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A Closer Examination of the Sequence of Definitions and Theorems Presented in this Book

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Abstract

In parallel with the design and implementation (at least as a prototype) of a proof-verification system based on set theory, the authors undertook the development of a large-scale proof scenario. Ideally, to demonstrate that the verifier can certify the correctness of a substantial body of mathematical analysis, this proof scenario should have culminated in the proof of the celebrated Cauchy integral theorem on analytic functions (whose statement is recalled at the end of this chapter). This presupposed proofs of the basic properties of the real and complex number systems defined in set-theoretic terms, the fundamental properties of limits, continuity and the differential and integral calculus.

This chapter shows the salient steps leading toward that (as yet) unachieved goal. A broad survey of main definitions and theorems is expanded.

Keywords

  • Proof Scenario
  • Cauchy Integral Theorem
  • Unachievable Goal
  • Ultimate Member
  • Single-valued Image

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    The largest proof scenario ever submitted to our verifier is available at http://setl.dyndns.org/EtnaNova/login/common_scenario.txt as raw text, and at http://setl.dyndns.org/EtnaNova/login/search_folder/scenario.pdf as a pretty-printed pdf-file.

  2. 2.

    To distinguish the image resulting from application of a map f to an element x from the image of the same x under a global function g, we will denote the former as f[x] and the latter as g(x).

  3. 3.

    Following Tarski [Tar24] we could have adopted the following alternative definition of finite sets, equivalent for any practical purpose to the definition given here:

    $$ {\rm Finite(}f{\rm )}{\kern 1pt}\leftrightarrow _{{\rm Def}} {\kern 1pt} \left( {\forall {\kern 1pt} g{\kern 1pt}\in {\kern 1pt} P\left( {P(f)} \right)\backslash \{ \emptyset\} {\kern 1pt} |{\kern 1pt} \left( {\exists m{\kern 1pt} |{\kern 1pt} g{\kern 1pt}\cap {\kern 1pt} P(m){\kern 1pt}= {\kern 1pt} \{ m\} } \right)} \right), $$

    i.e., f is finite if and only if every non-null set g constituted by subsets of f owns an inclusion-minimal element m. This will be shown in Sect. 7.5.

  4. 4.

    From this point on it is immaterial whether the reals have been introduced as Dedekind cuts or as equivalence classes of rational Cauchy sequences.

  5. 5.

    Let the view of real numbers as Dedekind cuts momentarily surface again. To be consistent with the approach that sees reals as equivalence classes of rational Cauchy sequences, in the formal specification given below we should apply an ad hoc ‘least upper bound’ operation (rather than the union operation) to a set of reals. Sloppiness on this point gives us the opportunity to signal a little advantage of the approach based on Dedekind cuts, which can represent the l.u.b. operation, −∞, and +∞ by \(\bigcup\), ∅, and ℚ, respectively.

References

  1. Bridges, D.S.: Foundations of Real and Abstract Analysis. Graduate Texts in Mathematics, vol. 174. Springer, Berlin (1997)

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  2. Cantor, G.: Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen. Math. Ann. 5, 123–132 (1872)

    MathSciNet  MATH  CrossRef  Google Scholar 

  3. Landau, E.: Foundation of Analysis. The Arithmetic of Whole, Rational, Irrational and Complex Numbers, 3rd edn. Chelsea, New York (1966)

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  4. Omodeo, E.G., Schwartz, J.T.: A ‘Theory’ mechanism for a proof-verifier based on first-order set theory. In: Kakas, A., Sadri, F. (eds.) Computational Logic: Logic Programming and Beyond—Essays in honour of Bob Kowalski, Part II, vol. 2408, pp. 214–230. Springer, Berlin (2002)

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  5. 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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Schwartz, J.T., Cantone, D., Omodeo, E.G. (2011). A Closer Examination of the Sequence of Definitions and Theorems Presented in this Book. In: Computational Logic and Set Theory. Springer, London. https://doi.org/10.1007/978-0-85729-808-9_5

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  • DOI: https://doi.org/10.1007/978-0-85729-808-9_5

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-807-2

  • Online ISBN: 978-0-85729-808-9

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