What's behind the word "size issues"?

Question

I occasionally hear the term "size issues", which refers to situations where a certain collection of objects "is too big to be a set". What does one mean by "too big to be a set"? Why is bigness contradictory (it's only contradictory if one assumes the axiom of replacement, right)? Is "bigness" a clear defined term? It doesn't seem so:

In the book Set theory: an introduction by Vaught, it says on page 12:

But the idea 'big' remains vague and our remarks about 'bigness' are to be taken only heuristic.

But using replacement and global choice, one can prove the following rigorous theorem:

A class is a proper class if and only if it's bijective to V, the class of all sets.

Isn't this theorem saying that "too big" really has a precise meaning?

When one talks about "bigness", does one assume the axiom of replacement? I think so, because otherwise the term wouldn't make so much sense in combination with contradictions.

Answer 1

Yes, if you have global choice, then you can prove the principle you speak about, which is known as Limitation of Size.

However, in most cases where the informal "too big to be a set" argument is used, one is generally assumed to be working in ZF(C), where neither global choice nor limitation of size can even be stated. Generally, what this informal argument means is something like

If the collection of all the things you're speaking about there constituted a set, then it follows that there would also need to be a set of all sets (or a set of all ordinals), in which case Russell's paradox (or Burali-Forti's) would obtain a contradiction. So your collection definitely does not specify a set.

Here, the step where you conclude "there would need to be a set of all sets" often involves Replacement, but not necessarily. For example, the collection of all singletons is "too big" just by virtue of the Axiom of Union -- and similarly, taking the union a small finite number of times will be enough to show that collections like "all groups" or "all small categories" are too big to be sets.

Answer 2

For small sets the property that a given object is in the set is very restrictive. For example, the property a given 'thing' is a a number between $7$ and $456$ is very discerning in a universal sense. However, the property that a set doesn't contain itself doesn't rule out too many objects, so in a loose sense, it allows enough elements that it's membership property is no longer consistent with our intuitive or axiomatic notions of membership.

The 'set' or class described in the latter example is the subject of russel's paradox, an inconsistency implied by the axiom of comprehension, the defunct axiom that any well formed property defines a set. So a distinction had to made between sets and classes, and sets are defined in a hierarchy from other sets so that "barber who only shaves..." classes won't become actual sets in your theory. The result is that some collections like 'the set of real valued functions on $\mathbb{R}$' are sets and collections like 'the set of all sets' and 'the set of all real valued maps on any $X$' are not.

Answer 3

The point is that Replacement guarantees that a class which can be put in bijection with a set, is a set. Here class means a definable collection, and a bijection means definable bijection.

If every set has a cardinality, then proper classes are "too big" to have a cardinality. Which is exactly the idea.

The thing is, that assuming global choice, every two proper classes are equipotent. So there is just "one size" of proper classes.

Without Replacement it is possible that there is a proper class which is countable, in the sense that there is a definable bijection between the natural numbers and the class. And if that is not a motivation to accept Replacement, I don't know what is...

Answer 4

I think people (who live alongside infinities) are reluctant to say that the idea of something being big is a real mathematical statement because it is known to be relative.

When presented as you did, it seems that "being too big" really means being as big as it is possible. But look at a countable model of ZFC. Its universe is externally as big as any infinite set in it (some of which might internally be regarded as finite), so then being too big is just allowing a correspondance with the universe: this needs not be an intrinsic property of the class or of the set of elements (according to the model) of this class, but this depends on the whole model.

When you look closer at why some things are "too big", you find out that the limitations are more subtle that you could think. One instance of this is the limitation that appears when a notion has a wide range that allows self reference in some way, and that a contrary to this notion exists. Famous and direct instances of this are:

-$V$ is "too big" to be a set because of the notion "element of $\ . \ $" -$\mathfrak{B}(\mathbb{N})$ is "bigger" than $\mathbb{N}$ because the notion "element of $f(.)$" -$Ord$ is too big to be a set for arguably the same reason.

Limitation of size synthesizes this but it is debatable whether the idea of size is really the key point in those matters.

Despite that, the image of size works because it is in accordance with the naïve (yet true by comprehension) idea that subclasses are tinier: a sublass of a set is a set.

I hope this is not too vague.