Re: A Logical Model for Lists as Relations

vc wrote:
> Mikito Harakiri wrote:
> > Kuratowski and Quine' constructions are not definitions. The ordered
> > pair definition is
> >
> > (x,y) = (a,b) if and only if x=a and y=b.
>
> You are confused. That's not a definition but rather a property an
> ordered pair should satisfy. True, there are alternative definitions
> of the ordered pair, but what's important is that the condition should
> hold. Besides, when you write (a,b) = (c,d), what exactly do you
> mean ? What *are* (a,b) and (c,d) ? The Kuratowsky (or anybody else's
> pair) tells you what it is in terms of sets.

Don't we introduce new symbols together with properties they must satisfy? When we are saying that group is a set of elements which satisfy some axioms, do we have to tell what those elements "are" in terms of set? E.g."Group is a set of permutations. Each permutation is an ordered tuple. An ordered tuple is a set ..." This is just unnecessary.

> > Any set construction that satisfies this property would do but, really,
> > this a pointless exercise just for the sake of representing round
> > brackets via curly ones.
>
> See above. The pair is actually a 'model' satisfying the pair equality
> axiom. Without a set construction, the expression is a meaningless
> string of characters.

Likewise group has many models and permutations is one of them. Group concept is meaninful without insisting on permutaions being group model.

> > OK, if an ordered pair is a set, then perhaps union and intersection of
> > ordered pairs make sense. No? An ordered pair is not a set (although it
> > can be considered as a set element, of course).
>
> Why not? it is a set all right, what else is it ?

Ok, lets see:

(a,b) = {{a}, {a,b}}
(b,c) = {{b}, {c,b}}

(a,b) \/ (b,c) = {{a}, {b}, {a,b}, {c,b}}

What is {{a}, {b}, {a,b}, {c,b}} may I ask?

> Does it also bother you that a von Neumann numeral is a set ?

Are you implying that people learn arithmetics via manipulations with Neumann numerals in elementary school?

> > That was not the point. Reduction to the sets doesn't buy us anything
> > (at least in case of ordered pair). A proposition "Everything is a set"
> > is just as silly
>
> The entire body (almost) of math stands on this "silly" foundation.
> If you have an alternative FOM suggestion, I doubt anyone will take
> you seriously if the only new idea you can offer is the ordered pair as
> a primitive notion.

Mathematics developed just fine long before "Principia Mathematica". Received on Fri May 12 2006 - 20:31:03 CEST