Re: limitation to induction on finite bounds

From: |-|erc (gotch_at_beauty.com)
Date: 07/09/04 

Date: Fri, 09 Jul 2004 01:18:37 GMT

"peter_douglass" <baisly@gis.net> wrote in >
> To facilitate referencing the rules which you have provided for reasoning
> about "occurence", I will give these rules names (which will appear in
> parentheses).
>
> (Occurence Introduction 1)
> sequence is a member -> sequence occurs
>
> (Occurence Extension 1)
> Given that <yy..> occurs in set S and <yy..> was
> contructed using Cantors diagonaliasion,
> we can validly conclude that set S is not demonstrably
> missing any sequence of digits. [That is to say, we can
> conclude that every sequence of digits occurs
> in S]

the brackets are out of context
Given : Cantor : <yy..> is missing -> S is missing some sequence
Conclude : S is not missing any (previously assumed to be) sequence

>
> (Occurence-Elimination 1)
> if digit-string occurs in S, S setminus digit-string = 0.
>
> (Non-Occurence Introduction 1)
> En e N, if( finite(length(d)), n<= length(d))
> Ai,
> !digitsmatchupton(i, d, n)
> -> d does not occur in S

that's just the version you asked for based on Occurance Defn:

d occurs in S
   <->
An e N , if (finite(length(d)), n<= length(d)) (i.e. regardless if its infinite or finite)
Ei,
digitsmatchupton(i, d, n)

>
> (Non-Occurence Elimination 1)
> sequence does not occur -> sequence is not a member
>
> Do you agree or disagree that rule Non-Occurence Introduction 1
> has a premiss that the length of d must be finite?

no, its within an if statement. I could push the range check down into
the function digitsmatchupton().

The definition is, FORALL N within the length of d. If d is infinite this
is equivalent to FORALL N.

>
> Do you agree or disagree that rule Non-Occurence Introduction 1
> cannot be used to introduce the non-occurence of
> an infinite sequence?

disagree

>
> You give an example
>
> HERC > try it out : S = 010203 and d = 0.3..
>
> [snip]
>
> HERC > does there exist a number where !digitsmatchupton(i, 0.3..., n)
> HERC > no, since for all lengths of the string 0.3333.. that digit sequence
> HERC > occurs. so the infinite string 0.33.. occurs in S.
>
> Do you agree or disagree that this example infers the occurence of a string,
> and does not infer it's non-occurence?

agree, the -> could work as <-> though, and I've now added the basic defn.

>
> Do you agree or disagree that Non-Occurence Introduction 1permits
> one to infer that a string does not occur, but does not permit one to
> infer (at least without the help of other rules) that a string does occur?

agree, but its what you originally asked for.

>
> Do you agree or disagree that this example does not an example
> of the rule Non-Occurence Introduction 1?
>
> HERC > If the string occurs, then *every digit* occurs in the correct
> HERC > digit position.
> HERC > Why would this be invalid for an infinite string?

disagree, this is just an explanation of digitsmatchupton. strictly its not an example usage.

>
> The argument you made is not an instance of Non-Occurence Introction 1,
> so it is not a counter-example to my claim that Non-Occurence Introduction 1
> is inadequate for inferring that an infinite sequence does not occur.
>
> Thank you for providing rule Occurence Elimination 1. Unfortunately,
> it makes use of set-minus, and I don't know the rules for using set-minus.
> Can you provide a rule similar to Occurence Elimination 1 which
> provides a conclusion for which the rules used to reason about this
> conclusion are well known? By well known, I mean available in
> text-books on the subject.
>
> If not, could you provide rules for reasoning about "set-minus"?
>

1
from Ghosty:
(or one can use |-|erc's SetMinus algorithm; A SetMinus r
= inf(dist(x,r), x in A))

2
S setminus r = d
 <->
 Ax e S, 0.3, 0.33, 0.333
 Ay e T,
 abs(x - r) = y, {0.03.., 0.003.., 0.0003..} = T

Az e T
Aa, a<=d<=z

3
A SetMinus n = min(a in A) abs(n - a)

Herc

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