# Re: Columns without names

From: JOG <jog_at_cs.nott.ac.uk>
Date: 21 Sep 2006 03:12:26 -0700

vc wrote:
> JOG wrote:
> > While I have your attention perhaps you might also clarify a
> > distinction that I previously had:
> >
> > I was under the impression that - given that the extension of a
> > predicate is the set of true propositions that can be formed by
> > substituting a term for each of its free variables
>
> In the mathematical context, a predicate extension is a collection of
> things in some universe for which the predicate holds. In other words,
> a predicate can be interpreted as a mathematical relation in some
> domain of interpretation, or one can say that a predicate defines a
> relation in some domain. The '<' predicate in the {1,2,3} domain
> defines the {(1,2), (1,3), (2,3)} relation which is the predicate
> extension.
>
> > - a predicate /always/ has an extension.
>
> It depend on your favorite set theory. In some, R = {x | not( x in x)}
> does not exist, in others it does.

Well, assuming that a set does exist (non-collectivizing predicates aside) I was asking if a predicate /always/ has an enumarable extension. I think you are telling me no, and in hindsight that's probably entirely obvious.

Either way I always viewed "not(x in x)" as the predicate, and the whole definition " x | not(x in x)" as the intension. Similarly are you saying that for a definition of rational numbers { m/n | is_integer(m) & is_integer(n) & non_zero(n) } the whole statement is the predicate, as opposed to just the part after the | . I appreciate the time spent answering
these questions.

>
> >In contrast a set defined such as { m/n :
> > m,n E Z & n!=0 } (rational numbers) is infinite and cannot be
> > enumerated.
> >
>
> This, I do not understand.
>
> > Is this distinction incorrect?

Just to offer you a quote from the Potter book:

"The idea of an extension of a property is standardly explained by examples
such as the properties of 'having a heart' and 'having a kidney'. Logic textbooks assure us that although they are different properties they apply
to the same objects (apart from pathological cases). Logicians express this
by saying that the two predicates have different intensions but the same
extension."

In light of vc's comments I am interested now if anyone else here who is mathematically inclined finds the term 'intension of a set' to be obscure. Jan? Received on Thu Sep 21 2006 - 12:12:26 CEST

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