Re: A Logical Model for Lists as Relations

Mikito Harakiri wrote:
> 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?

Symbols belong to the language of some theory. Interpretation of the language (FOL) utterances in modern math is usually set-theoretic.

Algebraic structures like groups describe a family of concrete sets satisfying certain axioms. As such, they can be considered in terms of some abstract set staisfying the axioms. The fact the structure (like rationals or whatever) exists is helpful for intuition and potential applications, but other than that it's unimportant, we do not care what those elements might be.

>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.

The algebraic group consists of the set and one operation with certain properties. The fact that there may be many different concrete sets is unimportant because in fact all of them are the same abstract set as seen from the group theory point of view (it's sort of obvious, no ?).

>Group
> concept is meaninful without insisting on permutaions being group
> model.

See above.

>
> > > 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?

It's a set, what else is it ? If you unite set of people and a set of pebbles, what is it ?

>
> > 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?

In the fancy language, your phrase above is called a non sequitur. Think about it.

>
> > > 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".

That's just too lame, and the remedy is the usual one: a trip to the nearest library (not Internet). Received on Fri May 12 2006 - 21:12:13 CEST