Date: Feb 27, 2013 12:51 PM Author: fom Subject: Re: a formal construction of Dedekind cuts On 2/25/2013 7:04 AM, Charlie-Boo wrote:

> On Feb 21, 8:21 pm, fom <fomJ...@nyms.net> wrote:

>> This is a formal construction.

>>

>> As an initial context, any set theory that

>> supports the use of braces, '{' and '}',

>> to stand for representation of its collection

>> finishing predicate will suffice.

>>

> > The iterated enclosure of a symbol by braces

> > shall be referred to as the Zermelo naming

> > predicate,

>

> In other words, gobbledy-gook phrase 1 is defined to be gobbledy-gook

> phrase 2?

>

> > x -> {x} -> {{x}} -> {{{x}}} -> {}{}{}

> >

> > For each symbol 'x',

> >

> > '{x} names x'

>

> If you really want to understand the relationship between x and {x},

> show the same concept in other contexts, especially formal ones (as is

> generally the case.)

"How do you define philosophy and mathematics, and how

can you have both? If you want mathematics, you have to

have formal primitives and ways to combine them to create

something new. Vague synonyms may be ok for philosophy,

but are not mathematics, are they?"

C-B to Zuhair

-----------------

Perhaps the best apology (explanation) concerning

the nature of mathematical logic on my bookshelves

is Veblen:

"The starting point of any strictly logical

treatment of geometry (and indeed of any branch

of mathematics) must then be a set of undefined

elements and relations, and a set of unproved

propositions involving them; and from these

all other propositions (theorems) are to

be derived from the methods of formal logic.

Moreover, since we assumed the point of view

of formal (i.e., symbolic) logic, the undefined

elements are to be regarded as mere symbols

devoid of content, except as implied by the

fundamental propositions. Since it is manifestly

absurd to speak of a proposition involving these

symbols as self-evident, the unproved propositions

referred to above must be regarded as mere

assumptions. It is customary to refer to these

fundamental propositions as axioms or postulates,

but we prefer to retain the term assumption as

more expressive of their real logical character.

"We understand the term a *mathematical science*

to mean any set of propositions arranged according

to a sequence of logical deduction. From the point

of view developed above such a science is purely

abstract. If any concrete system of things may

be regarded as satisfying the fundamental assumptions,

this system is a concrete application or representation

of the abstract science. The practical importance

or triviality of such a science depends simply on

the importance or triviality of its possible

applications. These ideas will be illustrated and

further discussed in the next section, where it

will appear that an abstract treatment has many

advantages quite apart from that of logical

rigor.

"The notion of a class is fundamental in logic

and therefore in any mathematical science."

============

Keeping in mind the last remark from Veblen, here

are a few remarks from Carnap concerning the nature

of logical construction,

"By a constructional system we mean a step-by-step

ordering of objects in such a way that the objects

of each level are constructed from those of the

lower levels. Because of the transitivity of

reducibility, all objects of the constructional

system are thus indirectly constructed from

objects of the first level.

"A theory is axiomatized when all statements of

the theory are arranged in the form of a deductive

system whose basis is formed by the fundamental

concepts. So far, much more attention has been

paid to the first task, namely, the deduction of

statements from axioms, than to the methodology

of the systematic construction of concepts."

[...]

"The aim of construction theory consists in

formulating a constructional system, i.e., a

stepwise ordered system of objects (or concepts).

The stepwise ordering is a result of the fact that

the objects on each level are "constructed" from

the objects of the lower levels in a sense to be

made precised later. In the formation of such

a system, the following four main problems are

encountered. To begin with, a basis must be chosen,

a lowest level upon which all others are founded.

Secondly, we must determine the recurrent forms

through which we ascend from one level to the

next. Thirdly, we must investigate how the objects

of various types can be constructed through repeated

applications of the ascension forms. The fourth

question concerns the overall form of the system as

it results from the stratified arrangement of the

object types. We call these four problems the

problem of basis, ascension form, object form,

and system form. The problems of basis, object

form, and system form are closely connected with

one another. Their solutions are dependent upon

one another, since the construction of the objects,

and thus the form of the system depends upon the

choice of the basis, while the basis is chosen so

as to allow the construction of all object types

from it."

[...]

"If in a constructional system of any kind, we carry

out step-by-step construction of more and more

object domains by proceeding from any set of basic

objects by applying in any order the class and

relation construction, then these domains, which

are all different spheres and of which each forms

a domain of quasi-objects relative to the preceding

domain, are called constructional levels. Hence,

constructional levels are object spheres which are

brought into a stratified order within the

constructional system by constructing some of these

objects on the basis of others."

