Date: Feb 8, 2013 4:24 PM
Author: fom
Subject: Re: Matheology § 210

On 2/8/2013 5:26 AM, Alan Smaill wrote:
> fom <> writes:

>> On 2/7/2013 7:54 AM, WM wrote:
>>> On 7 Feb., 09:10, William Hughes <> wrote:
>>>> On Feb 7, 9:00 am, WM <> wrote:
>>>> <snip>

>>>>> What does that mean for the set of accessible numbers?
>>>> That this potentially infinite set is not listable.

>>> Here we stand firm on the grounds of set theory.
>>> Once upon a time there used to be a logocal identity: The expression
>>> "Set X is countable" used to be equivalent to "Set X can be listed".

>> Incorrect.
>> Cantor understood that for a collection to be a
>> set, there was an underlying canonical well-ordered
>> form.

> What about the real numbers?

I am happy to go there. But, you will probably
not like my remarks.

The issue of well-order is model-theoretic. In
fact, the relevant interpretation of the axiom
of choice is that the Cartesian product of a
non-empty set is non-empty. The sign of identity
has no semantic import if this does not hold.

For Cantor, the presupposition lies with definitions
such as

"By an aggregate [...] of definite and separate
objects [...]"

The "definite and separate" part of this definition
is lost to modern set theory because of Russell.

It exists in Zermelo's 1908 paper where identity
of domain objects is given relative to the use
of the sign of equality between denotations. And,
if one reads that paper carefully, the "definite
identity" between objects is not that given by
the axiom of extension alone. Rather, Zermelo
specifically asserts the existence of singletons
and explains that the relationship of singletons
to denotations makes the sign of equality definite.

"According to axiom I, the elmentary sets
{a} and {a,b} are always uniquely determined
and there is only a single null set. The
question whether a=b or not is always
definite, since it is equivalent to the
question whether or not ae{b}."

Modern set theory has buried these matters
and speaks of its terms as objects. This, in
part, comes from Tarski whose correspondence theory
of truth speaks of a satisfaction map taking
symbols to objects without accounting for
how those objects have been named. But, the
source of confusion begins with Russell's logicism
and Russell's theory of description.

Also, Wittgenstein's rejection of the principle
of identity of indiscernibles further complicates

To continue this explanation, I will have to
switch back and forth through several disciplines.
I hope it does not get too confusing.

The real numbers are a particularly convoluted

Note that Zermelo's use of a singleton is standard
practice in textbooks on topology. For simplicity
of presentation, '{x}' is simply referred to as 'x'.
Almost no thinks about the fact that real analysis
is done with respect to the topological relations
of the number system. Implicit to real analysis
is the relation between Cantorian fundamental
sequences, Cantor's intersection theorem, and
Leibniz' principle of identity of indiscernibles.

Provided that my translations are not
being purposely deceptive or wholly
incompetent, what Leibniz actually wrote
concerning the principle of identity of
indiscernibles is given by:

"What St. Thomas affirms on this
point about angels or intelligences
('that here every individual is a
lowest species') is true of all
substances, provided one takes the
specific difference in the way that
geometers take it with regard to their

This is a far cry from the modern statement
of the principle in terms of quantification
over grammatical properties.

One does not see a correction in philosophical
contexts until Strawson challenges Russellian
description theory in "On Referring." And,
in Strawson book "Individuals" one finds the

"..., we can clearly distinguish
between the cases where we mean
to speak of qualitative identity
and the cases where we mean to
speak of numerical identity."

followed by an illustration including four
congruent squares among a collection of
other figures and two statements using
the phrase "the same." Naturally, the
expression conveying numerical identity
involved the geometric relations of the

In real analysis, and more generally in
metric spaces, this is expressed by Cantor's
intersection theorem. The theorem actually
takes two forms, which is convenient for
helping to delineate the related logical
issues. Relative to individuation, the
set diameter must vanish and I take this
use of distances to reflect what Leibniz
refers to as "specific difference in the
way that geometers take it"

Let X be a complete metric space, and
let {F_n} be a decreasing sequence of
non-empty closed subsets of X such that
d(F_n)->0. Then, taking F to be the
intersection over {F_n}, F contains exactly
one element.

Without a vanishing diameter, one simply
has a non-void intersection.

One of the historical issues in logic
is the indefinite "a" as distinguished
from the definite "the". To the extent
that Leibniz' principle of the identity of
indiscernibles is couched in the part-whole
relation of syllogistic classes, Cantor's
intersection theorem not only expresses
the "an individual is the lowest species",
but also delineates the indefinite reference
from the definite reference.

It is, however, not possible to see
these issues if one adheres to the belief
that logic is prior to mathematics.

Next, one needs to observe that Cantor's
intersection theorem occurs at a higher
logical type than the Cantorian fundamental
sequence. The fundamental sequences
are the real numbers by definition.

Once again, this is lost to modern set
theory. It is typical for set theorist
to use omega^omega when discussing the
real numbers. There is an isomorphism
of this set with the real numbers, but
that isomorphism is through the Baire
space via collections of rationals
formed relative to continued fractions.

