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Trial and error mathematics. I: Dialectical and quasidialectical systems. (English) Zbl 1384.03079
Summary: We define and study quasidialectical systems, which are an extension of Magari’s dialectical systems, designed to make Magari’s formalization of trial and error mathematics more adherent to the real mathematical practice of revision: our proposed extension follows, and in several regards makes more precise, varieties of empiricist positions à la Lakatos. We prove several properties of quasidialectical systems and of the sets that they represent, called quasidialectical sets. In particular, we prove that the quasidialectical sets are \(\Delta^0_2\) sets in the arithmetical hierarchy. We distinguish between “loopless” quasidialectal systems, and quasidialectical systems “with loops”. The latter ones represent exactly those coinfinite c.e. sets, that are not simple. In a subsequent paper we will show that whereas the dialectical sets are \(\omega\)-c.e., the quasidialectical sets spread out throughout all classes of the Ershov hierarchy of the \(\Delta^0_2\) sets.

MSC:
03A10 Logic in the philosophy of science
03A05 Philosophical and critical aspects of logic and foundations
03D55 Hierarchies of computability and definability
03D80 Applications of computability and recursion theory
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References:
[1] Annali di Matematica Pura ed Applicata. Series IV XCVIII pp 119– (1974)
[2] The Philosophy of Mathematical Practice (2008) · Zbl 1163.03001
[3] DOI: 10.1093/bjps/27.3.201 · Zbl 0364.00028 · doi:10.1093/bjps/27.3.201
[4] Proofs and Refutations (1976)
[5] Cognition 32 pp 137– (1986)
[6] DOI: 10.2307/2271872 · Zbl 0428.03043 · doi:10.2307/2271872
[7] Problems in the Philosophy of Mathematics pp 138– (1967)
[8] DOI: 10.2307/2270580 · Zbl 0203.01201 · doi:10.2307/2270580
[9] The nature of mathematical knowledge (1983) · Zbl 0519.00022
[10] Matematiche XXIX pp 1– (1974)
[11] The Logic of Reliable Inquiry (1996) · Zbl 0910.03023
[12] Computability: Turing, Gödel, Church, and Beyond pp 1– (2013)
[13] Journal of Philosophical Logic 4 pp 53– (1975)
[14] Computability Theory (2003)
[15] Language, Truth, and Logic in Mathematics pp 174– (1988)
[16] DOI: 10.1007/978-94-015-9558-2_12 · doi:10.1007/978-94-015-9558-2_12
[17] Bollettino della Unione Matematica Italiana. Series IV 9 pp 51– (1974)
[18] DOI: 10.1145/356914.356918 · doi:10.1145/356914.356918
[19] DOI: 10.1093/philmat/4.3.256 · Zbl 0881.03003 · doi:10.1093/philmat/4.3.256
[20] Recursively Enumerable Sets and Degrees (1987)
[21] DOI: 10.1016/j.tcs.2011.12.040 · Zbl 1279.68089 · doi:10.1016/j.tcs.2011.12.040
[22] DOI: 10.1305/ndjfl/1093637647 · Zbl 0645.03038 · doi:10.1305/ndjfl/1093637647
[23] Theory of Recursive Functions and Effective Computability (1967) · Zbl 0183.01401
[24] DOI: 10.2307/2270581 · Zbl 0193.30102 · doi:10.2307/2270581
[25] DOI: 10.2307/2274708 · Zbl 0763.03023 · doi:10.2307/2274708
[26] DOI: 10.1007/s001530050043 · Zbl 0854.03020 · doi:10.1007/s001530050043
[27] Deduction, Computation, Experiment pp 65– (2008)
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