
Re: a formal construction of Dedekind cuts
Posted:
Feb 25, 2013 8:04 AM


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.)
In Recursion Theory: There is a recursive function wr(x) such that for any natural number M the value of wr(M) is a computer program that outputs M and then halts.
In Set Theory: For anything x there is a set { x } that contains just the value of x.
See the relationship?
These are useful axioms.
CB
> '{{x},{{x}},{{{x}}},...} describes x' > > 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. > > In order to formulate a representation for > grounded von Neumann chains using only pairs of > matched braces, the context must also admit an > axiom of pairing over finished classes and an > axiom of union across finished classes. > > For each symbol 'x' that stands as representative > for a finished class, the operation stipulated by > > x :=> u{x,{{x}}} > > shall be referred to as von Neumann succession. > > For each symbol 'x' that stands as representative for > a finished class: > > 'x is the von Neumann predecessor of u{x,{{x}}}' > > 'u{x,{{x}}} is the von Neumann successor of x' > > '{u{x,{{x}}},{u{x,{{x}}},{{u{x,{{x}}}}}},...} > is the von Neumann chain of x' > > If 'x' is admissible as standing in representation for > a finished class, then the von Neumann successor of 'x' > is admissible as standing in reprsentation for a > finished class. > > If 'x' is admissible as standing in representation > for a finished class, then the von Neumann chain of 'x' > is admissible as standing in representation for a > finished class and 'x' is said to be its ground. > > If the concatenation > > {} > > is admissible as standing in representation for a > finished class, and, if every proper initial segment > of the concatenation > > {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},...} > > is admissible as standing in representation for a > finished class, then > > {{}},{{},{{}}},{{},{{}},{{},{{}}}},...} > > is the von Frassen supervaluation chain grounded by > > {} > > The contingent finished class > > {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},...} > > shall be called the Wittgenstein enumeration. > > This construction is intended to apply for for any > set theory that admits every proper initial segment > of the Wittgenstein enumeration and all of their proper > parts as finished classes. > > To ease the comprehensibility of the presented > material, the following stipulations shall be made: > > 0:={} > 1:={{}} > 2:={{},{{}}} > 3:={{},{{}},{{},{{}}}} > > and so on,... > > t:={{},{{}},{{},{{}}},...} > > and so on,.... > > The colloquial term for the suggested sequence > of marks > > 0,1,2,3,... > > shall be used. The marks of this sequence shall be > called the whole numbers and denoted by W. > > The colloquial term for the suggested sequence > of marks > > 1,2,3,... > > shall be used. The marks of this sequence shall be > called the natural numbers and denoted by N. > > The cyclic group on 10 particular inscriptions, > > 0>1>2>3>4>5>6>7>8>9>0 > > shall constitute the letters of an alphabet to be > called digits. An alphabet is a Dedekind system. > It may be classified as a Dedekind closedchain > system. > > The digits shall be put into relation with the whole > numbers according to the algorithm of long division > terminating with remainders. Each remainder shall > be a whole number. > > Each whole number shall be put in relation with digits > according to the formal relation > > y=(a_n*x^n + a_(n1)*x^(n1) + ... + a_1*x^1 + a_0*x^0) > > where the indeterminate terms are related to the > described alphabet by the stipulations, > > x^0:=0' /\ 0':={{0,1,2,3,4,5,6,7,8,9}} > > x^1:=0'' /\ 0'':={{{0,1,2,3,4,5,6,7,8,9}}} > > x^2:=0''' /\ 0''':={{{{0,1,2,3,4,5,6,7,8,9}}}} > > and so on, as needed. > > Thus, the alphabet is presumed to be a described > finished set. > > In any set theory that admits every proper initial > segment of the Wittgenstein enumeration and all of > their proper parts as finished classes, the alphabet may > be taken as any initial segment of the Wittgenstein > enumeration. > > In the formal relation given above, > > y=(a_n*x^n + a_(n1)*x^(n1) + ... + a_1*x^1 + a_0*x^0) > > each of the coefficients a_i are taken to be a letter > of the alphabet  that is, a digit. A lossless numeral > is defined as the orderisomorphic concatenation of > digits arranged sequentially according to > > <a_n,a_(n1),...,a_1,a_0> > > where the delimiters, '<' and '>' stand for a particular use > of the axiom of pairing by which ordered pairs may be > represented as finished classes using the brace notation for > the collection finishing predicate. > > The schema > > <x,y>:={{x},{x,y}} > > stipulates an ordered pairing for any symbols 'x' and 'y' > > The schemes > > <x_1,x_0>:=<x_1,x_0> > > <x_2,x_1,x_0>:=<x_2,<x_1,x_0>> > > <x_3,x_2,x_1,x_0>:=<x_3,<x_2,<x_1,x_0>>> > > and so on,... > > <x_n,x_(n1),...,x_1,x_0>:=<...