============

One thing that has amazed me on sci.math and sci.logic

are the professional participants who do not even seem

to know what constitutes the mathematics about which

they pontificate. When Kleene wrote "Introduction to

Metamathematics" in 1952, all of the basic axiomatics

had been well established. The following discussion

of the real number system is from his opening chapters:

"The theory of real numbers which is currently used

as the basis for analysis (except by critics of its

foundations) is the product of an earlier critical

movement initiated by Gauss, Cauchy, and Abel.

"This led late in the nineteenth century to the

arithmetization of analysis, so called, by Weierstrass,

Dedekind, and Cantor. Reliance on somewhat vague

geometrical intuitions was replaced by a definition

of the real numbers as certain objects constructed

out of natural numbers, integers, or rational

numbers. As Poincare said in 1900 'Today there

remain in analysis only integers or finite or

infinite systems of integers, interrelated by a

net of relations of equality or inequality'.

The definition of the real numbers from natural

numbers, integers or rationals can be given in

several ways. All lead to the same abstract structure

of the real number continuum. In other words, what

each of the definitions accomplishes is to provide

a representation of the real numbers by objects

constructed (directly or indirectly) out of

natural numbers"

[...]

"A representation which makes the ordering of the

real numbers especially perspicuous is that by Dedekind

cuts."

[...]

"The definition makes use of the presupposed system R

of rationals to construct the representatives of the

reals, without taking R into the resulting system C

as a subsystem. (If the members of R are individuals,

the members of C are sets of those individuals.)"

[...]

"The rationals among the reals form a subsystem C_R

of C which is isomorphic to the original system R

of rationals, as we verify each time we use the

representation to define a notion for the reals

which has previously been defined for the rationals."

================

I included the final two remarks because of specific

unwarranted "corrections" I have received on these

newsgroups.

================

As for other criticisms concerning a somewhat

terse presentation. The following is from "A

Theory of Sets" by Morse. To my knowledge,

these are fairly standard, but generally not

explicitly expressed.

"To discuss an object we must employ a name

for it and not the object itself. We agree

here that a name for an expression is formed

by placing the expression between single

quotation marks."

The above is preceded by,

"An expression is a linear array of symbols."

The above is preceded by,

"...c is a symbol if and only if c is a

mark which is not a quotation mark."

The above is preceded by,

"A mark is a more or less connected inscription."

So, for example, I explained a notation by

means of an unquoted use:

-----------

The iterated enclosure of a symbol by braces

shall be referred to as the Zermelo naming

predicate,

x -> {x} -> {{x}} -> {{{x}}} -> {}{}{}

-----------

Lacking quasi-quotation marks, I attempted to

indicate a general class of statements that

were to hold for any puported reference to

a class,

-----------

For each symbol 'x',

'{x} names x'

'{{x},{{x}},{{{x}}},...} describes x'

-----------

In the cases above, I could have included

an assertion such as "holds" or "is true".

But, such an assertion is superfluous by

the infinitary nature of such assertions.

The next set of expressions were not being

quoted in that they were relating the

primitive verb forms of the preceding statements

in relation to what it means for a symbol

to be representing a "finished class".

The phrase "finished class" is taken from the

history of Cantorian set theory where terms

like "set" and "class" are expressly stated

to *not* be the "extensions of concepts" that

they have come to mean because of the influence

of Russell.

In this case, it is referring to the construction

of relations one might require in a model through

the use of a description.

-----------

For each symbol 'x':

'x' stands as representative for a finished class

if and only if the description for 'x' implies that

'x' has a representable name.

if 'x' stands as representative for a finished

class, then 'x' is described by a Dedekind simply

infinite class generated through successive

iteration of the Zermelo naming predicate.

-----------

===============

No. What I did in this post was not extraordinarily

awful. It is just that no one even thinks about

what it actually might take to construct the

real numbers within a theory of classes.

There is no "description theory" in the language

of set theory by which the ordered relations

required to formulate "models" may be formed.

And, as I repeatedly point out to people, the

"theory of definability" that poses for

definition in the usual formulations of set

theory is inadequate.

That is what I conveyed in this post.

I formulated an idea of how the Dedekind cuts

might be formed if someone was actually

"working in ZFC". That is, of course, an

arguable methodology. But, the insinuations

in the responses do not address that, do

they?

The theory of real numbers is not the same

as the theory of transfinite numbers. And,

there are obvious problems with the definiteness

of infinite ordinals since forcing models

can manipulate cardinalities. Thus, the

"great questions" of set theory preclude the

transfinite sequence from being immediately

interpretable as real numbers.

Nor are the real numbers urelements, unless

there has been a complete revision of mathematics

to which I am unaware.

And, for all of the isomorphisms that may or

may not exist, omega^omega is not the real

number system either.

The identity of real numbers is obtained

by the ordering of the natural numbers retained

through each step of a formal construction.

It is not the identity relation of the set

theory axioms except in so far as the construction

does not violate the axiomatic identity relation

of the underlying set theory.

And, the objects of a given step are distinct

from the objects of the preceding step.