Re: limitation to induction on finite bounds

From: peter_douglass (baisly_at_gis.net)
Date: 07/08/04 

Date: 8 Jul 2004 07:55:48 -0700

responding to "|-|erc" <gotch@beauty.com>
   message news:6s3Hc.83895$sj4.70895@news-server.bigpond.net.au...

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]

(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

(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?

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

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?

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?

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?

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"?

--PeterD

Relevant Pages

  • Re: limitation to induction on finite bounds
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  • RE: Complications with formulas
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