Re: A puzzle for Cantorists


Jonathan Hoyle wrote:
Ross A. Finlayson wrote:

Your "first principles" and "direct reasoning" are logically proven to
be self-contradictory. Just because you believe them does not make
them true, Ross.


Jon, Not ZF.

Not ZF what?

As demonstrated to you on many occasions, this "cohesive structure" is
based upon false assumptions, which has been the basis of most of your
errors. I am convinced that some of your ideas could be salvaged
under another axiomatic system, but they do not apply to *our*
axiomatic system of mathematics.

Obviously you have some appreciation of the reasonings I present.

I do. Primarily your arguments fail because they are inconsistent
with your starting assumptions, such as ZFC. However, many of your
ideas could be consistently defined in other systems, albeit with some
modification.

There are no false assumptions in the null axiom theory.

One of your false assumptions is the belief that you can generate
theorems without axioms.

And Kronecker didn't think irrational numbers existed either.
Opinions ultimately don't matter...even the opinions of historical
mathematicians. What matters is what can be proven. And like it or
not, upon accepting the Axiom of Infinity, transfinite cardinals
exist. Now, if you don't like that, you are welcome to remove that
axiom and begin with a new set. But upon accepting ZFC, the choice is
no longer yours to "reject" or "accept" them.

Those are excellent mathematicians.

Perhaps, but at least in the case of Kronecker, a very sub-par human
being. As I state, opinions (as well as personality traits,
intelligence, looks, whatever) are totally irrelevant. What matters
only is what is proven.

I don't appreciate being called a crank...

I can't imagine anyone would be. I don't believe I have called you
one, at least in this posting.

...and am not one.

Let me ask you an honest question, Ross: If you were a crank, do you
think you could be able to recognize that fact? How would your
behavior differ? Also, why do you think that trained mthematicians
suggest that you are one?

Hope that helps,

Jonathan Hoyle
Eastman Kodak

That's a good question. I hope I could honestly assess myself. I
believe I've proven, using mathematical logic, that for example where
a set is defined by its elements and a set containing only and all
elements of the natural integers bijects to a set containing only and
all elements of the real numbers, N^G <-> R (Skolem), that they do.
Is a set defined by its elements? Does N^G contain only and all
elements in N? Does N^G biject to R? Then a set containing only and
all elements in N bijects to R. (There is no model of ZF, for it
would contain all the sets, which ZF denies in regularity.)

I say infinite sets are equivalent, "mathematicians" trained to
believe otherwise can not find in their belief system their teachings
incorrect, so in this realm of mathematics where there are right and
wrong answers, it defaults to that if they're right I'm not. It is
unfortunate for them that they don't know I've developed a
comprehensive line of argument about why and how infinite sets are
equivalent, in a post-Cantorian way, in the light of the Cantorian
results. That is to say, I am familiar with the Cantorian results.
If they devoutly ascribed to an axiom system that 2+2=5 it would still
be not so. (Transfinite cardinals have little or no utility.)

(It was well-known a hundred years ago that anything going sixty miles
an hour would spontaneously explode.)

About deriving theorems from the NAT, consider there is not one, then
there is nothing, via contradiction there is one, back and forth ad
infinitum generating a continuum of individua suitable as the
hierarchy. So, there is at least one, the empty set is a theorem, as
are a variety of others. Then, via properties of the continuum of
primary objects that are deducible from the uniquification of
individua, in mereology (consideration of boundaries) and so forth,
the properties of the natural numbers, as examplar objects, are self-
evident, as are as well those of all other natural constructions, for
example the space of all hypercomplex numbers. The null axiom theory
has theorems.

Jon, if there's a universe, then it is its own powerset. If there's
not, then there's no universal quantifier, for it would quantify over
all elements.

When I say Not ZF, I mean that ZF contains false axioms. There is a
universe, say, X, but not in ZF.

"Countable additivity" (summation of the unit's infinitesimals,
integration) is used much more than transfinite cardinals, in
application. Is it dishonest to say that integration is not the sum
of infinitesimals?

The null axiom theory is a naive set theory, it has much of naivete,
it's true in ignorance of it, in all systems.

When Fraenkel called transfinite cardinals a disease, I wonder why.

With regards,

Ross

--
Finlayson Consulting

.

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