# Philosophy Dictionary of Arguments

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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.
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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.

Author Concept Summary/Quotes Sources

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 (Ey1(y1ε x²) u ∀y1[y1 ε x² › Ez1(y1 ‹ z1 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.

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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:

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
In
From a Logical Point of View, , Cambridge, MA 1953

Quine VII (b)
W. V. A. Quine
Two dogmas of empiricism
In
From a Logical Point of View, , Cambridge, MA 1953

Quine VII (c)
W. V. A. Quine
The problem of meaning in linguistics
In
From a Logical Point of View, , Cambridge, MA 1953

Quine VII (d)
W. V. A. Quine
Identity, ostension and hypostasis
In
From a Logical Point of View, , Cambridge, MA 1953

Quine VII (e)
W. V. A. Quine
New foundations for mathematical logic
In
From a Logical Point of View, , Cambridge, MA 1953

Quine VII (f)
W. V. A. Quine
Logic and the reification of universals
In
From a Logical Point of View, , Cambridge, MA 1953

Quine VII (g)
W. V. A. Quine
Notes on the theory of reference
In
From a Logical Point of View, , Cambridge, MA 1953

Quine VII (h)
W. V. A. Quine
Reference and modality
In
From a Logical Point of View, , Cambridge, MA 1953

Quine VII (i)
W. V. A. Quine
Meaning and existential inference
In
From 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
In
Zur 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