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By “alternative set theories” we mean systems of set theory differing significantly from the dominant ZF (Zermelo-Frankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its variations, New Foundations and related systems, positive set theories, and constructive set theories.
An interest in the range of alternative set theories does not presuppose an interest in replacing the dominant set theory with one of the alternatives; acquainting ourselves with foundations of mathematics formulated in terms of an alternative system can be instructive as showing us what any set theory (including the usual one) is supposed to do for us. The study of alternative set theories can dispel a facile identification of “set theory” with “Zermelo-Fraenkel set theory”; they are not the same thing. Axiom of limitation of size: A class (C ) is proper if and only if there is a class bijection between (C ) and the universe. This elegant axiom is essentially due to von Neumann.
Although Elementary Set Theory is well-known and straightforward, the modern subject, Axiomatic Set Theory, is both conceptually more difficult and more interesting. Complex issues arise in Set Theory more than any other area of pure mathematics; in particular, Mathematical Logic is used in a fundamental way. An Introduction to Elementary Set Theory Guram Bezhanishvili and Eachan Landreth 1 Introduction In this project we will learn elementary set theory from the original historical sources by two key gures in the development of set theory, Georg Cantor (18451918) and Richard Dedekind (18311916).
A class bijection is a class of ordered pairs; there might be pathology here if we did not have enough pairs as sets, but other axioms do provide for their existence. It is interesting to observe that this axiom implies Replacement (a class which is the same size as a set cannot be the same size as the universe) and, surprisingly, implies Choice (the von Neumann ordinals make up a proper class essentially by the Burali-Forti paradox, so the universe must be the same size as the class of ordinals, and the class bijection between the universe and the ordinals allows us to define a global well-ordering of the universe, whose existence immediately implies Choice). Although Class Comprehension and Limitation of Size appear to tell us exactly what classes there are and what sets there are, more axioms are required to make our universe large enough. These can be taken to be the axioms of Z (other than extensionality and choice, which are not needed): the sethood of pairs of sets, unions of sets, power sets of sets, and the existence of an infinite set are enough to give us the world of ZFC. Foundation is usually added. The resulting theory is a conservative extension of ZFC: it proves all the theorems of ZFC about sets, and it does not prove any theorem about sets which is not provable in ZFC. For those with qualms about choice (or about global choice), Limitation of Size can be restricted to merely assert that the image of a set under a class function is a set.
We have two comments about this. First, the mental furniture of set theorists does seem to include proper classes, though usually it is important to them that all talk of proper classes can be explained away (the proper classes are in some sense “virtual”). Second, this theory (especially the version with the strong axiom of Limitation of Size) seems to capture the intuition of Cantor about the Absolute Infinite. A stronger theory with classes, but still essentially a version of standard set theory, is the Kelley-Morse set theory in which Class Comprehension is strengthened to allow quantification over all classes in the formulas defining classes. Kelley-Morse set theory is not finitely axiomatizable, and it is stronger than ZFC in the sense that it allows a proof of the consistency of ZFC. 5.2 Ackermann set theory The next theory we present was actually embedded in the set theoretical proposals of Paul Finsler, which were (taken as a whole) incoherent (see the notes on Finsler set theory available in the ).
Ackermann later (and apparently independently) presented it again. It is to all appearances a different theory from the standard one (it is our first genuine “alternative set theory”) but it turns out to be essentially the same theory as ZF (and choice can be added to make it essentially the same as ZFC). Ackermann set theory is a theory of classes in which some classes are sets, but there is no simple definition of which classes are sets (in fact, the whole power of the theory is that the notion of set is indefinable!) All objects are classes.
The primitive notions are equality, membership and sethood. The axioms are. Extensionality, Pairing, Union, Power Set, Foundation, Choice: As in our presentation of ZFC above. Separation, Replacement: As in our presentation of ZFC above, except that the standardness predicate cannot appear in the formula ( phi ). Definition: For any formula ( phi ), the formula ( phi ) st is obtained by replacing each quantifier over the universe with a quantifier over all standard objects (and each quantifier bounded in a set with a quantifier restricted to the standard elements of that set).
Idealization: There is a finite set which contains all standard sets. Transfer: For each formula ( phi(x) ) not mentioning the standardness predicate and containing no parameters (free variables other than (x) ) except standard sets, ( forall x phi(x) equiv forall x )( standard ((x) rightarrow phi(x)) ).
Standardization: For any formula ( phi(x) ) and standard set (A ), there is a standard set (B ) whose standard elements are exactly the standard elements (x ) of (A ) satisfying ( phi(x) ). Our form of Idealization is simpler than the usual version but has the same effect.
Transfer immediately implies that any uniquely definable object (defined without reference to standardness) is in fact a standard object. So the empty set is standard, ( omega ) is standard, and so forth. But it is not the case that all elements of standard objects are standard. For consider the cardinality of a finite set containing all standard objects; this is clearly greater that any standard natural number (usual element of ( omega) ) yet it is equally clearly an element of ( omega ).
It turns out to be provable that every set all of whose elements are standard is a standard finite set. Relative consistency of this theory with the usual set theory ZFC is established via familiar results of model theory. Working in this theory makes it possible to use the techniques of nonstandard analysis in a “elementary” way, without ever appealing explicitly to the properties of nonstandard models.
8.3 The multiverse view of set theory We examine the theory of the set theoretic multiverse proposed by Joel David Hamkins, whose purpose is to address philosophical questions about independence questions in standard set theory, but which when spelled out formally has some of the flavor of an alternative set theory. A set theoretic Platonist might say about the Continuum Hypothesis ( CH) that, since there is “of course” a single universe of sets, CH is either true or false in that world, but that we cannot determine which of CH and ( neg ) CH actually holds.
Hamkins proposes as an alternative (taking the same realist standpoint as the classical Platonist, it must be noted) that there are many distinct concepts of set, which we may suppose for the moment all satisfy the usual axioms of ZFC, each concept determining its own universe of sets, and in some of these universes CH holds and in some it does not hold.
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