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fom

Posts: 1,030
Registered: 12/4/12

Re: Matheology § 214
Posted: Feb 10, 2013 3:45 PM

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On 2/10/2013 3:38 AM, WM wrote:
> Matheology § 214
>
> What?s wrong with the axiom of choice?

It is what is required for the sign of
equality to have meaningful semantics
in so far that terms of the language are
taken to have singular reference.

You have already demonstrated such confusion
over the meaning of singular reference, that
I have been forced to make jokes about it.

> Part of our aversion to using the axiom of choice stems from our view
> that it is probably not ?true?.

If you understood the basics of logic and the
debates that have informed the present situation
in the philosophy of mathematics, then you would
know that Aristotle cautioned against negating
"substance".

The relationship of the axiom of choice with
the semantics of identity is the source of its
particular set of issues.

{{In fact it is true for existing sets

Please explain "existing set".

There are material objects in the sense of objects
identified via sensuous experience. This experience
seems to be representable in terms of "boundaries".
In the case of "bodies" those boundaries are understood
in terms of deontic logic--that is, it is not permitted
to walk through walls but it is permitted to walk through
water and air. Another set of representable boundaries
is understood in the field of vision. Since these
boundaries are generally associated with color, they
have been particularly vexing for analytical philosophy.

Such deconstruction has already been done by many
authors. Mach comes immediately to mind, but I think
it was fairly common. Poincare also wrote on the
relation to sensuous experience.

The extent to which a set exists is given by Cantor's
topological insights. As paradigm objects (see Strawson)
the plurality of material objects as parts of space is
dependent on the linear separations of space by its
hyperplanes. The finiteness of material objects is
dependent on complexes of hyperplanes that isolate
regions with compact closures.

Since one cannot assume that Euclidean geometry is
intrinsic to sensuous experience, one must introduce
metric relationships by other means. Russell was
willing to take projective geometry as corresponding
to sensuous experience, and, he observed that while
projective geometry had no metric structure, it express
multiplicity in unity.

But, Russell also concluded that "points" were not
the essential objects of sensuous experience.

So, you need to explain what you mean by an existing
set since you deny the infinity needed to separate a
space, you deny the infinity needed to distinguish
a boundary criterion, and you deny the infinity needed
to even consider your use of singular terms as
meaningful.

> - but there it is not required as an axiom but is a self-evident
> truth.}}

Self-evident truth is contrary to demonstrative
science.

The ground of "self-evidence" in the jargon of logic
arises from comments in Aristotle's "Topics" and not
from his discussion in "Posterior Analytics."

There is confusion in Aristotle by virtue of his
theory of "substance" and its relation to "essence".
The ultimate success of the classical approach to
mathematics has been to segregate "substance" from
logic. By classical, I mean the interpretation of
a system of symbols as calculi. And, of course, this
is the argument for treating the transfinite number
system as mathematics without regard to beliefs about
what is or what is not "self-evident."

What remains problematic is the confusion surrounding
"undefined language primitives" when it is applied to
symbols with intended interpretations as predicates,
functions, and constants. This confusion arises
precisely because of the problem of defining
Aristotelian substance in terms of essence.

> A theorem of Cohen shows that the axiom of choice is
> independent of the other axioms of ZF, which means that neither it nor
> its negation can be proved from the other axioms, providing that these
> axioms are consistent.

A model is not a theorem.

As noted above, the issues associated with the axiom of choice
are related to the problems of what constitutes the "material"
of mathematics and the Aristotelian admonition of negating substance.

In Leibniz one finds existence correlated with possibility rather
than actuality. In Hilbert one finds existence correlated with
non-contradiction. Relative to Kant's modal interpretation of
a contradiction, these are the same. And, Frege grounded the
existence of the natural numbers on the class of objects formed
from self-contradictory expressions.

So, mathematical existence is not a simple matter.

What Frege introduced into mathematics with his analysis of
negation and his objects "the True" and "the False" was a
fragment of language with a topological characteristic. With
a proper construction, it can be seen to be a minimal Hausdorff
topology.

It is a semi-regular topology. This is the same as the topology
on the Boolean algebras that are used for constructing forcing
language.

