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On the other hand, the results of classical set theory short of the paradoxes can be derived. See the table of Zermelo-Fraenkel axioms. It should be noted that this is not merely a logically necessary property of equality but an assumption about the membership relation as well. This is a restricted version of the principle of abstraction, now known as the principle of comprehension, for it provides for the existence of sets corresponding to formulas. Using tools of modern logic, the definition may be made as follows:.

Formulas are constructed recursively in a finite number of systematic steps beginning with the atomic formulas of I and proceeding via the constructions permitted in II. It is to be understood that a formula is a formal expression—i. A formula acquires meaning only when an interpretation of the theory is specified; i. It provides for the existence of sets by separating off certain elements of existing sets.

If this is not read very carefully, it may be seen as implying that we have adopted the comprehension axioms of naive set theory,. But we have not.

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Formulas of the untyped language of set theory in which it is possible to assign a type to each variable the same type wherever it occurs in such a way as to get a formula of TST are said to be stratified. The axioms of NF are strong extensionality no non-sets and stratified comprehension. For example, the Frege natural number 1, the set of all one-element sets, is provided by NF. No contradictions are known to follow from NF , but some uncomfortable consequences do follow.

The Axiom of Choice is known to fail in NF : Specker proved that the universe cannot be well-ordered. This might be thought to be what one would expect on adopting such a dangerous comprehension scheme, but this turns out not to be the problem. The problem is with extensionality. Jensen showed that NFU New Foundations with urelements , the version of New Foundations in which extensionality is weakened to allow many non-sets as described above under naive set theory is consistent, is consistent with Infinity and Choice, and is also consistent with the negation of Infinity which of course implies Choice.

Some other fragments of NF , obtained by weakening comprehension rather than extensionality, are known to be consistent. NF 3 , the version of NF in which one accepts only those instances of the axiom of comprehension which can be typed using three types, was shown to be consistent by Grishin He also demonstrated the consistency of the theory NFI in which one allows all instances of stratified comprehension in which no variable appears of type higher than that assigned to the set being defined bound variables of the same type as that of the set being defined are permitted, which allows some impredicativity.

NFP is equivalent to a weaker fragment of arithmetic, but does unlike NFU prove Infinity: this is the only application of the Specker proof of the negation of the Axiom of Choice to a provably consistent theory.


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It is of some interest whether this new fragment represents an independent way of getting a consistent fragment of NF. Of these set theories, only NFU with Infinity, Choice and possibly further strong axioms of infinity of which more anon is really mathematically serviceable. We examine the construction of models of this theory and the way mathematics works inside this theory.

A source for this development is Holmes A model of NFU can be constructed as follows. The proof that this is a model of NFU is not long, but it is involved enough that we refer the reader elsewhere. This construction is related to the construction used by Jensen, but is apparently first described explicitly in Boffa In any model of NFU , a structure which looks just like one of these models can be constructed in the isomorphism classes of well-founded extensional relations.

The theory of isomorphism classes of well-founded extensional relations with a top element looks like the theory of an initial segment of the usual cumulative hierarchy, because every set in Zermelo-style set theory is uniquely determined by the isomorphism type of the restriction of the membership relation to its transitive closure.

The surprise is that we not only see a structure which looks like an initial segment of the cumulative hierarchy: we also see an external endomorphism of this structure which moves a rank and therefore cannot be a set , in terms of which we can replicate the model construction above and get an interpretation of NFU of this kind inside NFU! See Holmes for the full discussion. This suggests that the underlying world view of NFU , in spite of the presence of the universal set, Frege natural numbers, and other large objects, may not be that different from the world view of Zermelo-style set theory; we build models of NFU in a certain way in Zermelo-style set theory, and NFU itself reflects this kind of construction internally.

But this turns out not to be the case and this means that the urelements, which seem to have no internal information, nonetheless have a great deal of structure in these models. The Frege construction of the natural numbers works perfectly in NFU. Any of the usual ordered pair constructions works in NFU.

This means that functions and relations are three types higher than the elements of their domains and ranges. This would make the type displacement between functions and relations and elements of their domains and ranges just one, the same as the displacement between the types of sets and their elements.