<x_n,<x_(n1),<...,<x_1,x_0> > > and so on,.... > > convey the intended extended use of the base schema to arrange > concatenations sequentially. > > Given any lossless numeral, > > <a_n,a_(n1),...,a_1,a_0> > > one may form a lossy numeral, > > <b_m,b_(m1),...,b_1,b_0> > > using an orderpreserving concatenation of digits > from > > <a_n,a_(n1),...,a_1,a_0> > > that are not syntactically equivalent with '0'. > > Marks are syntactically equivalent if they > can be placed in relation to one another > using the sign of equality under bare > quantification. For example, > > 2=2 > > expresses syntactic equivalence. Contingent to > to prior stipulations, one might consider > > 2=1+1 > > to be syntactically equivalent in some system. > But one might have a different system in which > > 0=1+1 > > expresses syntactic equivalence and '2' is in > no alphabet across which the sign of equality > expresses the relation of syntactic equivalence. > > Note, however, that syntactic equivalence is > distinct from inscriptional equivalence. The > expression > > 2=2 > > conveys inscriptional equivalence, whereas > > 2=1+1 > > does not. In order for a logicallyconstructed > system to be wellconstrued, every instance > of inscriptional equivalence must correspond > with syntactic equivalence. This is a pragmatic > presupposition governing proper use for the sign > of equality as a relation across an alphabet. > > Since the alphabet of digits has been constructed > using inscriptionally differentiated letters, a > correspondence between lossless numerals and > lossy numerals may be postulated. Given a > lossless numeral, > > <a_n,a_(n1),...,a_1,a_0> > > and its lossy numeral counterpart, > > <b_m,b_(m1),...,b_1,b_0> > > the ordered pair, > > {<a_n,a_(n1),...,a_1,a_0>,{<a_n,a_(n1),...,a_1,a_0>,<b_m,b_(m1),...,b_1,b_0>}} > > expressible as > > <<a_n,a_(n1),...,a_1,a_0>,<b_m,b_(m1),...,b_1,b_0>> > > may be formed. > > The grammar of logical construction dictates that the > use of ordered pairs represented with > > <x,y>:={{x},{x,y}} > > convey an instance of a foundational ground as being prior > to its relation with a derived counterpart. Thus, using > the correspondence between lossless and lossy numerals as > illustration, the lossless numeral, > > <a_n,a_(n1),...,a_1,a_0> > > is the foundational ground for the lossy numeral, > > <b_m,b_(m1),...,b_1,b_0>. > > To formulate this correspondence between lossless > and lossy numerals as the ground for a described > genus of numerals, it will be useful to simplify > the presentation. Let the stipulations > > a:=<a_n,a_(n1),...,a_1,a_0> > > b:=<b_m,b_(m1),...,b_1,b_0>. > > stand for representations of finished classes taken as > lossless and lossy numerals, respectively. > > Let > > <a,b>={{a},{a,b}} > > be a representation of its grounded construction. > > Then > > {<a,b>} names <a,b>. > > and > > {{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b> > > Moreover, > > u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b> > > and > > {u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} > is the von Neumann chain of <a,b> > > For any finished class, let its Russell ostension be > defined as the pair composed of its description and its > von Neumann chain. > > For each symbol 'x' that stands as representative for > a finished class: > > Let ZD(x) be its description. > > Let NC(x) be its von Neumann chain. > > Then, 'this(x) indicates {ZD(x),NC(x)}' > > and, the ordered pair > > <ZD(x),<NC(x),{ZD(x),NC(x)}>> > > is an instance of Russellian ostension. > > Given an arbitrary collection of Russellian ostensions > and any symbol 'x' that may stand as representative for > a finished class: > > 'this(x) chooses <ZD(x),<NC(x),{ZD(x),NC(x)}>>' > > is a substantiation predicate for the ordered pair, > > <x,<ZD(x),<NC(x),{ZD(x),NC(x)}>>> > > And, to say for any symbol 'x' > > "'x' has a representable name by virtue of the description > for 'x'" implies that 'this(x)' chooses well from a > finished class of finished Russellian ostensions. > > Given any lossless numeral a having a lossy counterpart > b, it is always assumed that > > 'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})' > > chooses well from a finished class of numerals. Such > a finished class is called a genus of numerals. > > In particular, the numerals whose digits constitute > the alphabet > > {0,1,2,3,4,5,6,7,8,9} > > are a genus. Each numeral is either a lossless numeral > or a lossy numeral. Thus, in relation to this > alphabet  that is, the digits of whole numbers  > there is a species of lossless numerals and a species > of lossy numerals. > > Suppose now that > > a:=<a_n,a_(n1),...,a_1,a_0> > > is a lossless numeral. > > Call any orderpreserving concatenation of digits > from the numeral a that does not include the digit a_n > a representative trailing segment of a. Say that a > representative trailing segment is full if there exists > a lossless numeral b such that the representative trailing > segment of a is ... > > read more »

