Abstract
The way in which Martin Davis conceived the first chapter of his book “Applied nonstandard analysis ” is a brilliant example of information hiding as a guiding principle for the design of widely applicable constructions and methods of proof. We discuss here a common trait that we see between that book and another writing of the year 1977, “Metamathematical extensibility for theorem provers and proof-checkers”, which Martin coauthored with Jacob T. Schwartz . To tie the said part of Martin’s study on nonstandard analysis to proof technology, we undertake a verification, by means of a proof-checker based on set theory, of key results of the non-standard approach to analysis.
The reader who remembers these key points will do well in what follows. In particular, it is now quite all right to entirely forget how the nonstandard universe was defined and to banish ultrafilters from our consciousness.
(Martin Davis, Applied Nonstandard Analysis, 1977)
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Notes
- 1.
- 2.
Landmark contributions of Martin to automatic theorem-proving in 1st-order predicate logic have been [10, 13, 19, 20, 25], historically occurring between Paul C. Gilmore’s and Dag Prawitz’ methods, on the one hand, and J. Alan Robinson’s resolution principle on the other. Concerning the linked conjunct method then proposed by Martin and his team at Bell Labs , see [29, 39].
- 3.
- 4.
See [8, pp. 5–6]. In a recent personal web-page, David Aspinall (Univ. of Edinburgh) defines Proof Engineering to mean the activity on construction, maintenance, documentation and presentation of large formal proof developments. Within Proof Engineering , according to Aspinall, “Software Engineering provides the techniques to develop large, structured and well-specified repositories of computer code; proof checking provides the mechanisms to provide a complete semantics and formal correctness as an absolute quality criterion.”.
- 5.
In particular, when \(\mathsf {D}=\mathbb {R}\), we get a field, \({^*\!{\mathbb {R}}}\), of entities called hyperreal numbers. In \({^*\!{\mathbb {R}}}\) there are positive numbers lying infinitely close to zero; the reciprocals of such infinitesimals must, of course, exceed any positive integer.
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- 7.
A slicker characterization of ultrafilters will be shown in Fig. 10.7.
- 8.
In a similar attitude, [11, p. 54] states that “one possible view is that the integers are atoms and should not be viewed as sets. Even in this case, one might still wish to prevent the existence of unrestricted atoms. In any case, for the ‘genuine’ sets, Extensionality holds and the other sets are merely harmless curiosities.”.
- 9.
When the need will arise, we will adjust this notation also to terms, indicating by \(t(\varvec{c})\) a term devoid of variables resulting from replacement of a variable of t by a constant \(\varvec{c}\).
- 10.
About Ref’s built-in operator \(\varvec{arb}\left( X\right) \) that occurs thrice in Fig. 10.1, suffice it to say for the time being that it selects an element of its operand X when \(X\ne \emptyset \), and that \(\varvec{arb}\left( \emptyset \right) =\emptyset \).
- 11.
In a passage echoing Abraham Robinson’s ‘provocative remark’ which we have recalled in the Introduction through Martin’s words, Jack says about this ability of Theory s [35, p. 9]: “\(\cdots \)definitions serve to ‘instantiate’, that is, to introduce the objects whose special properties are crucial to an intended argument. Like the selection of crucial lines, points, and circles from the infinity of geometric elements that might be considered in a Euclidean argument, definitions of this kind often carry a proof’s most vital ideas”. A typical case of this kind is, in arithmetic, the selection of the least natural number that meets some key property.
- 12.
This is a specialized variant of the Theory \(\mathsf {reachability}\) presented in [35, Sect. 7.3]. As seen here, the formal output parameters of a Theory always carry a subscript \(\varTheta \).
- 13.
What follows is not meant to imply that the definition of \(\mathbb {N}\) shown is the ideal one.
- 14.
A common definition of ordinals , owing to a simplification due to Raphael Robinson , is:
$$ \begin{aligned} \mathsf {Ord}(U)\ \,\leftrightarrow _{\text {Def}}\,\ \forall \,x\,(x\in U{\varvec{\,\rightarrow \,}}x\subseteq U) \, \& \,\forall \,x\,\forall \,y\big (\left\{ x, y\right\} \subseteq U{\varvec{\,\rightarrow \,}}(x\in y\vee y\in x\vee x=y)\big )\,. \end{aligned}$$ - 15.
Natural numbers will play an irreplaceable role in the informal arguments providing the rationale for the formal constructions that follow; within the formal treatment, their collection \(\mathbb {N}\) will act as a set whose infinitude is easiest to prove (and infinite sets will be crucial in Sect. 10.8).
- 16.
One way of implementing lists is discussed in [30, pp. 127–128].
- 17.
This way of representing formulae in conjunctive normal form is widely used today. In recent years [32] resorted to it, to give a Ref-based correctness proof for the DPLL satisfiability algorithm.
- 18.
A website reporting on our experiment is at http://www2.units.it/eomodeo/InitialSetupForNonStandardAnalysis.html, http://aetnanova.units.it/scenarios/InitialSetupForNonStandardAnalysis/.
- 19.
In Ref the well-foundendess of membership and statements of the axiom of choice easily result from the availability of the construct \(\varvec{arb}\) discussed in Sect. 10.6, thanks to the interplay of \(\varvec{arb}\) with abstract set formers; infinity is embodied by Ref’s built-in constant \(\varvec{\sigma }_{\!\infty }\).
- 20.
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Acknowledgements
Discussions with Francesco Di Cosmo helped in polishing this paper. The first author acknowledges partial support from the Polish National Science Centre research project DEC-2011/02/A/HS1/00395; and the second author from the project FRA-UniTS (2014) “Learning specifications and robustness in signal analysis.
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Cantone, D., Omodeo, E.G., Policriti, A. (2016). Banishing Ultrafilters from Our Consciousness. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_10
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