The uncountability of the reals

[Zuhair]

I remember in the lots of discussions present in this Usenet about uncountability of the reals, especially those raised against the poster WM, a point was repeatedly raised by "Virgil" and many others:

that if the reals were countable would they still constitute a real closed field?

Now in predicative type theories, one can have many types of reals, so we have reals of type n, reals of type n+1, etc.. and the set of all reals of each type i is always countable! and each of those sets constitutes a real closed field! apparently being a real closed field was the most important feature that reals were seen to possess, that any definition of the reals if it doesn't constitute a real closed field then it is deemed deficient and thus rejected. Now seeing that each tier of reals in predicative type theories is COUNTABLE and yet constitutes a REAL CLOSED FIELD, then any of these tiers can be taken to represent the reals since it has the sufficient structure needed for such representation. That said, I don't see what is the importance of
insisting that the reals must be uncountable?

[Ross A. Finlayson]

The challenge seems to be a theory that has both and explains why.

Though, it might be observed that complete ordered field is
because of all the sequences that are Cauchy and follows building
the field of rationals, which to this "countable model of R[0,1]"
(eg as countable additivity about measure 1.0) then is simply
lower order in two copies of the integers instead of the rationals
or algebraics as about all the quotients or ratios, the "sweep" or
"equivalency function" is just one limit of the quotients.

So, in structural terms and for apologetics and besides for a
priority of completeness again and courtesy Goedel, Russell,
et al. and the modern canon, then it's about a post-Cantorian
(and super-Cantorian) theory again then as post-axiomatic besides
as I've detailed here extensively for some "theory of truth".

[George Greene]

On Friday, November 3, 2017 at 4:04:30 PM UTC-4, Zuhair wrote:
> that if the reals were countable

The reals CAN'T BE countable, EVER, PERIOD, because EVERY COUNTING of them HAS an anti-diagonal. There is a first-order axiomatization of "the reals" that does not have a predicate for integers, and you can't diagonalize without THAT, so, maybe, FOR THAT LAME version.
https://math.stackexchange.com/questions/151000/tarskis-decidability-proof-on-real-closed-field-and-peano-arithmetic

[Zuhair]

I'm not speaking about that axiomatization of Traski, I'm speaking about much stronger theories that do have a predicate for integers, those are either theories that have a stepwise construction of sets, like the countable segment of Godel's constructible universe, or predicative type theories that parallel that segment. Those can have multiple tiers of reals, each is countable and each is a real closed field. Now any of those can be taken to represent the reals and again ALL of these tiers are COUNTABLE.

Indeed in most of these theories, the anti-diagonal can be constructed! but it would be a real of a HIGHER TYPE, so it won't belong to the diagonalized set of reals because those would all be of LOWER TYPE!!! In some of these theories, you can't even have the anti-diagonal, but those are too harsh.

So why insistent on the saying that the reals CAN'T be countable??? In those theories, they CAN! and all aspects of the reals are interpretable in them, what is the major aspect about the reals that these theories lack? I mean what are the mathematical benefits of uncountability of the reals, that those theories cannot capture by having a countable set of reals.

[Rupert]

Well, if you want them to be a complete ordered field...

[Zuhair]

hmmm..., so the set of all reals of rank omega+n for a suitable finite n in the constructible universe of Godol are not a complete ordered field?

Zuhair

[Ross A. Finlayson]

The natural/unit equivalency function or "sweep"
has only its own binary anti-diagonal, and, it's
always at the end.

(This uses arguments of symmetry as simply as those
of forward induction.)

There's a beginning and an end.

So, the complete ordered field, with field continuity,
sees no iteration of points making a line. It is then
instead with line continuity, that the points (of a model
of the real numbers or the linear continuum) are drawn to
the line segment and countable (and uniquely, and simply).

So, with regards to real analysis, LUB, and measure 1.0,
which are added to ZF to support RA, and "countable additivity",
where the "uncountable" part is removed via measure theory
from having ZF and RA talk past each other, instead:
LUB and measure 1.0 are built with line continuity,
and additivity in the sigma algebra or what is quite
simply framed in a foundational perspective from easily
lesser principles of a plainly logical development.

Nobody "uses" transfinite cardinals for real analysis
(or for that matter anything in physics). Real analysis
on the other hand is built with LUB and measure 1.0 for
the IVT and infinity is the usual lemniscate or "infinity".