The Benacerraf Problem

Benacerraf deems inappropriate, properties that set-theoretic objects intended to serve a specific mathematical function possess that are not relevant to that function. Thus in von Neumann’s explication:

we get the “inappropriate” 1 e 3. My tendency when faced with a philosophical problem regarding mathematics is to look to mathematical history and practice for help. Beginning in the 19th century, mathematicians were faced with a number of important equivalence relations and the need to see the equivalence as a kind of equality. Perhaps the first was Gauss’s use of the congruence relation, where a = b mod r is defined to mean that the natural number r, called the modulus, is a divisor of the integer b — a. The equivalence classes form a ring, and in the case that r is prime, a field. It has become customary to designate each such residue class by its least non-negative member. So for example, with the modulus 5, we have a finite field whose elements one writes as {0, 1, 2, 3, 4}. And we write such equations as 3 + 4 = 2 and 2 ? 3 = 1. In effect the equivalence classes are each represented by one of its members. But the Benacerraf “problem” applies here. The property 2 < 4 is “inappropriate” in just the same way as in the example above. It wouldn’t occur to a mathematician to be concerned with the question of whether these irrelevant properties show that e.g., the number 2 is not truly the class of numbers congruent to 2 modulo 5. Is there really a philosophical puzzle here?

The Frege-Russell attempt to define the cardinal numbers as, in effect, the equivalence classes corresponding to the relation between a pair of sets of the existence of a one-one correspondence between them. Of course the attempt failed because the classes were too large. What von Neumann did was to choose a member of each equivalence class to designate the class according to the elegant recursion:

so that the number n is designated by a set that does have exactly n members. This should trouble only philosophers who truly seek to know what a number “really” is.

 
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