Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.


fom
Posts:
1,969
Registered:
12/4/12


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


On 2/25/2013 7:04 AM, CharlieBoo 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, gobbledygook phrase 1 is defined to be gobbledygook > 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?"
CB 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 selfevident, 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 stepbystep 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 stepbystep 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 quasiobjects 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 quasiquotation 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.



