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