The use of identity in set theory is
logical identity. It is not the identity
of a metric space. So, to the extent
that Cantor used the metric properties
of the rational numbers to define the
real numbers, within set theory the
identity of such collections have their
singular nature with respect to a
different criterion.

I am beginning to realize that most
people have never considered how the
real numbers are obtained from the natural
numbers. The natural numbers have an order.
At each level of the construction, the order
relation is inherited from the underlying
set. So, in the construction of the integers,
an integer is an infinite collection of
pairs whose order is inherited from the
natural numbers from which the pairs had
been formed. It is likewise with the
rationals. Thus, the identity of the
real numbers in relation to the logical
hierarchy of definition is obtained from
from the order relation,

x=y <-> (x<=y /\ x>=y)

which, ultimately, is inherited from
the order relation of the natural

In set theory, however, identity is currently
taken to be based on the principles of
logicism coming from the tradition of
Russell, Carnap, Quine, etc.

So, the next problem is to understand how
one attaches a metric structure to a
collection whose individuation is no
longer based on the principle of identity
of indiscernibles except insofar as it
is implemented grammatically with quantifiers
and the biconditional.

The answer to this question lies with
topological uniform spaces. Uniform
spaces generalize the investigation of
uniform topological properties of the
real numbers. But, their definition
does not require any notion of number.
Rather, they are defined relative to a
system of relations called a uniformity.
These relations are all binary in the
sense that they are subsets of the
Cartesian product of an underlying
set, and thus precisely what is required
for dealing with identity as a relation.

For a metric space, the relevant
axiom is

x=y <-> d(x,y)=0

The axiom we need is

x=y -> d(x,y)=0

and this weakening of the metric
space axioms corresponds to what is
called a pseudometric.

Fortunately, the metrization of uniform
spaces is done with respect to
pseudometrics. You can find a proof of
the metrization lemma in "General Topology"
by Kelley. But, Kelley's proof requires
modifications for foundational purposes.
Naturally, there are uses of the real
numbers within the proof that must be
revisited in terms of rational numbers.
What is most interesting, is that the
proof establishes a foundational
relationship between Dedekind cuts and
the metrization of a logical use of
identity. Use of the least upper
bound property is an essential part of
the proof.

For this discussion, however, there is
a different aspect of the proof toward
which attention must be directed. One
of the assumptions of the proof is the
containment of a product of relations in
the prior member of a countably infinite
sequence of relations


In terms of ordered pairs,


Now, there are constraints on these
relations -- in particular, they must
all contain the (model-theoretic)
diagonal. And, the strategy of the
proof is to introduce an auxilliary
function that will reflect the
closeness of points retained in a
sequence of set differences.

Oddly, the end result of this process
is introduced into logic by Tarski in
1971 when he turns his attention to
algebraic logic and the relation of
first-order logic to geometric interpretations
of sequences of variables as sequences of
coordinates. Among his axioms for
cylindrical algebras, one finds

AxAy(x=y <-> Ez(x=z /\ z=y))

which, if translated into a statement
in terms of ordered pairs using symmetry
to exchange symbols yields


So, presumably, the exposition is
at a point where the numbers can be defined
within set theory and be related to one
another using a pseudo-metric on the basis
of relations whose underlying identity
criterion is not based on a metric.

This has not yet met the challenge of
your question.

There is a profound issue with definability
in modern set theory. Begin by considering
a different challenge to Russellian
description theory more closely related to
model theory. Among his detractors
had been Abraham Robinson. It is in
his criticisms that one returns to an
understanding that the completion of
an incomplete symbol in the sense of
Frege requires names. Robinson writes:

"We still have to clarify the role of
identity. One correct definition of
the identity from the point of view
of first-order model theory is undoubtedly
to conceive of it as the set of diagonal
elements of MxM, i.e., as the set of
ordered pairs from M whose first and
second pairs coincide. The symbol "="
then denotes this relation and it is
correct that (M |= a=b) if "a" and "b"
are constants which denote the same
individual in M, or, more generally,
that (M |= s=t) if "s" and "t" are terms
which denote the same individual in
M. But, the identity may also be
*introduced* by this condition so that
(M |= s=t), *by definition* if "s"
and "t" denote the same individual
under the correspondence C, which is
again assumed implicitly, and this
seems more apposite in connection
with the discussion of sentences which
involve both descriptions and

For Frege, an expression of the


is incomplete in the sense that
it cannot be assigned a truth
value. Replacing the variable
with the name of a number
completes the expression as


which has a truth value relative
to the usual interpretation of
those symbols.

Since Robinson had to explain how
his version of handling descriptions
would force model-theory to reconsider
its treatment of the identity relation,
it is clear that there is an entire
complex of issues intertwined here.

Authors like Quine actually argued
for the use of descriptions to eliminate
names from formal discourse. And,
Russell, viewing naming as an extra-logical
process wrote "Principia Mathematica"
in the full generality afforded to him
by his interpretation of descriptions.
So, the idea of names as simple abbreviation
is ubiquitous. In turn, this comes
from a theory of identity based solely
on subsititutivity.

