Date: Feb 8, 2013 4:24 PM Author: fom Subject: Re: Matheology § 210 On 2/8/2013 5:26 AM, Alan Smaill wrote:

> fom <fomJUNK@nyms.net> writes:

>

>> On 2/7/2013 7:54 AM, WM wrote:

>>> On 7 Feb., 09:10, William Hughes <wpihug...@gmail.com> wrote:

>>>> On Feb 7, 9:00 am, WM <mueck...@rz.fh-augsburg.de> 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

matters.

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

situation.

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

figures."

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

statement:

"..., 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

figure.

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

numbers.

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

U_(n+1)*U_(n+1)*U_(n+1)cU_n

In terms of ordered pairs,

(w,x)*(x,y)*(y,z)=(w,z)

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

(z,x)*(x,y)*(y,z)=(z,z)

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

identity."

For Frege, an expression of the

form

x+2=5

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

in

3+2=5

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

argument.

In

news://news.giganews.com:119/5bidnemPpsnq13zNnZ2dnUVZ_sOdnZ2d@giganews.com

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

|Set(V_0)

|Name(V_0)=R

|v_1 EQ v_1

||Set(v_1)

||Name(v_1)=L

Or, more elaborately, to handle the context

of a universe and a null,

v_0 EQ v_0

|Name(V_0)=V()

|v_1 EQ v_1

||-(v_1 = v_0)

|||Name(v_1)=null()

|||v_2 EQ v_2

||||((-(v_2 = v_0)) /\ (-(v_2 = v_1)))

|||||Set(v_2)

||||||(null() c v_2)

||||||Set(v_1)

|||||||Name(V_2)=R

|||||||v_3 EQ v_3

||||||||(((-(v_3 = v_0)) /\ (-(v_3 = v_1))) /\ (-(v_3 = v_2)))

|||||||||Set(v_1)

||||||||||Name(V_3)=L

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

x=x

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

Frege:

"... 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

between

*A set is determined by its elements*

and

*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

counter-example.

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

given.

"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.