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From: ladkin@KESTREL.ARPA (Peter Ladkin)
Newsgroups: mod.ai
Subject: Re: The Analog/Digital Distinction
Message-ID: <14133@kestrel.ARPA>
Date: Mon, 3-Nov-86 18:40:28 EST
Article-I.D.: kestrel.14133
Posted: Mon Nov  3 18:40:28 1986
Date-Received: Thu, 6-Nov-86 21:34:02 EST
References: <20@mind.UUCP>
Sender: daemon@ucbvax.BERKELEY.EDU
Organization: Kestrel Institute, Palo Alto, CA
Lines: 49
Keywords: analog, digital, continuous, discrete
Approved: ailist@sri-stripe.arpa

(weinstein quoting goodman)
> >	A scheme is syntactically dense if it provides for infinitely many
> >	characters so ordered that between each two there is a third.
(harnad)
> I'm no mathematician, but it seems to me that this is not strong
> enough for the continuity of the real number line. The rational
> numbers are "syntactically dense" according to this definition. 

Correct. There is no first-order way of defining the
real number line without introducing something like countably
infinite sequences and limits as primitives.
Moreover, if this is done in a countable language, you are 
guaranteed that there is a countable model (if the definition
isn't contradictory). Since the real line isn't countable,
the definition cannot ensure you get the REAL reals.

Weinstein wants to identify *analog* with *syntactically dense*
plus some other conditions. Harnad observes that the rationals
fit the notion of syntactic density. 
The rationals are, up to isomorphism, the only countable, dense,
linear order without endpoints. So any syntactically dense scheme
fitting this description is (isomorphic to) the rationals, 
or a subinterval of the rationals (if left-closed, right-closed, 
or both-closed at the ends).

One consequence is that one could define such an *analog* system
from a *digital* one by the following method:

Use the well-known way of defining the rationals from the
integers - rationals are pairs (a,b) of integers,
and (a,b) is *equivalent* to (c,d) iff a.d = b.c. 
The *equivalence* classes are just the rationals, and
they are semantically dense under the ordering

    (a,b) < (c,d) iff there is (f,g) such that f,g have
    the same sign and (a,b) + (f,g) = (c,d)

where (a,b) + (c,d) = (ad + bc, bd), and the + is factored
through the equivalence.

We may be committed to this kind of phenomenon, since every 
plausible suggested definition must have a countable model, 
unless we include principles about non-countable sets that 
are independent of set theory. And I conjecture that every 
suggestion with a countable model is going to be straightforwardly 
obtainable from the integers, as the above example was.

Peter Ladkin
ladkin@kestrel.arpa