The particular problem for mathematics
is that its "objects" are abstract. The
only sense by which its objects are
named is through description theory.
The well-ordering of the reals in any
sense that might constitute a proof
can never be done. Descartes put the
numbers onto the coordinate axes
arbitrarily and the only way that
logic can make that logical is to
understand Aristotle's epistemic
argument for distinguishing a
demonstrative science from rhetorical



I discuss this with some context. But,
viewed in the context of demonstration,
naming would proceed along the lines of
something like

v_0 EQ v_0
|v_1 EQ v_1

Or, more elaborately, to handle the context
of a universe and a null,

v_0 EQ v_0
|v_1 EQ v_1
||-(v_1 = v_0)
|||v_2 EQ v_2
||||((-(v_2 = v_0)) /\ (-(v_2 = v_1)))
||||||(null() c v_2)
|||||||v_3 EQ v_3
||||||||(((-(v_3 = v_0)) /\ (-(v_3 = v_1))) /\ (-(v_3 = v_2)))

In other words, well ordering is not
a mathematical voodoo. It is imposed
by the semiotic constraint of naming
individuals as individuals. The
latter part of that statement requires
that the axiom


have a semantical interpretation conveying
that the symbol 'x' will have a uniform
interpretation throughout the discourse.

Cantor understood that the kind of
set theory we currently use to ground
our model theory is inappropriate.
This is clear in his criticism of

"... to take 'the extension of a
concept" as the foundation of the
number concept. He overlooks the
fact that in general the 'extension
of a concept' is something quantitatively
completely undetermined. Only in
certain cases is the 'extension
of a concept' quantitatively determined,
then it certainly has, if it is
finite, a definite natural number,
and if infinite, a definite power.
For such quantitative determination
of the 'extension of a concept'
the concepts 'number' and 'power'
must previously be already given
from somewhere else, and it is a
reversal of proper order when one
undertakes to base the latter concepts
on the concept 'extension of a concept.'"

To see that modern set theory has
gone astray by eliminating the denotations
of Zermelo's 1908 paper, one need only
look to Frege:

"But, although the relation of
equality can only be thought for
holding of objects, there is an
analogous relation for concepts.
[...] We say that an object A is
equal to an object B (in the sense
of completely coinciding with it)
if A falls under every concept under
which B falls, and conversely. We
obtain something corresponding
to this if we switch the roles of
concept and object. We could then
say that the relation we had in
mind above holds between the concept
PHI and the concept XI, if every
object that falls under PHI also
falls under XI, and conversely."

In other words, the axiom of extension
is not object identity.

To address the very next thing you must
be thinking, there is a difference

*A set is determined by its elements*


*A set is a collection taken as an object*

In Aristotle, this is addressed. But, the
minimalism of foundational investigations
has obliterated Aristotle's distinctions.

"A distinctive property is one that
does not reveal what the subject is,
though it belongs only to that subject
and is reciprocally predicated of it."

"A definition is an account that
signifies the essence."

The historical problem for Aristotle is
that "essence" and "substance" are
problematic in relation to "foundation."
By couching any collection of definite
descriptions into the sequenced order of
a derivational schema, the first description
for some object constitutes its essential
definition. This also preserves the
historical relationship between identity
and definition, wherein a definition (and,
thus, the existence or uniqueness assertion of
its description) can be destroyed by

But, first one must have an identity
relation and a notion of definability
that is appropriate to the task.

In case you have a problem with Cantor
distinguishing between "number" and "power"
in his criticism of Frege, recall that
Cantor's definition of a cardinal
number is based upon a theory of
units rejected by Frege:

"We will call by the name 'power'
or 'cardinal number' of M the general
concept which, by means of our active
faculty of thought, arises from the
aggregate M when we make abstraction
of the nature of its various elements m
and of the order in which they are

"We denote the result of this double act
of abstraction, the cardinal number or
power of M, by

|M| (Cantor used a double-bar above)

Since every single m, if we abstract from
its nature, becomes a 'unit,' the cardinal
number is a definite aggregate composed of
units, and this number has in our mind
as an intellectual image or projection
of the given aggregate M."

But, with all deference to Frege, he
retracted his logicism,

"The more I have thought the matter
over, the more convinced I have become
that arithmetic and geometry have developed
on the same basis--a geometrical one in
fact--so that mathematics in its entirety
is really geometry."

And, in modern mathematics, the lattice
of equivalence relations on a domain is a geometric
lattice. The common name for them is a matroid,
and, one of their principal motivating investigations
had been the notion of dependent and independent sets
of vectors abstracted from their numerical definition.

Cantor did not have the results of
the last century to definitively discern
how to view the problem of well-ordering the
reals. That we still talk about it as if it
is something that can be proved in a metaphysical
sense merely reflects how universities allow
their faculties to "preach to the choirs" instead
of participating in interdisciplinary research.