How Godel Transformed Set Theory (original) (raw)
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Gödel's Argument for Cantorian Cardinality
Noûs, 2018
On the first page of “What is Cantor's Continuum Problem?”, Gödel argues that Cantor's theory of cardinality, where a bijection implies equal number, is in some sense uniquely determined. The argument, involving a thought experiment with sets of physical objects, is initially persuasive, but recent authors have developed alternative theories of cardinality that are consistent with the standard set theory ZFC and have appealing algebraic features that Cantor's powers lack, as well as some promise for applications. Here we diagnose Gödel's argument, showing that it fails in two important ways: (i) Its premises are not sufficiently compelling to discredit countervailing intuitions and pragmatic considerations, nor pluralism, and (ii) its final inference, from the superiority of Cantor's theory as applied to sets of changeable physical objects to the unique acceptability of that theory for all sets, is irredeemably invalid.
The motives behind Cantor's set theory
The celebrated “creation” of transfinite set theory by Georg Cantor has been studied in detail by historians of mathematics. However, it has generally been overlooked that his research program cannot be adequately explained as an outgrowth of the mainstream mathematics of his day. We review the main extra-mathematical motivations behind Cantor’s very novel research, giving particular attention to a key contribution, the Grundlagen of 1883, where those motives are articulated in some detail. Evidence from other publications and correspondence is pulled out to provide clarification and a detailed interpretation of those ideas and their impact upon Cantor’s research. Throughout the paper, a special effort is made to place Cantor’s scientific undertakings within the context of developments in German science and philosophy at the time (philosophers such as Trendelenburg and Lotze, scientists like Weber, Riemann, Vogt, Haeckel), and to reflect on the German intellectual atmosphere during the 19th century.
The actual infinity in Cantor's set theory
The actual infinity Aristotle-Cantor , potential infinity . The origins of Cantor’s infinity, aleph null, the diagonal argument The natural infinity , continuum The mathematical infinity A first classification of sets Three notable examples of countable sets The 1-1 correspondence, equivalent sets, cardinality . The theory of transfinite numbers Existence and construction, existence proofs
Scientific American, 1967
T he abstract t h e o~ of sets i5 currently in a state of change that ill meral wa!"~ IS ana10gou~ to the 19th-centup m~olution m g t w n e q . 145; m any revolution, phticd or scientific, it i s d B d t for those parhcipthp in the revolution or w5hlessi1lp it to foretell its ultimate mnsequences, except perhaps that they \\*ill be profound. One thing that can b done i s to to use the past as a guide to the future. It is an unreliable guide, to be sure. but ktter than none.
Considerations contra cantorianism
2011
With the avoidance of Russell’s paradox and its cognates as one paramount motivation, and the avoidance of ungrounded mathematical objects as another, the twentieth century from early on saw the initiation of various foundational theories which altogether avoided an invocation of infinite power sets. This is famously the case in the predicativist tradition going back to Herman Weyl’s Das Kontinuum, and further investigated later principally by Solomon Feferman, but also by others. This was clearly also an important motivational aspect of the perhaps less rigorously formulated original intent of Luitzen Brouwer’s intuitionist program, and it is presently manifest much more precisely within parts of the intuitionist tradition in that Per Martin Lof’s constructive Type Theory lacks an analogue of the infinitary power-set operation, as does Peter Aczel’s constructive set theory CZF. The reverse mathematics program initiated by Harvey Friedman seemed to have established that only a very ...
I present some strands of Gödel's thought in the light of Cantor's philosophy of mathematics. Gödel's Cantorianism is transparent in three areas: 1) Cantor's distinction between immanent/transient mathematical existence, and the requirement that the mathematician be only concerned with the immanent one, is reflected by Gödel's belief in the objectivity of concepts (conceptual realism), whereas belief in their trans-subjective existence is also a constituent of Gödel's Platonism. 2) Cantor's suggestion that mathematical concepts can be non-arbitrarily expanded is resumed by Gödel and re-cast in a phenomenological fashion. In particular, Gödel seems to think that there is an objective conceptual development of set theory and, thereby, a unique notion of set in the same way as Cantor thought that there was only one correct realisation of the notion of actual in finite. 3) Cantor also defined extrinsic criteria for introducing new axioms: consistency, success, fruitfulness. Gödel describes analogous criteria while discussing his programme for fi nding new set-theoretic axioms.""
Cantor's Proof in the Full Definable Universe
Australasian Journal of Logic 2010 9: 10-25, 2010
"Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the scope of quantifiers reveals a natural way out."
Set Theory from Cantor to Cohen
Philosophy of Mathematics, 2009
Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf. What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The whole transfinite landscape can be viewed as having been articulated by Cantor in significant part to solve the Continuum Problem. Zermelo's axioms can be construed as clarifying the set existence commitments of a single proof, of his Well-Ordering Theorem. Set theory is a particular case of a field of mathematics in which seminal proofs and pivotal problems actually shaped the basic concepts and forged axiomatizations, these transmuting the very notion of set. There were two main junctures, the first being when Zermelo through his axiomatization shifted the notion of set from Cantor's range of inherently structured sets to sets solely structured by membership and governed and generated by axioms. The second juncture was when the Replacement and Foundation Axioms were adjoined and a first-order setting was established; thus transfinite recursion was incorporated and results about all sets could established through these means, including results about definability and inner models. With the emergence of the cumulative hierarchy picture, set theory can be regarded as becoming a theory of well-foundedness, later to expand to a study of consistency strength. Throughout, the subject has not only been sustained by the axiomatic tradition through Gödel and Cohen but also fueled by Cantor's two legacies, the extension of number into the transfinite as transmuted into the theory of large cardinals and the investigation of definable sets of reals as transmuted into descriptive set theory. All this can be regarded as having a historical and mathematical logic internal to set theory, one that is often misrepresented at critical junctures in textbooks (as will be pointed out). This view, from inside set theory and about itself, serves to shift the focus to