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Sunday, June 16, 2013

Grice -- the axiom of infinity and the von Neumann-Bernays-Gödel axioms -- infinity formulated so as to imply the existence of the empty set (cfr. Grice on "Vacuous Names")

Speranza

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, y \cup \{y\} is also a member of x.

Infinity can be formulated so as to imply the existence of the empty set.

REFERENCES

 

  • 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 .
  • 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.
  • Müller, Gurt, ed. (1976), Sets and Classes: On the Work of Paul Bernays, Amsterdam: North Holland .
  • Potter, Michael, (2004), Set Theory and Its Philosophy. Oxford Univ. Press.
  • 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 .

1 comment:

  1. 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.

    In 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

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