```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 howcan you have both?  If you want mathematics, you have tohave formal primitives and ways to combine them to createsomething new.  Vague synonyms may be ok for philosophy,but are not mathematics, are they?"C-B to Zuhair-----------------Perhaps the best apology (explanation) concerningthe nature of mathematical logic on my bookshelvesis Veblen:"The starting point of any strictly logicaltreatment of geometry (and indeed of any branchof mathematics) must then be a set of undefinedelements and relations, and a set of unprovedpropositions involving them; and from theseall other propositions (theorems) are tobe derived from the methods of formal logic.Moreover, since we assumed the point of viewof formal (i.e., symbolic) logic, the undefinedelements are to be regarded as mere symbolsdevoid of content, except as implied by thefundamental propositions.  Since it is manifestlyabsurd to speak of a proposition involving thesesymbols as self-evident, the unproved propositionsreferred to above must be regarded as mereassumptions.  It is customary to refer to thesefundamental propositions as axioms or postulates,but we prefer to retain the term assumption asmore expressive of their real logical character."We understand the term a *mathematical science*to mean any set of propositions arranged accordingto a sequence of logical deduction.  From the pointof view developed above such a science is purelyabstract.  If any concrete system of things maybe regarded as satisfying the fundamental assumptions,this system is a concrete application or representationof the abstract science.  The practical importanceor triviality of such a science depends simply onthe importance or triviality of its possibleapplications.  These ideas will be illustrated andfurther discussed in the next section, where itwill appear that an abstract treatment has manyadvantages quite apart from that of logicalrigor."The notion of a class is fundamental in logicand therefore in any mathematical science."============Keeping in mind the last remark from Veblen, hereare a few remarks from Carnap concerning the natureof logical construction,"By a constructional system we mean a step-by-stepordering of objects in such a way that the objectsof each level are constructed from those of thelower levels.  Because of the transitivity ofreducibility, all objects of the constructionalsystem are thus indirectly constructed fromobjects of the first level."A theory is axiomatized when all statements ofthe theory are arranged in the form of a deductivesystem whose basis is formed by the fundamentalconcepts.  So far, much more attention has beenpaid to the first task, namely, the deduction ofstatements from axioms, than to the methodologyof the systematic construction of concepts."[...]"The aim of construction theory consists informulating a constructional system, i.e., astepwise ordered system of objects (or concepts).The stepwise ordering is a result of the fact thatthe objects on each level are "constructed" fromthe objects of the lower levels in a sense to bemade precised later.  In the formation of sucha system, the following four main problems areencountered.  To begin with, a basis must be chosen,a lowest level upon which all others are founded.Secondly, we must determine the recurrent formsthrough which we ascend from one level to thenext.  Thirdly, we must investigate how the objectsof various types can be constructed through repeatedapplications of the ascension forms.  The fourthquestion concerns the overall form of the system asit results from the stratified arrangement of theobject types.  We call these four problems theproblem of basis, ascension form, object form,and system form.  The problems of basis, objectform, and system form are closely connected withone another.  Their solutions are dependent uponone another, since the construction of the objects,and thus the form of the system depends upon thechoice of the basis, while the basis is chosen soas to allow the construction of all object typesfrom it."[...]"If in a constructional system of any kind, we carryout step-by-step construction of more and moreobject domains by proceeding from any set of basicobjects by applying in any order the class andrelation construction, then these domains, whichare all different spheres and of which each formsa domain of quasi-objects relative to the precedingdomain, are called constructional levels.  Hence,constructional levels are object spheres which arebrought into a stratified order within theconstructional system by constructing some of theseobjects on the basis of others."============One thing that has amazed me on sci.math and sci.logicare the professional participants who do not even seemto know what constitutes the mathematics about whichthey pontificate.  When Kleene wrote "Introduction toMetamathematics" in 1952, all of the basic axiomaticshad been well established.  The following discussionof the real number system is from his opening chapters:"The theory of real numbers which is currently usedas the basis for analysis (except by critics of itsfoundations) is the product of an earlier criticalmovement initiated by Gauss, Cauchy, and Abel."This led late in the nineteenth century to thearithmetization of analysis, so called, by Weierstrass,Dedekind, and Cantor.  