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

On Oct 9, 6:32 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Oct 9, 6:09 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

David, to argue against (some) results of ZFC I present arguments as
to why, in my opinion, ZFC is inconsistent.

And your "arguments" as to your opinion that ZFC is inconsistent are
not proofs that ZFC is inconsistent.

To argue against
Goedelian results I present arguments as to why that there are no
strong, consistent, complete (and concrete) theories is inconsistent.

Whatever "concrete" means, Godel didn't claim that there aren't
arithmetically sufficient, consistent, complete theories. Rather, he
(as improved by Rosser) proved that there are no RECURSIVELY
axiomatized, arithmetically sufficient, consistent, complete theories.

And whatever "arguments" you have about this matter are not proofs
that incompleteness is inconsistent with whatever you think it is
inconsistent with.

To argue against results of transfinite cardinals I present arguments
as to why they are not consistent.

Your arguments are not proofs.

So, I don't carry on in the face of obvious counterexamples and
reasonable theorems.

You carry on in face of the obvious fact that your word jumbles are
not proofs.

Instead, I offer justifications as to why these
contentious issues (like there isn't a universe, true doesn't mean
provable, infinite sets lack elements and aren't infinite) may
conscientiously be nullified, because of their own inconsistencies.
_Then_ I feel comfortable in the consideration of various nonstandard
theories in mathematical foundations.

Whatever that is supposed to be about, in none of it do we find proofs
of anything.

Consider the physical universe, and map mathematical objects to
physical objects. Then, (all) functions among those represent
physical objects, as do functions among those, ad infinitum. Then,
there are infinitely many objects in the universe, which
mathematically is its own powerset. Then, there is realizable
evidence ("contrary evidence") that it is reasonable to discredit the
powerset result.

No proof of anything in the above.

I found my opinion around the use of mathematical proofs to illustrate
that what I say is so.

You don't use mathematical proof.

With regards to the duck test, proper classes are defined by their
elements, as are sets. Are not proper classes defined by their
elements, containing some specified elements, as are sets, similarly
walking, quacking, etcetera?

Objects are defined by having a property that is had only by that
object.

What's your opinion about the topic under discussion: is the
specified transfinite recursion schema not defined for ordinals up to
the cardinality of the irrationals?

In ZFC there is no upper bound to the transfinite recursion schemata.

If it's not, the irrationals
aren't uncountable. If it is, they aren't either.

You've not proven that.

MoeBlee

The property that there remains an uncountable number or irrationals
left in the interval (p_lambda, 0) would only be so up to and
including any ordinal equivalent to the irrationals, were ZFC
consistent. For each element of an uncountable set there is a
distinct element of a countable set, then as there are 1-1 functions
either way there is a 1-1 and onto function, thus ZFC is inconsistent.

Do you just choose to disbelieve that part?

Arguments, a selection of supporting reasons sufficient to draw a
particular conclusion, are proofs of a sort. Proofs are simply
mathematical argument. That such sweeping arguments can be expressed
in so few words is a sign of their elegance not inapplicability.

Speaking of inapplicability, where there are nonstandard measure
theories, there aren't any applications of transfinite cardinals in
physics, say. Where as well via a simple argument the existence of a
physical universe along the lines of a mathematical universe
illustrates a stark counterexample (proof contrary) to the powerset
result, reasonable people may well find via a clear argument impetus
for the discovery of theories of the infinite more suitably evidenced
in reality. Concrete means real.

About the impromptu Goedel argument, that a recursively axiomatized
theorem strong enough to be Goedelianly incomplete thus has anonymous
axioms detailing properties of its objects, thus as well has another
ghost axiom further specifying that, as does that augmented set of
axioms etcetera, then a recursively axiomatized theory strong enough
to be Goedelianly incomplete isn't recursively axiomatized. In fact,
given the very first anonymous axiom, which presupposes properties
about the objects not otherwise specified in the tangible, explicitly
stated axioms, where those properties are unknown, then the theorem
isn't recursively axiomatizable, because there are as well elements of
the language unknown to any lector.

Proofs are arguments. Mathematical arguments are proofs, of varying
levels of rigor and explicitness.

Now, back to the argument at hand, it has to do with well-ordering a
subset of the irrationals. As described above using transfinite
induction, or a transfinite recursion schema indicating a well-
ordering of a subset of the irrationals, there is thus described in
ZFC using standard definitions of the irrational numbers a set of
irrationals. This set, which is constructible in ZFC via the well-
ordering principle, if it can't be uncountable, has that there aren't
uncountably many irrationals. Otherwise, it can be uncountable, in
ZFC, and is constructed so as to be. That done, it is illustrated
that due the denseness of the rationals in the reals, and reverse
normal ordering on the set, that for each element of the uncountable
set of irrationals there exists a particular, distinct rational
number. Otherwise the rationals aren't dense in the reals.

Then, via Cantor-Schroeder-Bernstein and a trivial injection the other
way, there is a bijection between the irrationals and rationals, due
to the denseness of each of the rationals and irrationals in the
reals.

So, then I've proven, some write proved, that given that the
irrationals are uncountable, in ZFC there exists (any of a wide
variety of) particular uncountable subsets of the irrationals such
that for each element there is a distinct rational, and that means ZFC
is inconsistent.

I have another proof of the existence of an injection from the
irrationals to rationals as referenced earlier in the thread, you
might consider that as well. Also, I argue that in theories with
expanded comprehension compared to ZFC, that ZFC's universe is the
Russell set, thus irregular, thus ZFC is inconsistent, as part of a
slate of arguments giving reasonable people justification to search
for novel mathematical foundations that aren't necessarily consistent
with the standard.

Address the arguments, as you haven't.

Ross

--
Finlayson Consulting


.

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