Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Charlie-Boo Posts: 1,635 Registered: 2/27/06
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, 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.)

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.

C-B

> '{{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 ...
>

Date Subject Author
2/21/13 fom
2/22/13 Shmuel (Seymour J.) Metz
2/23/13 Charlie-Boo
2/23/13 Charlie-Boo
2/23/13 Charlie-Boo
2/25/13 Charlie-Boo
2/27/13 fom
2/27/13 Shmuel (Seymour J.) Metz