???

5. The effects of weak versions of the axiom of choice.

In the previous parts of the paper, we left open whether the bicategory AnaCat is Cartesian closed. In this section, we show that a very weak version of the AC, one that is consistent with the negations of most of the usual special cases of the AC, is sufficient to ensure that the said conclusion holds.

This section is somewhat incomplete; since the first version of this paper was written, further, and partly better, results have been found, in a collaboration of Robert Par? and the author; it is planned that they will be described in [M/P]. On the other hand, it incorporates substantial improvements that were kindly communicated to me by the Referee.

In this section, we sometimes use classical logic throughout; the marking (CL) indicates that the result in question depends on classical logic (the principle of excluded middle).

@

For sets A , B , A

@

B abbreviates that there is a bijection A

A?@

B . I propose the following axiom of class-set theory.

Small Cardinality Selection Axiom (SCSA). There is a class-function assigning, to each set

@

A , a set

k

A

k

and a bijection

i

:A

A?@k

A

k

such that, for each set B , the class A
{

k

A

k

: A

@

B} is a set.

Of course, under the Global AC (there is a class-function that assigns to each inhabited set a member of that set), and by using classical logic (whose validity is a consequence of the AC, by the well-known argument of R. Diaconescu (see [Jo], 5.23, p. 141)), we have the SCSA in the strong form when

?

B

?

= {

k

A

k

: A

@

B} is a singleton; now,

k

A

k

is the usual cardinality of A ; the global version of choice is needed for the function A

ffl@i

.
A

As the Referee has pointed out, and as will be explained below, the SCSA is related to A. Blass? Axiom of Small Violations of Choice (SVC) (see [Bl], section 4., p.41). This axiom is as follows.

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