Re: abundance of irrationals!)

Hi,

In examining Cantor's first proof, if you have a construction of the
reals where they are, say, nonstandard in the sense of being at once a
sequence of monotonically increasing scalar valued-points as well as a
field, with reasonable restrictions on statements about a given real
number in one of those senses vis-a-vis another, then a conclusion can
be made rather simply that given Cantor's premises of his first proof,
that a bijection between the naturals and reals would be this monotonic
function, where EF is an example of that monotonic, "not a real",
function.

If you say that the set of real numbers comprises a set, which many do,
where often the verb conjugation between singular and plural subject is
implicitly mutable in the sophisticated plain language expression,
where the real numbers are a set then the Zorn's Lemma/Axiom of
Choice/well-ordering principle implies that the set is well-orderable.


With the unary or infinite radix, any antidiagonal is an element of the
set, via induction.

Induction is itself a very key consideration of the consideration of
things infinite. As the realm of infinite quantities, eg ordinals, has
been seen to transcend the finite, induction carries over.

I think one thing that is a thorn in the side of many casual and even
dedicated people who think about infinite sets is the difficulty of
proving something in the real world using the transfinite cardinals.
One reason why it might seem contentious to argue for the validity of
the transfinite cardinal is that the physical universe is a physical
object, and the physical universe contains infinitely many physical
objects, where functions between them are physical objects, thus the
physical universe of physical objects is a counterexample.

That leads to the same consideration that has always affected the
consideration of the infinite, basically called Burali-Forti or the
order type of all ordinals would be an ordinal, the set of all sets
would be its own powerset, etcetera. Luckily, there are philosophical
parallels that can be codified into a mathematical logic, ranging from
simple, ancient justification of the rationale of being to insightful
and highly technical philosophers of more modern times.

These concepts of the infinite find themselves highly intertwined with
other basic and fundamental questions of mathematical logic and the
philosophies of mathematical logic.

There's always one more. Skolemize, your model is countable. There is
no cosmic daylight savings time. Half of the integers are even.

Quantification implies a universal set. ZF, Zermelo-Fraenkel Set
Theory, is inconsistent, because of regularity.

Regards,

Ross F.

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