Reliance on somewhat vaguegeometrical intuitions was replaced by a definitionof the real numbers as certain objects constructedout of natural numbers, integers, or rationalnumbers.  As Poincare said in 1900 'Today thereremain in analysis only integers or finite orinfinite systems of integers, interrelated by anet of relations of equality or inequality'.The definition of the real numbers from naturalnumbers, integers or rationals can be given inseveral ways.  All lead to the same abstract structureof the real number continuum.  In other words, whateach of the definitions accomplishes is to providea representation of the real numbers by objectsconstructed (directly or indirectly) out ofnatural numbers"[...]"A representation which makes the ordering of thereal numbers especially perspicuous is that by Dedekindcuts."[...]"The definition makes use of the presupposed system Rof rationals to construct the representatives of thereals, without taking R into the resulting system Cas 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_Rof C which is isomorphic to the original system Rof rationals, as we verify each time we use therepresentation to define a notion for the realswhich has previously been defined for the rationals."================I included the final two remarks because of specificunwarranted "corrections" I have received on thesenewsgroups.================As for other criticisms concerning a somewhatterse presentation.  The following is from "ATheory of Sets" by Morse.  To my knowledge,these are fairly standard, but generally notexplicitly expressed."To discuss an object we must employ a namefor it and not the object itself.  We agreehere that a name for an expression is formedby placing the expression between singlequotation 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 amark 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 bymeans of an unquoted use:-----------The iterated enclosure of a symbol by bracesshall be referred to as the Zermelo namingpredicate,x -> {x} -> {{x}} -> {{{x}}} -> {}{}{}-----------Lacking quasi-quotation marks, I attempted toindicate a general class of statements thatwere to hold for any puported reference toa class,-----------For each symbol 'x','{x} names x''{{x},{{x}},{{{x}}},...} describes x'-----------In the cases above, I could have includedan assertion such as "holds" or "is true".But, such an assertion is superfluous bythe infinitary nature of such assertions.The next set of expressions were not beingquoted in that they were relating theprimitive verb forms of the  preceding statementsin relation to what it means for a symbolto be representing a "finished class".The phrase "finished class" is taken from thehistory of Cantorian set theory where termslike "set" and "class" are expressly statedto *not* be the "extensions of concepts" thatthey have come to mean because of the influenceof Russell.In this case, it is referring to the constructionof relations one might require in a model throughthe use of a description.-----------For each symbol 'x':'x' stands as representative for a finished classif and only if the description for 'x' implies that'x' has a representable name.if 'x' stands as representative for a finishedclass, then 'x' is described by a Dedekind simplyinfinite class generated through successiveiteration of the Zermelo naming predicate.-----------===============No.  What I did in this post was not extraordinarilyawful.  It is just that no one even thinks aboutwhat it actually might take to construct thereal numbers within a theory of classes.There is no "description theory" in the languageof set theory by which the ordered relationsrequired to formulate "models" may be formed.And, as I repeatedly point out to people, the"theory of definability" that poses fordefinition in the usual formulations of settheory is inadequate.That is what I conveyed in this post.I formulated an idea of how the Dedekind cutsmight be formed if someone was actually"working in ZFC".  That is, of course, anarguable methodology.  But, the insinuationsin the responses do not address that, dothey?The theory of real numbers is not the sameas the theory of transfinite numbers.  And,there are obvious problems with the definitenessof infinite ordinals since forcing modelscan manipulate cardinalities.  Thus, the"great questions" of set theory preclude thetransfinite sequence from being immediatelyinterpretable as real numbers.Nor are the real numbers urelements, unlessthere has been a complete revision of mathematicsto which I am unaware.And, for all of the isomorphisms that may ormay not exist, omega^omega is not the realnumber system either.The identity of real numbers is obtainedby the ordering of the natural numbers retainedthrough each step of a formal construction.It is not the identity relation of the settheory axioms except in so far as the constructiondoes not violate the axiomatic identity relationof the underlying set theory.And, the objects of a given step are distinctfrom the objects of the preceding step.
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