# What is the status of the Axiom of limitation of size? (adrift for almost a century now)

On reviewing the wiki article

Axiom of limitation of size

I come away with the impression that there are issues surrounding this 'maybe too powerful' principle/heuristic/doctrine/axiom that haven't been completely settled. Yes, the axiom has been 'digested' into Von Neumann–Bernays–Gödel (NBG) set theory, where Von Neumann gets the first letter 'N', but it looks like a compromise (see next section).

Have logicians given up on making Von Newumann's idea (not necessarily the specific technical formulation) a centerpiece for reasoning about infinity?

I am not a logician, but one 'goal' would be to combine the following concepts,

Powerset Operator
Infinity
Limitation of size

into a cohesive 'amalgam' and part of a logical system for reasoning.

Extracts from the wiki article

The first sentence:

In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes.

So we are nearing a century since its introduction.

The last segment of the article:

Gödel found von Neumann's axiom to be "of great interest":

$$\text{ }$$"In particular I believe that his [von Neumann's] necessary and sufficient condition which a property must satisfy, in order to define a set, is of great interest, because it clarifies the relationship of axiomatic set theory to the paradoxes. That this condition really gets at the essence of things is seen from the fact that it implies the axiom of choice, which formerly stood quite apart from other existential principles. The inferences, bordering on the paradoxes, which are made possible by this way of looking at things, seem to me, not only very elegant, but also very interesting from the logical point of view. Moreover I believe that only by going farther in this direction, i.e., in the direction opposite to constructivism, will the basic problems of abstract set theory be solved."

(From a Nov. 8, 1957 letter Gödel wrote to Stanislaw Ulam)

• I don't quite understand why you think of NBG as a compromise, can you elaborate? – Noah Schweber Jan 15 at 20:59
• @NoahSchweber Reading Gödel's letter to Ulam, my impression is that he wasn't satisfied with the NBG 'product', saying how interesting the axiom is "from the logical point of view". In the same section, Abraham Fraenkel: "The rather arbitrary character of the processes which are chosen in the axioms of ZFC as the basis of the theory, is justified by the historical development of set-theory rather than by logical arguments." . – CopyPasteIt Jan 15 at 22:35
• @NoahSchweber ... So we can look at specifying the axioms of Set theory as a 'system design' problem. So maybe limitation isn't the key. but the 'next time around' there really should be a 'centerpiece' to the theory. – CopyPasteIt Jan 15 at 22:36
• Who says that there isn't a centerpiece to ZF? – Noah Schweber Jan 15 at 22:44
• Well, I think the point is that the centerpiece doesn't need to be a single axiom, but a motivating idea. The motivating idea for ZF is (or has become over time) transfinite recursion; replacement of course is the key axiom (scheme) for developing transfinite recursion, but regularity also plays a role in ensuring that the universe is precisely the stuff that can be "recursively built." – Noah Schweber Jan 15 at 22:48

Interest in alternative systems never dies. Although ZF-style set theory (or more precisely in my opinion, "cumulative-hierarchy-style set theory") is by far dominant, there's no inherent reason for that to remain the case forever, and there's certainly no reason to abandon the study of alternative set theories in general. Limitation of size does play an important role in such theories, so I'd say that the answer to your question is a weak "no."

However, I think this also misses the broader question: why did limitation of size (at least as such) fade away in the first place? We have to understand that before we decide what role limitation of size should play in the next set theory we cook up.

First, we remember that there are really two pieces to limitation of size. The first is that any class which surjects onto the universe of sets is a proper class. You mention that limitation of size is arguably too strong; well, this half of limitation of size is too weak to be useful on its own (although it's an important motivating force - e.g. behind replacement). The worryingly strong direction is the converse, which says that any class which does not so surject is a set.

The intuitive point now is that essentially as long as we have regularity and enough replacement - and I'll call this "cumulative-hierarchy-style" class or set theory - we can show that any proper class surjects onto the ordinals. Namely, sending $$x$$ to the rank of $$x$$ gives a surjection onto a cofinal class of ordinals, and the Mostowski collapse turns this into a surjection onto the whole class of ordinals. So by composing surjections, limitation of size holds iff there is a surjection from the ordinals to all sets. This in turn is equivalent to the existence of a well-ordering of the universe of all sets, aka global choice.

Now the key point is the above makes sense in mere set theory (in particular, ZF). Of course, on the face of it that's nonsense since we talked explicitly about classes, which we can't do in ZF. Instead, in ZF everything is about (parameter-)definable classes. But the above argument still essentially goes through, and we can prove that given a model $$M$$ of ZF (or indeed much less), if every parameter-definable class in $$M$$ is either a set in $$M$$ or definably surjects onto $$M$$, then there is a parameter-definable surjection of the $$M$$-ordinals onto $$M$$, and the converse holds as well.

It turns out that this can be collapsed into a single first-order(!) statement: namely, that there is some set $$A$$ such that every set is definable from $$A$$ together with an ordinal. (This isn't hard to see - we just say "$$x$$ is the $$\alpha$$th element of the well-ordering of $$V$$ induced by $$s$$," where $$s$$ is our $$\{A\}$$-definable surjection from the ordinals to $$V$$.) This can be written as "$$V=$$ HOD[A] for some set $$A$$." In case we have a parameter-freely definable surjection from the ordinals to $$V$$, we get $$V=$$ HOD. Just like in the case of the axiom of constructibility it's not immediately clear that this is actually first-order expressible, but a neat trick with the reflection principle shows that it is. So limitation of size for definable classes is a first-order principle even in mere set theory (or at least, in ZF). Now HOD and its variants are extremely important concepts in modern set theory even ignoring foundational considerations, so the "HOD-language" tends to win out (and certainly wins out when looking at ZF or its extensions).

The final piece of this picture is the shift in interpretation. Initially we may have thought of limitation of size as a maximizing principle (anything that could be a set, is), but in light of its equivalence with $$V=$$ HOD (ignoring parameters for now for simplicity) it threatens to take on the opposite character in cumulative-hierarchy-style set theory: which is more restrictive, that every set have some ordinal defining it or that there be no ordinal which lets us define some fixed set? The cumulative hierarchy idea pushes against the "obviously maximizing" nature of limitation of size. So it's hard to justify using limitation of size as a centerpiece of a set theory if we're committed to the centrality of the cumulative hierarchy idea and to the value of maximization of mathematical concepts, and these seem more deeply entrenched.

• I think that even HOD[A] is enough for limitation of size to kick in. In any case, I think that also after Gödel and Cohen research in foundation moved from "true" to "consistent". – Asaf Karagila Jan 15 at 21:18
• Can you clarify the connection between limitation of size and V = HOD? I'm with you until the second-to-last paragraph. – Alex Kruckman Jan 15 at 21:52
• @AsafKaragila Whoops, apparently I couldn't decide between "parameter-freely definable" and "parameter-definable." Fixing ... – Noah Schweber Jan 15 at 21:55
• @AlexKruckman Does my edit (+ the correction Asaf indicated) help? – Noah Schweber Jan 15 at 22:00
• @NoahSchweber Yes, it's clear now, thanks! – Alex Kruckman Jan 15 at 22:24