## Philosophy Dictionary of ArgumentsHome | |||

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Infinity Axiom: The infinity axiom is an axiom of set theory, which ensures that there are infinite sets. It is formulated in e.g. such a way that a construction rule is specified for the occurrence of elements of a described set. If {x} is the successor of x, the continuation is formed by the union x U {x}. See also set theory, successor, unification, axioms._____________ Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. | |||

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W.V.O. Quine on Infinity Axiom - Dictionary of Arguments IX 205 Def Infinity Axiom/Quine: an infinite number of elements in types should be possible. One possibility is e.g. : Tarski: that there is a non-empty class x², such that each of its elements is a subclass of another element. Russell: for each x² e N ³ there is a class y1 with x² elements: short L² ε N³. (1) Ex² (Ey ^{1}(y^{1}ε x²) u ∀y^{1}[y^{1} ε x² › Ez^{1}(y^{1} ‹ z^{1} ε x²)]).Vs: some thought that the question of whether there were infinitely many individuals was more a question of physics or metaphysics. It is inappropriate to let arithmetic depend on it. Russell and Whitehead regretted the infinity axiom and the axiom of choice, and both made special cases dependent on them, as I do most comprehension assumptions. Frege's Natural Numbers/Quine: are plagued by the necessity of infinity axioms, even if we allow type theory, liberalization and cumulative types, or finally heterogeneous classes. Because within each type there is a finite barrier to how large a class can be, unless there are infinitely many individuals. Zermelo's concept of numbers would be a solution here, but brings problems with complete induction. IX 206 Real Numbers/Quine: for them and beyond, however, infinity axioms are always necessary. Infinity Axiom/Zermelo: (5) Ex[Λ ε x u ∀y(y ε x › {y} ε x)]. It postulates a class to which at least all natural numbers in Zermelo's sense belong. It is equivalent to "N ε ϑ" because N is itself an x that satisfies (5), and vice versa, if x satisfies (5), then N < x., and thus "N ε ϑ" according to the exclusion scheme. Unlike Russell's, this infinity axiom says nothing about the existence of individuals. But it separates the last connections to type theory. Zermelo's and Neumann's numbers are even antithetic to cumulative type theory, because such a class breaks the boundaries of all types. Axioms of Infinity/Russell: was caused by the law of subtraction "S'x = S'y > x = y". In other words, it was used so that the natural numbers would not break off. Similarly for the real numbers. But its meaning goes even further: each subsequent type is the class of all subclasses of its predecessor and thus, according to Cantor's theorem, larger than its predecessor. To accept infinitely many individuals therefore means to accept higher infinities without end. For example, the power class in (7) says that {x:x < N} ε ϑ, and this last class is greater than N after the theorem of Cantor. And so it goes further up. Infinity Axiom/Zermelo: breaks the type limits. Quine pro: this frees us from the burden comparable to the type indices, because even in type theory with universal variables we were forced to Frege's version of the natural numbers, which meant recognition of a different 5 in each type (about classes of individuals) of a different 6 in each type, a different N in each type, etc. In addition there is, throughout the whole hierarchy, a multiplication of all details of the theory of real numbers. 3/5 is something different in every following type and also π, Q, R. For all these constants it is practically necessary to keep the type indices. In Zermelo's system with its axiom of infinity such multiplications do not occur with the task of type boundaries. Zermelo's protection was that he avoided classes that were too large. For the reverse assurance that classes cannot exist only if they were larger than all existing classes, very little provision has been made in its segregation scheme. IX 208 Fraenkel and Skolem first did this in their axiom scheme of substitution. II 93 Infinity Axiom/QuineVsRussell: the Principia Mathematica ^{(1)} must be supplemented by the axiom of infinity when certain mathematical principles are to be derived. Axiom of infinity: ensures the existence of a class with an infinite number of elements - New Foundations/Quine: instead comes with the universal class of ϑ or x^ (x = x).>Infinity/Quine, >Classes/Quine. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. _____________ Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz InZur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |

Ed. Martin Schulz, access date 2024-03-01