Aug 26, 2:12 pm, Virgil <vir...@xxxxxxxxxxx> wrote:

In article <1188161306.949947.21...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:
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.

But you yourself just wrote in the previous post:

Beth_ 0 = aleph_0, dimwit, and is smaller than c, as can be seen in all
reasonable references

And since beth_0 = aleph_0, it follows that 2^beth_0 = 2^aleph_0, and
of course 2^aleph_0 = c. And indeed, in the link you _yourself_ just
provided earlier to Wikipedia:

"Note that the second beth number beth_1 is equal to c, the
cardinality
of the continuum...."

Thus, rather than clarifying the definition of aleph and beth to RF
and others, you confused RF even more about standard set theory.

Of course, it's understandable that you would be confused, since it's
easy to be confused about this sort of thing. But actually it turns
out that "beth_1 = c" is a theorem of ZFC, and very easy proved, I
might add.

Not being a contrarian, still, no, that's not so. Perhaps it
contributed to the confusion of others. That's one reason why this is
an interesting forum for the discussion of these kinds of issues, in
that, there is a community of experts who readily notice and for many
readily correct perceived errors, in mathematics where there are by
and large indisputably right, and correspondingly barely to totally
flawed wrong, or "not even wrong", answers. One plus one equals two,
half of the integers are even.

Another point of interest is the ability to discuss those parts of
mathematics and mathematical logic that, mostly in the foundations of
the mathematical logic, that have been known since antiquity and
maintain in the contemporary, that are contendable issues, contentious
issues, and in the perception of some, contended issues.

So, here that Beth_1 is defined to equal c, and 2^Aleph_0, where I
wrote once above Aleph_1 which would presume CH, which many do, are
not contentious issues. As matters of common convention they are to
simply be noted in correction, not being particularly pathological
definitions, although I do seem to recall the usage of the Beth
numbers beginning at c in the thinking behind the writing, following
another's usage instead of my own previous.

I don't care so much because I don't find the transfinite cardinals
very compelling, being gross qualitative comparisons where many more
exact and applicable comparisons of the relative sizes of sets are
possible.

So, with that out of the way, then there is the consideration of ZF as
a theory and the notion that it fulfills the properties of being a
set. For any subset of any set in ZF, it's a subset of ZF, for any
element or set in ZF, it's an element of ZF. ZF is defined by its
elements, ZF is a set. While these might simply meet the necessary
requirements for ZF to be a set, then via quantification over all the
elements of ZF, also known as universal quantification, then as has
been noted here many times, it seems that a predicate always resolving
to true as the specification of a set would see the set so defined be
the entirety of the sets in ZF, or else no predicate in first order
logic always evaluates to true, and ZF is a set. Then, as the Russell
set, in matching as well that specification, thus ZF E ZF, and from
that ZF is inconsistent, as that would violate ZF's axiom of
regularity.

Then, in the discussion here there was the notion of universe,
physical universe, in mapping the elements of it to mathematical
elements, illustrates a counterexample to the non-existence of a
mapping, via identity, between set and powerset. In exact analog to
Cantor's paradox, as noted a resolution would have that a) the
universe doesn't exist, or b) Cantor's theorem doesn't generally
hold. If the universe which by definition contains everything didn't
exist then nothing else would either (which would eventually have that
it does exist).

Another notion was that of the consideration of a uniform probability
distribution over the naturals, said probability distribution
described by the equivalency function as cumulative distribution
function. Then, it would be more reasonable to say something, for
example, along the lines of that the probability of two integers being
coprime, which is a known constant, has a particular value.

Then another was in consideration of well-ordering the reals, and how
the range of EF is a set dense in the unit interval, well-ordered by
the real numbers' natural ordering.

Ross

--
Finlayson Consulting

.

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