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

On Oct 17, 6:47 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:
...
That is where the desired property for a given ordinal (that there are
more elements in the interval (0, p_alpha) for ordinal alpha less than
the cardinality of the irrationals) holds for ordinals less than or
equal to the cardinality of the irrationals, where for higher ordinals
the property would not consistently hold, but it is not necessary that
it does. So, in the course of values over all ordinals, for ordinals
less than or equal to the cardinality of the irrationals there are at
least that many remaining in the interval (p_alpha, 0). (Otherwise,
there wouldn't be that many in the interval.) For ordinals greater
than the cardinality of the irrationals, they as well satisfy the
property in being greater than the cardinality of the irrationals.
Then, there are as many elements p_alpha as there are ordinals alpha
that are less than or equal to the cardinality of the irrationals.
Then, that holds for sufficiently many irrationals, for each of which
can be displayed a distinct rational, that theorem contradicts another
in the theory.

Basically for each partition of the irrationals intersecting the
interval (p_alpha, 0) into (p_alpha, p_alpha+) and (p_alpha+, 0), each
partition has the same cardinality.

The rationals and irrationals are each dense in the reals.

So, ZFC is inconsistent.

What, no witty rejoinder?

So, unless there is some reason why there aren't uncountably many
irrationals in any interval (0, p_alpha), there aren't uncountably
many irrationals in (0, p_alpha).

Then, this thread ends in a similar fashion as to how the rational/
irrational thread of late 2005 ended: illustrating a perceived
inconsistency in ZFC from the uncountability of irrationals and
denseness of rationals and irrationals in the reals. (http://
groups.google.com/group/sci.math/msg/f218848b9bbe4830)


No, this thread ends with everyone giving up on you
ever giving even a semi-comprehensible explanation
from why you think it is possible to have
a strictly decreasing sequence of reals with an uncountable
number of elements.

-William Hughes



.

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