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Let's define an iterative function $V^F$ to indicate a iterative hierarchy building function that iterates a function $F$ starting from $\emptyset$ after a well ordering relation set $R$ whose domain is a set $S$, as follows:

$$V^F_i= \emptyset ................................................... i \in S: \not \exists s \in S (s \ R \ i)$$

$$ V^{F}_{i +1^R} = F (V^F_i).......... \forall j=i+1^R \iff j \in S \wedge \ i \ R \ j \ \wedge \not \exists k \in S (i \ R \ k \ R \ j) $$

$$V^{F}_j = \underset {i\ R \ j} {\bigcup}(V^F_i)....................................\forall j \in S: \not \exists i \in S (j=i+1^R)$$

[Note: a complete exposition of the function $V$ must include symbols $S$ and $R$ in it like as: $_{SR} V_i^F$, but this is not written here for convenience purposes.]

Now if we add to $\text{Z}$ the following axiom schema:

Iterative Conception: If $F$ is a function symbol, then:

$$\forall S \ \forall R \ [\text{$R$ is a well ordering on $S$} \to \exists x \ (x= \underset {i \in S} {\bigcup} \ _{\small SR} \large {V} ^ {\small F}_{\small i})]$$

Now wouldn't that be equivalent to $\text{ZF}$?

The rationale behind this question is that if we take the iterative conception of sets in the very 'abstract' sense, then the particulars of indexing of the stages (i.e. the domain of the iterative function) would be immaterial, the only thing needed is that it must be well ordered, accordingly we can index the iterative stages with Von Neumann ordinals or with Scott ordinals, or actually with any well ordered set $S$ after some well-ordering relation $R$ on it as defined above. Not only that! The iterative conception can be generalized as to work over any "function", it need not be limited to the "power set" operator. I think the above principle meets the 'abstract' notion of the iterative conception of sets, and to me, it appears that it interprets the whole of $\text{ZF}$.

Now I've read the following quote from Boolos, posted in Mathoverflow in one of the answers to a posting titled: Is $V$, the Universe of all Sets, a fixed object? I'll quote it again here:

There is an extension of the stage theory from which the axioms of replacement could have been derived. We could have taken as axioms all instances of a principle which may be put, 'If each set is correlated with at least one stage (no matter how), then for any set $z$ there is a stage $s$ such that for each member $w$ of $z$, $s$ is later than some stage with which $w$ is correlated'.

This bounding or cofinality principle is an attractive further thought about the interrelation of sets and stages, but it does seem to us to be a further thought, and not one that can be said to have been meant in the rough description of the iterative conception. For that there are exactly $\omega_1$ stages does not seem to be excluded by anything said in the rough description; it would seem that $V_{\omega_1}$ (see below) is a model for any statement that can (fairly) be said to have been implied by the rough description, and not all of the axioms of replacement hold in $V_{\omega_1}$. (*) Thus the axioms of replacement do not seem to us to follow from the iterative conception.

(*) Worse yet, $V_{\delta_1}$ would also seem to be such a model. ($\delta_1$ is the first uncomputable ordinal.)

This is from Boolos - The iterative conception of a set, p. 228.

So my question again, how that conforms to what Boolos is saying? Here the very abstract notion of the iterative conception of sets is taken and the stage $V_{\omega_1}$ is indeed provable to exist in this theory, so it doesn't serve as a model of this theory. Actually, I think this theory is just another reformulation of $\text{ZF}$. Accordingly what it is presented here seems to go against Boolos and against the conclusion that the axioms of replacement do not seem to follow from the iterative conception of sets?

I mean obviously one can indeed define weaker forms of iterative hierarchies, by restricting oneself to indexing by von Neumann's and taking $F$ to be the power set operator, by that one can go down even to $V_{\omega+\omega}$ being the model of it. But this doesn't seem to conform to the abstract notion of the iterative conception of sets. On the other hand I've seen attempts to use the iterative conception of sets to define theories having a universal set in them and of course Replacement won't follow for the general case of sets in these theories, like the attempt made by Thomas Forster in The iterative conception of set, but still Replacement works within the iterative hierarchy of those theories, so it is still coupled tightly with the iterative conception of sets!

Note: This is an afternote: I think that if the root intuition behind the iterative conception of sets is related to a stepwise definition of sets from prior ones in stages, then it seems that adding the condition that the definition of prior sets must not use quantification that range over sets belonging to later stages, i.e. predicativity restrictions need to be added in order to conform to the spirit of iterative conception, then Boolos's remark becomes correct! Since the stages would be subjected to similar conditions to the stages of the Constructive universe of Godel, and so $L_{\omega_1}$ would be unreachable from below, and so unreachable by iterating stages defined after predicatively defined functions from below.

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  • $\begingroup$ The text in question actually seems to appear on the 14th, not the 15th, page of the article Boolos - The iterative conception of a set, which is logical p. 228. I changed your link accordingly. $\endgroup$ – LSpice Mar 5 '18 at 22:47
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In my blog post Transfinite recursion as a fundamental principle in set theory, I prove that the principle of transfinite recursion is equivalent to the replacement axiom.

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  • $\begingroup$ fabulous, actually what you said in your blog is almost identical to the ideas of mine given here, though of course, you have the full formal proofs. In principle, I like to present matters in terms of the iterative conception of sets, and I think it is actually identical to replacement (even in absence of choice), so I think either I misunderstood Boolos, or he was simply speaking about less abstract versions of the iterative conception of sets, not sure really. Thanks for the nice answer. $\endgroup$ – Zuhair Al-Johar Mar 2 '18 at 19:07

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