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Topic: § 433 The reason for calling matheology matheology
Replies: 24   Last Post: Feb 22, 2014 5:23 PM

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 fom Posts: 1,968 Registered: 12/4/12
Re: § 433 The reason for calling ma
theology matheology

Posted: Feb 20, 2014 12:39 PM

On 2/20/2014 5:39 AM, mueckenh@rz.fh-augsburg.de wrote:
>
>
> Zeitgeist on the odds of my students to understand transfinite set theory:" if they can be convinced that what their senses tell them may be Not be the whole picture, then they may have a chance."
>
> Virgil, appearing as Wisely Non-Theist: "There are 'more' real numbers than there are finite definitions to define them with, so most reals can only be defined collectively, not individually."
>
> Ben Bacarrisse emphasized: "They are not 'entire undefinable'. The set of them can be defined." And he added: "You can know things about the set. For example, that it can't be bijected with N."
>
> William P. Hughes: "A subcollection of a listable collection may not be listable."
>
> Alan Smaill: "After all, since matheology accepts undefinable real numbers, then why are you trying to suggest that it does not accept undefinable enumerations?"
>
> That is true. I never got a grasp of this idea: Why should undefinable definable enumerations (aka lists) be exemptet from the list of unlisted exemptions?
>
> Regards, WM
>

I am not entirely certain of your
question. But, I will give you
two answers. Here is the first.

Consider, for a moment, the origins
of the problem.

Whatever deficiencies may have existed
in Euclid, magnitudes and multitudes
been related to the notion of "part",
and, the definition of "point" had been
based on the negative criterion of
having no parts.

mathematics, suddenly Viete and Descartes
appear. Viete conflates magnitudes and
multitudes with his algebra. Descartes
conflates magnitudes and multitudes with
his analytic geometry.

a certain aesthetic for logical principles.
And, seemingly unrelated, he devised a
proof for the intermediate value theorem
specifically directed at eliminating geometric
intuition from the explanation of how
algebraically defined curves intersected.

The proof involves a fixed point argument
involving two continuous functions over
a compact set. Because of the presumed
boundary conditions, there must be some
point on the interior of the set at which
the two functions have the same value.

Now, between Bolzano and Cantor, functions
had been generalized. I believe that this
is attributed to Dirichlet. With Cantor
and Peano, it is recognized that notions
such as dimension fail to be invariant with
respect to this definition of function.
Only later, when topology becomes organized,
is it recognized that continuity is an
essential aspect to the notion of dimension
invariance.

To the chagrin of many, including yourself,
Cantor argues successfully for a metaphyical
existence of limits as numbers and for a
transfinite arithmetic. He is supported
in this by the Hilbert school. But, because
of contradictions arising from the inexactness
of natural language, the effort to rescue
set theory leads to a heavy formalization
of mathematics at the hands of logicians.

In the midst of this, predicative mathematics
arises. Under the presumption that "objects"
are primary, the formulation of set-based
mathematics becomes dominated by a typed hierarchy
in which functions do not arise from definitions
in the sense of Bolzano. To the contrary, they
correspond with their extensional definition in
some typed structure.

So, whether or not one may "speak of a function,
a relation, or a listing" depends upon whether or
not it has an extensional definition in a definite
interpretation of the language. If one cannot,
then it is undefinable (relative to a model).
What is generally true, on the basis of Cantor's
diagonal proof, is that it is not possible to have
a model in which everything is "definable simpliciter".

I started this little story with Viete and Descartes
because the abstract treatment of languages without
interpretation presents itself in universal algebra
and the problem of associating every geometric point
with a name begins with analytic geometry.

But, the strict constraints arising from predicative
mathematics is grounded in the very kinds of objections
which you, yourself, make. And, that is the very
perspective which demands that functions are logically
posterior to objects. This "defeats" Bolzano's proof
in the sense that the names of the real numbers are
prior to the two functions upon which his proof
relies.