Re: Is the meaning of order intrinsic ?

>From: "Emery, Pat" <pemery@grci.com>
>
> > To put things in an order is to arrange them in a sequence.
>
>Yes of course.  A sequence is an order; and order is a sequence.  I don't
>know how to distinguish the two.  I think for our purposes we can consider
>those words synonyms.
>
> >A sequence is a
> > function.  I think all that you need for an intrinsic or non-intrinsic
> > definition of order is to be able to define a function.
>
>I don't know what you mean by "intrinsic definition".  To me something that
>is intrinsic cannot be defined, or rather does not need to be defined.  In
>model theoretic semantics they don't define Truth.

It depends what you mean by 'define'. There is no definition of the 
truth-value 'Truth', of course. But there is a (fairly elaborate) 
definition of satisfaction of a set of sentences in an 
interpretation. Being mathematics, it does not provide metaphysical 
definitions of its basic terminology; but then mathematics never does.

>I suppose Tarski tried,
>but I for one was not satisfied that he said anything.

My goodness, I guess we will have to re-think the foundations of our 
subject in the face of such a devastating critique.

>I don't see how you can use the mechanism of a function to define sequence
>either.  If I say f(a, b) =/= f(b, a) then I suppose I have defined that
>(a,b) is an ordered pair in relationship to the function f.  But if sequence
>was not already prior to the mechanism of our system the expression f(a, b)
>would not even be distinguishable from the expression f(b,a) .. so such a
>definition chases its own tail ... doesn't it?

Actually, sequence can be defined in terms of functional application, 
or of set membership, using a well-known trick , the common idea of 
using trees to encode sequences. You need the notion of a pair and 
the ability to *distinguish* the two things that are paired, but that 
distinguishing need not be described in terms of an order. We 
conventionally use order in the syntax to indicate the two parts of a 
pair, eg by writing (A . B) to indicate the pair consisting of A and 
B, but that is really just a lexicographic accident arising from the 
linear notation which we use in writing text. Abstractly, a pair is 
just a structure with two distinguishable substructures. Then a 
sequence can be defined by iterating pair construction in the usual 
way, which would look like this using dotted-pair notation:
(A1 . (A2 . (A3 . (..... (An .  END)...))) The same kind of thing can 
be done with functions, or sets, or whatever. Eg using functions, one 
encodes an n-ary function - whichis interchangeable with an n-ary 
sequence at this level of abstraction -  as a single-argument 
function whose value is an (n-1)-argument function (a trick invented 
by a logician called Curry, and hence known as currying.)

Pat

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Received on Thursday, 24 May 2001 17:26:33 UTC