Re: A Logical Model for Lists as Relations

Mikito Harakiri wrote:
> vc wrote:
> > Mikito Harakiri wrote:
> > [...]
> > > Don't you need ordered pair definition as well? I refuse to accept
> > > Kuratovski set trick as ordered pair representation in terms of sets.
> > >
> >
> > It sounds capricious. The pair does the required job, and besides
> > there are alternative definitions of the ordered pair (Quine' for
> > example).
>
> 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.

>
> 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.

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

>
> > > Alternatively, a sequence can be defined axiomatically via Kleene
> > > algebra.
> >
> > Why proliferate unnecessarily the number of primitives ?
>
> 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.

>as "Everything is object".
Received on Fri May 12 2006 - 19:52:20 CEST