[DL] Semantics of Number restriction: small issue (?)

[Yevgeny Kazakov]

Dear Umberto,

On Fri, Mar 9, 2012 at 6:39 PM, Umberto Straccia
<umberto.straccia at isti.cnr.it> wrote:
> More specifically,  the standard set theoretic semantics of e.g.,
>
> (\geq n R)
>
> i.e.,
>
> (\geq n R)^I = \{ x | #\{ y \in \Delta^I | (x,y) \in R^I \} \geq n\}
>
> where we usually write that #S is the "cardinality of S" may be somewhat
> troubling (unless we use of continuum hypothesis, axioms of choice ...).

I do not see anything troubling here.
Cardinality of a set is a well-defined notion (essentially equivalence
classes w.r.t. bijections between sets). Comparison of cardinalities
is also well defined. Continuum hypothesis? Why it has anything to do
with that? As far as the axiom of choice is concerned, there are lots
of places where it is used in DLs, e.g., for proving the tree model
property of ALC. I do not see how it is relevant here.

Since DLs are expressible in FO logic, by Herbrand theorem, it is
sufficient to consider only countable interpretations anyway.

> then I suggest the equivalent set theoretic expression
>
> (\geq n R)^I = \{ x | \exists S \subset \{ y \in \Delta^I | (x,y) \in R^I \}
> such that #S = n\}

I do not see how this is principally different from the original
definition. You still have to use the cardinality function on subsets
S.

Yevgeny