The Iterative Conception and Set Theory

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all.

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George Boolos. The iterative conception of set. The journal of philosophy, vol. 68 (1971), pp. 215–231. - Dana Scott. Axiomatizing set theory. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 207–214. - W. N. Reinhardt. Remarks on reflection principles, large cardinals, and elementary embeddings. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 189–205. - W. N. Reinhardt. Set existence principles of Shoenfield, Ackermann, and Powell. Fundament a mathematicae, vol. 84 (1974), pp. 5–34. - Hao Wang. Large sets. Logic, foundations of mathematics, and computahility theory. Part one of the proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 309–333. - Charles Parsons. What is the iterative conception of set?Logic, foundations of mathematics, and computahility theory. Part one of the proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 335–367.

Journal of Symbolic Logic ◽

10.2307/2274241 ◽

1985 ◽

Vol 50 (2) ◽

pp. 544-547 ◽

Cited By ~ 2

Author(s):

John P. Burgess

Keyword(s):

Philosophy Of Science ◽

Set Theory ◽

American Mathematical Society ◽

Publishing Company ◽

Foundations Of Mathematics ◽

Reidel Publishing Company ◽

Pure Mathematics ◽

Iterative Conception ◽

The University ◽

Axiomatic Set Theory

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Set theory, different systems of

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y025-1 ◽

2018 ◽

Author(s):

Michael Potter

Keyword(s):

Twentieth Century ◽

Mathematical Logic ◽

Set Theory ◽

The Other ◽

Von Neumann ◽

Class Theory ◽

Iterative Conception ◽

The One ◽

Limitation Of Size ◽

As If

To begin with we shall use the word ‘collection’ quite broadly to mean anything the identity of which is solely a matter of what its members are (including ‘sets’ and ‘classes’). Which collections exist? Two extreme positions are initially appealing. The first is to say that all do. Unfortunately this is inconsistent because of Russell’s paradox: the collection of all collections which are not members of themselves does not exist. The second is to say that none do, but to talk as if they did whenever such talk can be shown to be eliminable and therefore harmless. This is consistent but far too weak to be of much use. We need an intermediate theory. Various theories of collections have been proposed since the start of the twentieth century. What they share is the axiom of ‘extensionality’, which asserts that any two sets which have exactly the same elements must be identical. This is just a matter of definition: objects which do not satisfy extensionality are not collections. Beyond extensionality, theories differ. The most popular among mathematicians is Zermelo–Fraenkel set theory (ZF). A common alternative is von Neumann–Bernays–Gödel class theory (NBG), which allows for the same sets but also has proper classes, that is, collections whose members are sets but which are not themselves sets (such as the class of all sets or the class of all ordinals). Two general principles have been used to motivate the axioms of ZF and its relatives. The first is the iterative conception, according to which sets occur cumulatively in layers, each containing all the members and subsets of all previous layers. The second is the doctrine of limitation of size, according to which the ‘paradoxical sets’ (that is, the proper classes of NBG) fail to be sets because they are in some sense too big. Neither principle is altogether satisfactory as a justification for the whole of ZF: for example, the replacement schema is motivated only by limitation of size; and ‘foundation’ is motivated only by the iterative conception. Among the other systems of set theory to have been proposed, the one that has received widespread attention is Quine’s NF (from the title of an article, ‘New Foundations for Mathematical Logic’), which seeks to avoid paradox by means of a syntactic restriction but which has not been provided with an intuitive justification on the basis of any conception of set. It is known that if NF is consistent then ZF is consistent, but the converse result has still not been proved.

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Dynamic Set Theory

10.1093/oso/9780199641314.003.0012 ◽

2018 ◽

Author(s):

Øystein Linnebo

Keyword(s):

Set Theory ◽

Dynamic Approach ◽

Dynamic Setting ◽

Basic Law ◽

Iterative Conception ◽

The Law

The dynamic abstractionist approach to set theory canvassed in Chapter 3 is properly developed. We begin with a plural version of Frege’s Basic Law V. While this law is inconsistent in the ordinary static setting, it becomes consistent when transposed to a dynamic setting. Thus transposed, the law ensures that any objects whatsoever can be used to define a set. This is an intuitive and highly explanatory principle of set formation, which traces its roots back to Cantor. The resulting dynamic approach to set theory justifies all of ordinary ZF set theory and provides an explication of the celebrated iterative conception of sets.

