29 research outputs found

    Troubles with (the concept of) truth in mathematics

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    In the paper the problem of definability and undefinability of the concept of satisfaction and truth is considered. Connections between satisfaction and truth on the one hand and consistency of certain systems of omega-logic and transfinite induction on the other are indicated

    Church's thesis and its epistemological status

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    The aim of this paper is to present the origin of Church's thesis and the main arguments in favour of it as well as arguments against it. Further the general problem of the epistemological status of the thesis will be considered, in particular the problem whether it can be treated as a definition and whether it is provable or has a definite truth-value

    Decidability vs. undecidability. Logico-philosophico-historical remarks

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    The aim of the paper is to present the decidability problems from a philosophical and historical perspective as well as to indicate basic mathematical and logical results concerning (un)decidability of particular theories and problems

    Truth vs. provability – philosophical and historical remarks

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    Since Plato, Aristotle and Euclid the axiomatic method was considered as the best method to justify and to organize mathematical knowledge. The first mature and most representative example of its usage in mathematics were Elements of Euclid. They established a pattern of a scientific theory and in particular a paradigm in mathematics

    On the depth of G\"{o}del's incompleteness theorem

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    In this paper, we use G\"{o}del's incompleteness theorem as a case study for investigating mathematical depth. We take for granted the widespread judgment by mathematical logicians that G\"{o}del's incompleteness theorem is deep, and focus on the philosophical question of what its depth consists in. We focus on the methodological study of the depth of G\"{o}del's incompleteness theorem, and propose three criteria to account for its depth: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of G\"{o}del's incompleteness theorem.Comment: 23 pages, revised version. arXiv admin note: text overlap with arXiv:2009.0488

    Finding the limit of incompleteness I

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    In this paper, we examine the limit of applicability of G\"{o}del's first incompleteness theorem (G1\sf G1 for short). We first define the notion "G1\sf G1 holds for the theory TT". This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which G1\sf G1 holds. To approach this question, we first examine the following question: is there a theory TT such that Robinson's R\mathbf{R} interprets TT but TT does not interpret R\mathbf{R} (i.e. TT is weaker than R\mathbf{R} w.r.t. interpretation) and G1\sf G1 holds for TT? In this paper, we show that there are many such theories based on Je\v{r}\'{a}bek's work using some model theory. We prove that for each recursively inseparable pair ⟨A,B⟩\langle A,B\rangle, we can construct a r.e. theory U⟨A,B⟩U_{\langle A,B\rangle} such that U⟨A,B⟩U_{\langle A,B\rangle} is weaker than R\mathbf{R} w.r.t. interpretation and G1\sf G1 holds for U⟨A,B⟩U_{\langle A,B\rangle}. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree 0<d<0′\mathbf{0}< \mathbf{d}<\mathbf{0}^{\prime}, there is a theory TT with Turing degree d\mathbf{d} such that G1\sf G1 holds for TT and TT is weaker than R\mathbf{R} w.r.t. Turing reducibility. As a corollary, based on Shoenfield's work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which G1\sf G1 holds.Comment: 18 pages. Accepted and to appear in Bulletin of Symbolic Logi

    On the relationship between meta-mathematical properties of theories

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    In this work, we understand incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationship between the following twelve important metamathematical properties of theories: Rosser, EI (Effectively inseparable), RI (Recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, 0′\mathbf{0}^{\prime} (theories with Turing degree 0′\mathbf{0}^{\prime}), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties PP and QQ of these properties, we examine whether the property PP implies the property QQ, and whether the property QQ implies the property PP.Comment: 23 page

    Current research on G\"odel's incompleteness theorems

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    We give a survey of current research on G\"{o}del's incompleteness theorems from the following three aspects: classifications of different proofs of G\"{o}del's incompleteness theorems, the limit of the applicability of G\"{o}del's first incompleteness theorem, and the limit of the applicability of G\"{o}del's second incompleteness theorem.Comment: 54 pages, final accepted version, to appear in The Bulletin of Symbolic Logi
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