Re: RAF: Rational numbers, irrational numbers: each dense in real numbers

On Oct 22, 7:58 am, William Hughes <wpihug...@xxxxxxxxxxx> wrote:
On Oct 22, 12:43 am, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:



On Oct 21, 6:54 pm, William Hughes <wpihug...@xxxxxxxxxxx> wrote:

On Oct 21, 9:05 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

With regards to the proof (largely in sketch, but assembled piece by
piece) of the existence via choice of a transfinite "sequence" of
descending irrationals, I would like to hear more arguments as to why
that is not possible, so they can be addressed.

Let X be an ordinal, and for every x in X, let r_x
be a real number. 0 < r_x < 1

Let C={r_x| x in X}

Note that if x is in X, then x has
a successor, call it x+1.

Assume that 0<r_(x+1)<r_x.

C is a strictly descending "sequence".

Let N be the set of natural numbers.
Let n be an element of N.

Let U_n = {r_x| x in X, 1/n >= (r_x-r_(x+1)) > 1/(n+1)}

U_n has fewer than n+1, elements, so U_n is finite.

C = union n in N U_n

C is the union of a countable number of finite sets, so
C is countable.

Thus any strictly descending "sequence" is countable.

Note that all we need for this proof is that every element
of the "sequence" is strictly greater than its successor.
Since every element has a successor, we do not have
to distinguish betweeen limit and non-limit ordinals.

- William Hughes

<snip stuff which does not address argument>

You did not address this argument,
(if a set of non-zero
intervals has finite length it is countable)
you gave your confused partial construction again.

If you wish to continue in this line please use the
following form.

Let X be an uncountable ordinal.

If x is an element of X an algorithm for
obtaining r_x is ...

Note two things.

x can be any element of X. This means that
x may be a limit ordinal, may be larger than
a limit ordinal, may be larger than two limit
ordinals, may be larger than a countable number
of limit ordinals. Any method that assignes
a value to r_x, must work if x is larger than
a countable number of limit ordinals.

if your method depends on defining the "next"
element, you will never get to (let alone past)
the first limit ordinal (transfinite induction
does not use the concept "next"). You need a method
that assigns a value to r_x, if you have
the values for every r_y where y < x (there may not be
a last such y).

- William Hughes

That's like saying: don't just prove there exists a well-ordering of
the reals, give an explicit well-ordering of the reals.

I'll further elaborate what I mean by constructing various subsets of
well-orderings of the reals in terms of limit ordinals etcetera
later. For now, I just want to remind that there is the proof of
existence of a well-ordering of the reals, in ZFC, and nobody can
identify r_x, the x'th element of a well-ordering of R in terms of an
ordinal X, there is no algorithm for it, or the reals aren't
standard. As an illustrated consequence in the above development, the
reals aren't standard.

Ross

--
Finlayson Consulting

.

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