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Topic: a formal construction of Dedekind cuts
Replies: 7   Last Post: Feb 27, 2013 4:14 PM

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Charlie-Boo

Posts: 1,588
Registered: 2/27/06
Re: a formal construction of Dedekind cuts
Posted: Feb 23, 2013 6:58 AM
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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.


Is any of this formally defined? You are the only person to ever use
the phrase "supports the use of braces" on the internet other than
dentists and sports medicine doctors. http://tinyurl.com/supportstheuseofbraces

C-B

> The iterated enclosure of a symbol by braces
> shall be referred to as the Zermelo naming
> predicate,
>
> x -> {x} -> {{x}} -> {{{x}}} -> {}{}{}
>
> For each symbol 'x',
>
> '{x} names x'
>
> '{{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 closed-chain
> 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_(n-1)*x^(n-1) + ... + 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_(n-1)*x^(n-1) + ... + 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 order-isomorphic concatenation of
> digits arranged sequentially according to
>
> <a_n,a_(n-1),...,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_(n-1),...,x_1,x_0>:=<...<x_n,<x_(n-1),<...,<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_(n-1),...,a_1,a_0>
>
> one may form a lossy numeral,
>
> <b_m,b_(m-1),...,b_1,b_0>
>
> using an order-preserving concatenation of digits
> from
>
> <a_n,a_(n-1),...,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 logically-constructed
> system to be well-construed, 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_(n-1),...,a_1,a_0>
>
> and its lossy numeral counterpart,
>
> <b_m,b_(m-1),...,b_1,b_0>
>
> the ordered pair,
>
> {<a_n,a_(n-1),...,a_1,a_0>,{<a_n,a_(n-1),...,a_1,a_0>,<b_m,b_(m-1),...,b_1,b_0>}}
>
> expressible as
>
> <<a_n,a_(n-1),...,a_1,a_0>,<b_m,b_(m-1),...,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_(n-1),...,a_1,a_0>
>
> is the foundational ground for the lossy numeral,
>
> <b_m,b_(m-1),...,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_(n-1),...,a_1,a_0>
>
> b:=<b_m,b_(m-1),...,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_(n-1),...,a_1,a_0>
>
> is a lossless numeral.
>
> Call any order-preserving 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 »





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