The problems you are attaching to the axiom of choice are, in
fact, problems with the axiom of extension because that is
the axiom which governs the question of set identity.

The multiplicity of set-theoretic models arises for the same
reasons that a donut and a coffee cup (with a closed handle)
are topologically "the same".

Because of this, the only meaningful notion of truth in
a set theory based on the axiom of extension is truth
persistence under forcing.

> Thus as far as the rest of the standard axioms
> are concerned, there is no way to decide whether the axiom of choice
> is true or false.

That is not true.

There is a difference between decision and conclusion.

If you even had a clue concerning the assertions of
"the standard axioms" you would be writing different
statements about a different axiom.

> This leads us to think that we had better reject the
> axiom of choice on account of Murphy?s Law that ?if anything can go
> wrong, it will?.

Well, that must be the source of why no one
had been able to reduce Aristotle's relation
between substance and essence to first principles.

There were no Murphys in ancient Greece.

> This is really no more than a personal hunch about
> the world of sets.

Aristotle's "Topics" -- argument from common belief

Aristotle's "Posterior Analytics" -- argument from principles

That is "the basics"

> We simply don?t believe that there is a function
> that assigns to each non-empty set of real numbers one of its
> elements.

But, this difficulty comes from the philosophical questions
of how mathematics relates to sensuous experience in so far
as its ability to model real-world situations.

Look at the discussion of "sense" in Emile Borel's book
"Space and Time". As soon as you even begin to put numbers
on space, you have broken symmetry. As soon as you begin
to consider truth and falsity asymmetrically, you have
broken symmetry.

There are numerous equivalent formulations of the
axiom of choice. The most relevant is the one that
asserts that the Cartesian product of non-empty sets
is non-empty. The reason the this is the *most*
relevant is because it applies to the model theory
for the sign of equality. After that, one should look
to other equivalent formulations in mathematics and
see how it relates to those disciplines.

> While you can describe a selection function that will work
> for ?nite sets, closed sets, open sets, analytic sets, and so on,
> Cohen?s result implies that there is no hope of describing a de?nite
> choice function that will work for ?all? non-empty sets of real
> numbers, at least as long as you remain within the world of standard
> Zermelo-Fraenkel set theory.

Right. But look at the word you used -- "definite"

Where else do we see that word? Oh yes... "definite description"

How does a forcing language work? It establishes a name that cannot
be correlated with a name from the ground model. It manipulates the
namespace and the identity relation.

Does definability in standard set theory correspond with the
notion of definability accorded to definite descriptions? No.

What else is involved with forming a forcing language? Oh
yes... one takes a separative partial order and forms a
complete Boolean algebra without a bottom for the express
purpose of reversing the sense of the order relation.

You are using the exact mathematical methods to which you
object so strenuously to argue for your position against
those methods.

> And if you can?t describe such a
> function, or even prove that it exists without using some relative of
> the axiom of choice, what makes you so sure there is such a thing?

As a demonstrative science in the sense of
Aristotle, mathematics is about facticity rather
than truth. Proof is used to establish relation
between presumed facts. The facts are presumed
because argument from principles is not argument
from self-evident truth.

No certainty. But a great platform for modeling.

> Not that we believe there really are any such things as in?nite sets,
> or that the Zermelo-Fraenkel axioms for set theory are necessarily
> even consistent.

Aristotle's "Topics" -- argument from common belief

Aristotle's "Posterior Analytics" -- argument from principles

That is "the basics"

> Indeed, we?re somewhat doubtful whether large natural
> numbers (like 80^5000, or even 2^200) exist in any very real sense,
> and we?re secretly hoping that Nelson will succeed in his program for
> proving that the usual axioms of arithmetic?and hence also of set
> theory?are inconsistent. (See E. Nelson. Predicative Arithmetic.
> Princeton University Press, Princeton, 1986.)

All that would prove is that the programs of
research conducted in response to Kant had
failed. Such a result leaves Kantian views
about mathematics wholly unscathed.

Oh, wait. Frege rejected his own logicism
in the end. It would leave his views unscathed
as well.

> All the more reason,
> then, for us to stick with methods which, because of their concrete,
> combinatorial nature, are likely to survive the possible collapse of
> set theory as we know it today.

Well, there is always the many-world interpretation
of quantum mechanics...