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The Problem of Conceptual Understanding

Mathematical Knowledge and the Interplay of Practices ◽

10.23943/princeton/9780691167510.003.0010 ◽

2015 ◽

Author(s):

José Ferreirós

Keyword(s):

Real Number ◽

Set Theory ◽

Conceptual Understanding ◽

Philosophy Of Mathematics ◽

Axiom System ◽

The Real ◽

Iterative Conception ◽

Universe Of Sets ◽

The Universe ◽

The Web

This chapter considers one of the most intriguing questions that philosophy of mathematics in practice must, sooner or later, confront: how understanding of mathematics is obtained. In particular, it examines how issues of meaning and understanding in relation to practice and use relate to the question of the acceptability of “classical” or postulational mathematics, a question usually formulated in terms of consistency. The chapter begins with a discussion of the iterative conception of the universe of sets and its presuppositions, analyzing it from the standpoint of the web of practices. It then addresses the issue of conceptual understanding in mathematics, as exemplifid by the theory Zermelo–Fraenkel axiom system (ZFC). Finally, it looks at arguments based on the idea of the real-number continuum as a source of justification for the axioms of set theory.

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Set theory, philosophy of

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y092-1 ◽

2018 ◽

Author(s):

Michael Potter

Keyword(s):

Set Theory ◽

The Other ◽

Second Stage ◽

Other Hand ◽

Iterative Conception ◽

Power Set ◽

After Effect ◽

Almost All ◽

The Universe ◽

Philosophy Of Set Theory

The various attitudes that have been taken to mathematics can be split into two camps according to whether they take mathematical theorems to be true or not. Mathematicians themselves often label the former camp realist and the latter formalist. (Philosophers, on the other hand, use both these labels for more specific positions within the two camps.) Formalists have no special difficulty with set theory as opposed to any other branch of mathematics; for that reason we shall not consider their view further here. For realists, on the other hand, set theory is peculiarly intractable: it is very difficult to give an unproblematic explanation of its subject matter. The reason this difficulty is not of purely local interest is an after effect of logicism. Logicism, in the form in which Frege and Russell tried to implement it, was a two-stage project. The first stage was to embed arithmetic (Frege) or, more ambitiously, the whole of mathematics (Russell) in the theory of sets; the second was to embed this in turn in logic. The hope was that this would palm off all the philosophical problems of mathematics onto logic. The second stage is generally agreed to have failed: set theory is not part of logic. But the first stage succeeded: almost all of mathematics can be embedded in set theory. So the logicist aim of explaining mathematics in terms of logic metamorphoses into one of explaining it in terms of set theory. Various systems of set theory are available, and for most of mathematics the method of embedding is fairly insensitive to the exact system that we choose. The main exceptions to this are category theory, whose embedding is awkward if the theory chosen does not distinguish between sets and proper classes; and the theory of sets of real numbers, where there are a few arguments that depend on very strong axioms of infinity (also known as large cardinal axioms) not present in some of the standard axiomatizations of set theory. All the systems agree that sets are extensional entities, so that they satisfy the axiom of extensionality: ∀x(xЄa ≡ xЄb) → a=b. What differs between the systems is which sets they take to exist. A property F is said to be set-forming if {x:Fx} exists: the issue to be settled is which properties are set-forming and which are not. What the philosophy of set theory has to do is to provide an illuminating explanation for the various cases of existence. The most popular explanation nowadays is the so-called iterative conception of set. This conceives of sets as arranged in a hierarchy of stages (sometimes known as levels). The bottom level is a set whose members are the non-set-theoretic entities (sometimes known as Urelemente) to which the theory is intended to be applicable. (This set is often taken by mathematicians to be empty, thus restricting attention to what are known as pure sets, although this runs the danger of cutting set theory off from its intended application.) Each succeeding level is then obtained by forming the power set of the preceding one. For this conception three questions are salient: Why should there not be any sets other than these? How rich is the power-set operation? How many levels are there? An alternative explanation which was for a time popular among mathematicians is limitation of size. This is the idea that a property is set-forming provided that there are not too many objects satisfying it. How many is too many is open to debate. In order to prevent the system from being contradictory, we need only insist that the universe is too large to form a set, but this is not very informative in itself: we also need to be told how large the universe is.

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