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We will assume that ordered pairs are of the same type as their projections in the sequel, but we will not present the rather complicated definition of the Quine pair. Once pairs are defined, the definition of relations and functions proceeds exactly as in the usual set theory. The definitions of integers and rational numbers present no problem, and the Dedekind construction of the reals can be carried out as usual. We will focus here on developing the solutions to the paradoxes of Cantor and Burali-Forti in NFU , which give a good picture of the odd character of this set theory, and also set things up nicely for a brief discussion of natural strong axioms of infinity for NFU.

It is important to realize as we read the ways in which NFU evades the paradoxes that this evasion is successful: NFU is known to be consistent if the usual set theory is consistent, and close examination of the models of NFU shows exactly why these apparent dodges work. Two sets are said to be of the same cardinality just in case there is a bijection between them.

This is standard. The Frege natural numbers are the same objects as the finite cardinal numbers. Strongly cantorian sets are important because it is not necessary to assign a relative type to a variable known to be restricted to a strongly cantorian set, as it is possible to use the restriction of the singleton map and its inverse to freely adjust the type of any such variable for purposes of stratification.

Axiomatic Set Theory

The strongly cantorian sets are can be thought of as analogues of the small sets of the usual set theory. Ordinal numbers are defined as equivalence classes of well-orderings under similarity. There is a natural order on ordinal numbers, and in NFU as in the usual set theory it turns out to be a well-ordering—and, as in naive set theory, a set!

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As is the case with Zermelo set theory, natural extensions of this theory make it much stronger. We give just one example. Like Zermelo set theory, NFU has advantages and disadvantages. An advantage, which corresponds to one of the few clear disadvantages of Zermelo set theory, is that it is possible to define natural numbers, cardinal numbers, and ordinal numbers in the natural way intended by Frege, Russell, and Whitehead.

The fact that there are fewer singletons than objects in spite of an obvious external one to one correspondence takes getting used to. This author can testify that it is perfectly possible to develop good intuition for NFU and work effectively with stratified comprehension; part of this but not all of it is a good familiarity with how things are done in TST , as one also has to develop a feel for how to use principles that subvert stratification.

As Sol Feferman has pointed out, one place where the treatments in NFU at least those given so far are clearly quite involved are situations in which one needs to work with indexed families of objects. We also observe that strong principles which minimize the nonstandard behavior of the ordinals turn out to give surprisingly strong axioms of infinity in NFU ; the nonstandard structure of the ordinals allows insight into phenomena associated with large cardinals.

Some have thought that the fact that NFU combines a universal set and other big structures with mathematical fluency in treating these structures might make it a suitable medium for category theory.

http://adiryxohiz.tk Although we have some inclination to be partial to this class of set theories, we note that there are strong counterarguments to this view. It is true that there are big categories, such as the category of all sets as objects and functions as the morphisms between them , the category of all topological spaces and homeomorphism, and even the category of all categories and functors. Moreover, if one restricts to the s.

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Contemplation of the models only confirms the impression that the correct analogue of the proper class category of sets and functions in ZFC is the proper class category of s. There may be some applications for the big set categories in NFU , but they are not likely to prove to be as useful as some have optimistically suggested.


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  7. See Feferman for an extensive discussion. An important point is that there is a relativity of viewpoint here: the NFU world can be understood to be a nonstandard initial segment of the world of ZFC which could be arranged to include its entire standard part!

    ZFC might be viewed as motivated by a generalization of the theory of sets in extension as generalizations of the notion of finite set, replacing the finite with the transfinite and the rejected infinite with the rejected Absolute Infinite of Cantor while the motivation of NFU can be seen as a correction of the theory of sets as intensions that is, as determined by predicates which led to the disaster of naive set theory. But the impression of a fundamental contrast should be tempered by the observation that the two theories nonetheless seem to be looking at the same universe in different ways!

    We are not very familiar with this theory, so have no additional comments to make about it; do notice that the translations of formulas with set abstracts in them into first order logic without abstracts are definitely not positive in our more restricted sense, and so one may expect some kind of trouble! The motivation for the kinds of positive set theory we are familiar with is topological.

    We are to understand the sets as closed sets under some topology. Compactness of the topology suffices. Moreover, we now need to be aware that formulas with several parameters need to be considered in terms of a product topology.