The axiom of infinity is also one of the von Neumann–Bernays–Gödel axioms.
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC.
A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC.
The ONTOLOGY (as Grice would call it) of NBG includes:
proper classes,
objects having members but that cannot be members of other entities.
NBG's principle of class comprehension is predicative.
Quantified variables in the defining formula can range only over sets.
Allowing impredicative comprehension turns NBG into Morse-Kelley set theory (MK). NBG, unlike ZFC and MK, can be finitely axiomatized.
There exists an inductive set, namely a set x whose members are
(i) the empty set;
(ii) for every member y of x,
is also a member of x.Infinity can be formulated so as to imply the existence of the empty set.
REFERENCES
- Adámek, Jiří; Herrlich, Horst, and Strecker, George E (2004) [1990]. Abstract and Concrete Categories (The Joy of Cats) (PDF). New York: Wiley & Sons. ISBN 0-471-60922-6.
- Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.
- Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought (2nd revised ed.), Basel, Switzerland: Birkhäuser, ISBN 3-7643-8349-6.
- Gödel, Kurt (1940), The Consistency of the Continuum Hypothesis, Princeton University Press.
- Hallett, Michael (1984), Cantorian Set Theory and Limitation of Size, Oxford: Clarendon Press.
- Kanamori, Akihiro (2009), "Bernays and Set Theory", Bulletin of Symbolic Logic 15: 43–69. (Page numbering in Notes refers to online article whose numbering starts at 1.)
- Kanamori, Akihiro (2012), "In Praise of Replacement", Bulletin of Symbolic Logic 18: 46–90.
- Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, North-Holland, ISBN 0-444-85401-0.
- Mendelson, Elliott, (1997), An Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall. ISBN 0-412-80830-7. Pp. 225–86 contain the classic textbook treatment of NBG, showing how it does what we expect of set theory, by grounding relations, order theory, ordinal numbers, transfinite numbers, etc.
- Mirimanoff, Dmitry (1917), "Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles", L'Enseignement Mathématique 19: 37–52.
- Richard Montague, (1961), "Semantic Closure and Non-Finite Axiomatizability I," in Infinitistic Methods: Proceedings of the Symposium on Foundations of Mathematics, (Warsaw, 2–9 September 1959). Pergamon: 45-69.
- Muller, F. A., (2001), "Sets, classes, and categories," British Journal of the Philosophy of Science 52: 539-73.
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- Pudlak, P., (1998), "The lengths of proofs" in Buss, S., ed., Handbook of Proof Theory. North-Holland: 547-637.
- von Neumann, John (1923), "Zur Einführung der transfiniten Zahlen", Acta litt. Acad. Sc. Szeged X. 1: 199–208. English translation: van Heijenoort, Jean (1967), "On the introduction of transfinite numbers", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 346–354.
- von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik 154: 219–240. English translation: van Heijenoort, Jean (1967), "An axiomatization of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 393–413.
- von Neumann, John (1928), "Die Axiomatisierung der Mengenlehre", Mathematische Zeitschrift 27: 669–752.
- von Neumann, John (1929), "Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre", Journal für die Reine und Angewandte Mathematik 160: 227–241.
Posts
I find this observation rather puzzling when talking of NBG as a conservative extension of ZFC, since the existence of the empty set is usually taken to be an axiom of ZFC even though it can be derived from other axioms of ZFC.
ReplyDeleteIn ZFC the existence of the empty set follows from the separation axiom schema, and from the replacement scheme (from which separation can be obtained).
To understand what is intended here, to make the claim intelligible. one really needs to know in what context infinity suffices to derive the existence of the empty set, since in the context of ZFC or NBG - {empty set, infinity} the existence of the empty set is already provable.
RBJ