Abstract
In [3] a general logical framework for formalizing set theories of different strength was suggested. We here employ that framework, focusing on the exploration of computational theories. That is, theories whose set of closed terms suffices for denoting every concrete set (including infinite ones) that might be needed in applications, as well as for computations with sets. We demonstrate that already the minimal computational level of the framework, in which only a minimal computational theory and a minimal computational universe are employed, suffices for developing large portions of scientifically applicable mathematics.
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Notes
- 1.
Already in [9] it was argued that “a main asset gained from Set theory is the ability to base reasoning on just a handful of axiom schemes which, in addition to being conceptually simple (even though surprisingly expressive), lend themselves to good automated support”. More recently, H. Friedman wrote (in a message on FOM on Sep 14, 2015): “I envision a large system and various important weaker subsystems. Since so much math can be done in systems much weaker than ZFC, this should be reflected in the choice of Gold Standards. There should be a few major Gold Standards ranging from Finite Set Theory to full blown ZFC”.
- 2.
- 3.
The thesis that \(J_{2}\) is sufficient for core mathematics was already put forward in [33].
- 4.
- 5.
Due to page constraints, all proofs in the paper were omitted, and will appear in an extended version of the current paper.
- 6.
Though the official language does not include \(\forall \) and \(\rightarrow \), since we assume classical logic we take \(\forall x_{1}...\forall x_{n}\left( \varphi \rightarrow \psi \right) \) as an abbreviation for \(\lnot \exists x_{1}...\exists x_{n}\left( \varphi \wedge \lnot \psi \right) \).
- 7.
\(RST^{FOL}\) can be shown to be equivalent to the system obtained from Gandy’s basic set theory [20] by adding to it the Restricted \(\in \)-induction schema.
- 8.
Rudimentary functions are obtained by omitting the recursion schema from the usual list of schemata for primitive recursive set functions (see, e.g., [14]).
- 9.
\(v\left[ x:=a\right] \) denotes the x-variant of v which assigns a to x.
- 10.
The use of n-ary predicates can standardly be reduced, of course, to unary predicates.
- 11.
Two other ideas that appear in the sequel were adopted from [33]: treating the collection of reals as a proper class, and the use of codes for handling certain classes. It should nevertheless be emphasized that the framework in [33] is exclusively based on semantical considerations, and it is unclear how it can be turned into a formal theory like ZF or PA (and it is certainly not suitable for mechanization as is).
- 12.
In this paper, as in standard mathematical textbooks, the term “function” is used both for collections of ordered pairs and for set-theoretical operations (such as \(\cup \)).
- 13.
We abbreviate by \(z\check{=}\left\langle x,y\right\rangle \) and \(\left\langle x,y\right\rangle \check{\in }z\) the two formulas that are provably equivalent to \(z=\left\langle x,y\right\rangle \) and \(\left\langle x,y\right\rangle \in z\) and are safe w.r.t. \(\left\{ x,y\right\} \) which were introduced in [5].
- 14.
The “basic properties” of a certain object is of course a fuzzy notion. However, it is not difficult to identify its meaning in each particular case, as will be demonstrated in several examples below.
- 15.
It can be shown that the power of full induction over \(\mathbb {N}\) (i.e. for any formula \(\varphi \)) can be achieved by adding to \(RST_{HF}^{FOL}\) the full \(\in \)-induction scheme.
- 16.
As noted in Footnote 4, some of claims in the sequel have counterparts in [5]. However, the minimality restriction on the universe employed in this paper, which in turn requires the use of classes, makes a crucial difference.
- 17.
Notice that \(\mathbb {Q}^{+}\) is a \(\succ \)-set and \(\mathbb {R}^{+}\) is a \(\succ \)-class.
- 18.
It should be noted that the least upper bound principle is not derivable for all subsets also in Weyl’s approach [34]. We next use similar coding techniques to the ones employed by Weyl to obtain the principle for standard mathematical objects.
- 19.
In [33] such codings are called “proxies”.
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Acknowledgements
The second author is supported by: Fulbright Post-doctoral Scholar program; Weizmann Institute of Science – National Postdoctoral Award program for Advancing Women in Science; Eric and Wendy Schmidt Postdoctoral Award program for Women in Mathematical and Computing Sciences.
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Avron, A., Cohen, L. (2018). A Minimal Computational Theory of a Minimal Computational Universe. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2018. Lecture Notes in Computer Science(), vol 10703. Springer, Cham. https://doi.org/10.1007/978-3-319-72056-2_3
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