In a couple of very well known papers, George Boolos argued that “the axioms of replacement do not follow from the iterative conception”. Was he right? Or can Replacement be justified on (some core version of) the iterative conception? This is the topic of the particularly interesting §3.6 (pp. 90–100) of Conceptions of Set, ‘The Status of Replacement’.
Luca discusses three lines of argument to be found in the literature for the thought that the iterative conception does warrant Replacement. I’ll comment on two in this post.
The first he calls Gödel’s Argument. Two quotes from Gödel: (i) “From the very idea of the iterative concept of set it follows that if an ordinal number α has been obtained, the operation of power set P iterated α times leads to a set Pα(∅)” And then (ii) “the next step will be to require that any operation producing sets out of sets can be iterated up to any ordinal number.”
In response, Luca makes the following central points:
- Tait and Koellner have argued that elaborating Gödel’s claim (i) requires appeal to Choice. But not so. For we can work with the Scott-Tarski definition of an ordinal, and then, without needing an assumption of Choice, Gödel’s thought will at least warrant adding to Z+ the Axiom of Ordinals — the axiom that there is a level Vα for every ordinal α.
- This theory with the Axiom of Ordinals is rich, much more powerful than Z+, and in fact buys us the nice results that Boolos claimed for Replacement. However, the Axiom of Ordinals is weaker than full Replacement.
- A version of Gödel’s second claim (ii) is needed to get us from the iterative conception to full Replacement, and it isn’t clear why (ii) should be thought of as part of the iterative conception.
On (1), accepting Gödel’s (i), Luca’s discussion seems spot on. On (2) quite a few readers (those familiar with ZFC but who haven’t read Potter’s book) might well have welcomed rather more at this point on the Axiom of Ordinals, on its virtues and mathematical consequences. They might reasonably ask: if the Axiom of Ordinals in fact gives us (a good deal of) what we want, just why — other than conservative adherence to tradition! — should we buy full-power Replacement? On (3), we could I suppose go back to pause over (i) to wonder if the idea of the universe of sets being layered by iterating the set of operation has to go along with the idea that those iterations should be ordinal-many (for any ordinal we can obtain). But leave that aside: Luca it is surely right that it is one thing to build into the iterative conception the idea that the core set of operation should in some sense be iterated ‘as far as possible’, it is another thing to require that other operations be iterated as far as possible too.
The second line of argument for Replacement to be discussed is what Luca calls The Argument from Co-finality. Thinking in terms of stages of the hierarchy, Shoenfield (in his famous Handbook article) suggests that for any collection of stages S, there will be one after it “provided that we can imagine a situation in which all of the stages in S have been completed”. But then assume that we have a set x and for every y in x there is a stage Sy correlated somehow or other with y. Then
Suppose … we take each y in x … and complete the stage Sy. When we reach the stage at which x is formed, we will have formed each y in x and hence completed each stage Sy.
So if S is the collection of stages Sy, we can imagine a situation in which all of the stages in S have been completed and there will be a stage after it.
Now, if we buy this cofinality principle, then Replacement is immediate (since Replacement tells us that the image of x under a function f will itself be a set; for each y in x take Sy to be the stage at which f(y) is formed, and then by Shoenfield’s principle there will be a stage after all those at which we can gather together all the f(y) into a set …).
What are we to make of all this? Luca surely hits the nail on the head! As he neatly notes, Shoenfield’s argument
seems to assume that if a process can be completed, and we replace each stage of the process with a process that can be completed, then the maxi-process consisting of all these processes can itself be completed. But this is just the Axiom of Replacement reformulated in terms of stages and processes.
The supposed defence of Replacement is therefore too close to being circular.
So we haven’t got here an independent argument from the iterative conception to Replacement. Luca concludes, however, on a (surprisingly?) sympathetic note: “the cofinality principle certainly seems to harmonize well with the iterative conception, and can perhaps be seen as one way of spelling out the idea that the cumulative process through which the hierarchy is obtained should be iterated as far as possible.” But equally couldn’t we spin it the other way? — the iterative conception itself doesn’t take us as far as Replacement, and it takes a further independent thought to justify that principle. Which leaves us with the question of what rival further thoughts (equally harmonizing with the basic iterative conception) might be on the cards.
To be continued on a third line of argument for Replacement, via reflection principles.