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Notes to Infinitary Logic

1. Observe, however,
that while the formation rules for
(,)
allow the deployment of infinitely many quantifiers, each preformula
can contain only finitely many *alternations* of
quantifiers. Languages permitting infinite quantifier alternations
have been developed in the literature, but we shall not discuss them
here.

2. This remark loses its force when the base language contains predicate symbols with infinitely many argument places. However, this possibility is excluded here since our base language is a conventional first-order language.

3. I.e., such that no
contradictions can be derived from
using the deductive machinery in
** P**.

4. If *A* is a
set,
| *A* denotes the membership
relation on *A*, i.e., {<*x, y*>
*A*
× *A*: *x*
*y*}.

5. Strictly speaking,
this is only the case when
is *regular*, that is, not the limit of
< cardinals each of which is
< . In view of the fact that "most"
cardinals are regular, we shall take this as read.

6. It should be pointed
out, however, that there are languages
(,)
apart from
(,)
and (_{1},)
which are complete; for example, all languages
(^{+},)
and (,)
with inaccessible .

7. This is just a
consequence of the fact that a first-order deduction is a finite
sequence, hence a member of
*H*().

8. Take to be any logically false sentence!

9. A set
*A* is *transitive* if *x*
*y*
*A*
*x*
*A*.

10. For the
definition of admissible set, see the Supplement at the end of
§5.

John L. Bell

*First published: January 23, 2000*

*Content last modified: January 23, 2000*