Math Forum Discussions

fom

Posts: 1,033
Registered: 12/4/12

Re: a formal construction of Dedekind cuts
Posted: Feb 27, 2013 12:51 PM

Plain Text

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.