We have reached the last few pages of §3.6; we are still considering whether the iterative conception delivers Replacement. Luca has critically considered two proposed routes from the one to the other; he now turns to discuss a third, an Argument from Reflection.
Naively put, reflection principles tell us that, for any property — or at least, any kosher property of the right kind! — belonging to the universe of all sets, we can find a level of the hierarchy which already has that property. In other words, kosher properties of the whole universe are “reflected” down to a set-sized sub-universe. Three interrelated questions arise:
- Does the iterative conception sanction any form of reflection principle like this, and if so why?
- How are we to spell out, more formally, some acceptable form(s) of reflection principle? What are the kosher properties that can get reflected down?
- Which formal reflection principles entail Replacement?
On (1), Luca outlines a supposed link between (i) the idea that iteration goes on ‘as far as possible’ with the idea (ii) that the hierarchy should be ‘absolutely infinite’ in the sense that it resists unique characterization by any non-trivial property. And from (ii) it is supposed to follow that any property we can correctly assign to the universe must fail to pin down the full universe, so (iii) that property will already exemplified by some initial part of the hierarchy.
But I don’t really get the supposed link between (i) and (ii). At the end of his discussion, Luca claims that “absolute infinity is a natural way of understanding the idea that the iteration process is to be carried out as far as possible.” But I could have done with rather more explanation of why it should seem natural. And indeed I could have done with a more about the step from (ii) to (iii).
But ok, let’s grant that for some class of kosher properties, there is an intuitively appealing chain of reasoning that leads naturally enough from the iterative conception to reflection for those properties.
We now need to go formal: how do we formally capture the relevant reflection principles? The discussion of (2) and (3) inevitably becomes more technical. I won’t attempt to summarize here Luca’s already rather compressed exploration (which also touches on closure principles as a weaker alternative to reflection principles). In fact, I suspect many readers of the book will find this episode pretty tough going. Those beginners who needed e.g. to be reminded about cardinals and ordinals at the end of Chapter 1 are surely not going to easily follow this discussion which turns, inter alia, on distinguishing which order of higher-order variables are allowed to occur parametrically in formal versions of reflection principles. But yes, a strong enough formal reflection principle will entail Replacement. Still, I suppose someone might well pause to wonder whether the required open expressions with higher order parameters express kosher properties of the kind that were being countenanced in the intuitive considrations at stage (1).
Overall, I found Luca’s discussion of what he calls the Argument for Reflection intriguing (it got me re-reading some of the literature he mentions), but inconclusive. But then he too in the end is cautious. He says that “if the idea of iteration as far as possible is perhaps not part of the iterative conception, it harmonizes well with it. Understood or augmented in this way, the iterative conception sanctions the Axiom of Ordinals and possibly the Replacement Schema”. [my emphasis]
Which gets us to the end of Chapter 3, responding to some initial ‘Challenges to the Iterative Conception’. So let’s pause here. The rest of the book looks at various rivals to the iterative conception, starting in Chapter 4 with the naïve conception again — can we, after all, rescue it from its apparently damning inconsistency by departing from classical logic? I’ll discuss the whole chapter in one bite in the next posting!