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A Self-contained Beginning for Ref’s Main Proof Scenario

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Abstract

This chapter presents in full a group of formalized proofs that reaches in a small number of pages many results about ordinals, various properties of the transitive closure operation, transfinite induction, and then Zorn’s lemma. The proofs of a few basic facts concerning finite sets, including a finite induction principle, are also included.

In preparation for the study of ordinals, the notion of reachability in a ‘big’ graph, which is a system whose nodes and arcs might form proper classes, is treated formally.

To begin developing an acceptable formal treatment of finiteness without much preparatory work, the following definition is adopted: a set F is finite if every nonnull family of subsets of F owns an inclusion-minimal element.

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  • DOI: 10.1007/978-0-85729-808-9_7
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References

  1. Aczel, P.: Non-Well-Founded Sets. CSLI Lecture Notes, vol. 14. CLSI, Stanford (1988)

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  2. Peddicord, R.: The number of full sets with n elements. Proc. Am. Math. Soc. 13(5), 825–828 (1962)

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  3. Tarski, A.: Sur les ensembles fini. Fundam. Math. VI, 45–95 (1924)

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Correspondence to Domenico Cantone or Eugenio G. Omodeo .

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© 2011 Springer-Verlag London Limited

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Schwartz, J.T., Cantone, D., Omodeo, E.G. (2011). A Self-contained Beginning for Ref’s Main Proof Scenario. In: Computational Logic and Set Theory. Springer, London. https://doi.org/10.1007/978-0-85729-808-9_7

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  • DOI: https://doi.org/10.1007/978-0-85729-808-9_7

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-807-2

  • Online ISBN: 978-0-85729-808-9

  • eBook Packages: Computer ScienceComputer Science (R0)