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TINY PROPER CLASSES
Cantor believed there are classes of objects that add up to a completed totality and classes of objects unable to do so. The former are the ones we now call sets. Cantor called the latter inconsistent multiplicities but they are nowadays more frequently known as proper classes. In the philosophy of set theory, it is usually assumed that inconsistent multiplicities or proper classes are simply too big to be sets. Those multiplicities that paradoxes have proven inconsistent (the multiplicity of all sets, the multiplicity of all well-founded sets, of all non self-membered sets, of all ordinals, of all cardinals) are in fact huge. And this conception of the difference between consistent and inconsistent multiplicities is represented in some axiomatics of set theory by axioms like the Axiom of Limitation of Size (in some presentations of the axiomatic NBG) or the Axiom of Replacement in the axiomatic ZF.
In set theory, two sets A and B are the same size iff there is a one-to-one function from each other. If A can be so paired with a subset of B but not with B, A is said to be smaller or less numerous than B. Roughly put, the Axiom of Limitation of Size states that all proper classes are the same size (the size of the universe) and the Axiom of Replacement states that whatever is the size of some set is a set.
We wish to suggest an alternative interpretation of the very existence of proper classes, that is, an interpretation of the difference between sets and proper classes based not on size but on the notion of availability: the members of a set are available once for all whereas the members of a proper class are not. Proper classes are indefinitely extensible or open ended: whenever a portion of them is available as a completed totality, it is possible to use this availability to define an object of the class that is not in the portion. This type of construction is known as diagonalization and, we suggest, it reveals the reason why not all members of a proper class can be simultaneously available: it is always possible to render new members available by diagonalization. This is why Russell (1907) called such classes self-reproductive classes (Russell, B. 1905. On some difficulties in the theory of transfinite numbers and order types. Proceedings of the London Mathematical Society s2-4(1): 29-53).
Assume, for instance, we have a set S of sets. We can immediately define a set not in S, namely, the set RS of all non self-membered sets of S (the possibility of this construction is guaranteed by the Axiom of Separation). If RS were in S, it would be self-membered if and only if it were not; so, it is not in S. RS diagonalizes out of S; and as S was any set of sets, there can be no set of all sets.
The following example should contribute a bit of evidence for the availability interpretation of proper classes.
Consider a person, for instance, Andrew Wiles and define the class CW of all sets Wiles will define in his lifetime. Whatever it is, CW is finite. Let us assume that CW is a set and let hand it to Wiles. Then Wiles could easily define the set RCW of all non self-membered sets in CW and RCW would not be in CW though it would have been defined by Wiles in his lifetime. This would reveal that CW was not the set of all sets Wiles will define in his lifetime. It seems that the class of all sets that Wiles will define along his lifetime is not available to Wiles as a completed totality. And it seems it is so for the same reason the class of all sets is not available to anyone as a completed totality: in each case, the purported completed totality could be diagonalized out of and exposed as an indefinitely extensible class.
CW might well not exist as a set, even if it would have only a finite number of members; at least, not for Wiles. For, if it is not a question of size but of availability, one could conjecture that while CW is not a set for Wiles, it could well be such for, say, Grigori Perelman. Perelman could not diagonalize out of CW: if Perelman defined a set not in CW, this would not extend CW as it would occur if Wiles did. However, if CW were available as a set to Perelman, then the class CP of all sets Perelman will define during his lifetime would surely be a set for Wiles. But, as CW is supposed to be a set for Perelman, CW could be a member of CP (Perelman could define it), which in turn would be available to Wiles as a completed totality, for Wiles could in turn define the set of all sets Perelman will define in his lifetime. Then CW, which could not be a set for Wiles, would be a member of what is possibly a set for him. This appears to be impossible, since proper classes are prohibited from being members of any classes.
It seems, therefore, that CW and CP will be finite proper classes for all of us while Andrew Wiles and Grigori Perelman, respectively, are still living and will only become sets for us afterwards. This is not as weird as it could prima facie appear to be. Consider that it is only while these great mathematicians are still alive that those tiny proper classes can be diagonalized out of. In any case, the existence of classes that are not sets though they are tiny, that is, much much smaller than the size of the universe (finite indeed!), would lend support to the interpretation of proper classes in terms of availability and not size which we have merely outlined here.
Laureano Luna
IES Doctor Francisco Marin
Philosophy.
Siles. Spain.
laureanoluna@yahoo.es
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