A Philosopher's Dozen

Selections from the writings of Stephen Yablo

Wanted: Dumb Genie
—Excerpt from "Circularity and Paradox"

THE UNIVERSE OF SETS IS BUILT UP BIT BY BIT, recursively as it were; a set's members come in before the set itself. Kit Fine has a nice device for making this vivid. He asks us to imagine that we have a genie at our disposal. We give the genie instructions and s/he carries them out. Instructions can be simple—take the successor of this number—or iterative—keep on taking successors until further applications produce nothing new. The advice Kit favors in set theory is: take power sets at successor stages and unions at limit stages, until further application yields no new accessible cardinals.

I would like to change this advice in two ways. The first change is that I want my genie to be as dumb as possible, so that the sets are built up by repeated application of the simplest possible instructions. Power sets and unions at limit ordinals are more than my genie can handle. He is given a single, easily comprehensible instruction: whenever you have made some things, make the set of them.

The second change is this. Fine has the genie stopping when all sets of a certain infinite size have been reached. This detracts from overall simplicity because the notion of an accessible cardinal is rather sophisticated. And it adds arbitrariness because the genie could equally well have stopped elsewhere.

The one principled reason I can think of for stopping precisely here 
would be that you have now got too many objects to form a set. But I agree with Hellman (and Putnam) in finding it incomprehensible how that could happen. What could the obstacle possibly be to gathering the sets created so far into a new set? It conflicts with 'something too deeply rooted in our use of set-like operations to renounce the possibility of "going beyond" any definite totality' (Hellman, Mathematics Without Numbers, Oxford 1989, p. 74)

This suggests an instruction more like what Fine gives for the natural 
numbers: keep on going until further applications produce nothing new. The difference is that this time, further applications always do produce something new. The set of Xs is always additional to the Xs. So in the set case, the genie's work is never done. This is not because set creation is so intrinsically time consuming, but because whatever you might propose as the stopping point affords the genie materials for adding something new. The instruction more fully stated is: whenever you have made some things, form their set, continuing forever.

"Continuing forever" cannot just mean, "don't ever stop making new sets." That instruction a lazy genie could follow by making the empty set at t = 0, its singleton at t = 1, and so on through the rest of time. This would produce only (Zermelo's version of) the natural numbers 0,1,2,3,.... "Continuing forever" means, and you can consider this stipulative, "anything a faster-moving genie could make, our genie eventually does make."

The lazy genie is not continuing forever in this sense, because a faster-moving genie could collect all the natural numbers into a set, and the lazy genie never does. A genie that continues forever is interesting for this reason: it ensures the truth of "whatever the Xs may be, there is a set containing exactly them"—which is a form of the naive comprehension principle supposedly refuted by Russell's paradox. •






Soundings, Spring 2010