Recursive Functions and Metamathematics

Murawski, Roman

doi:10.1007/978-94-017-2866-9

Cited by 17 publications

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“…6 Proofs of those theorems can be found in (Murawski, 1999a and1999b). 7 In fact it can be proved even in the fragment of Peano arithmetic with induction for Σ 0 1 formulas only.…”

Section: If γ Is a Class Of Formulas Of The Language L(a −mentioning

confidence: 99%

“…Detailed information on the arithmetization and on the arithmetical counterparts of various metamathematical notions can be found, e.g., inMendelson (1964),Shoenfield (1967) orMurawski (1999b).…”

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confidence: 99%

“…The proof of this theorem can be found, e.g., inMurawski's (1999b). 5 More information on this problem can be found in(Woleński, 1991) and(Murawski, …”

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confidence: 99%

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Troubles with (the concept of) truth in mathematics

Murawski

2007

LLP

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“…6 Proofs of those theorems can be found in (Murawski, 1999a and1999b). 7 In fact it can be proved even in the fragment of Peano arithmetic with induction for Σ 0 1 formulas only.…”

Section: If γ Is a Class Of Formulas Of The Language L(a −mentioning

confidence: 99%

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confidence: 99%

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Troubles with (the concept of) truth in mathematics

Murawski

2007

LLP

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“…So, informally, f would be considered computable if we had an algorithm or decision procedure such that, for each number k, gives us the value of f (k) if k is in the domain of f , and runs forever if k is not in the domain of f . Sets (of natural numbers) 9 I omit here the definitions of these notions regarding number functions. We have connection: a set (a relation) is strongly representable (definable) in Σ iff its charcteristic function is representable (definable) in Σ.…”

Section: Computability Recursiveness and Representabilitymentioning

confidence: 99%

Analogy and diagonal argument

Zbigniew

2006

LLP

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Abstract. In this paper, I try to accomplish two goals. The first is to provide a general characterization of a method of proofs called -in mathematics -the diagonal argument. The second is to establish that analogical thinking plays an important role also in mathematical creativity. Namely, mathematical research make use of analogies regarding general strategies of proof. Some of mathematicians, for example George Polya, argued that deductions is impotent without analogy. What I want to show is that there exists a direct line leading from Cantor's diagonal argument to constructions that underlies of the proofs of several important theorems of the mathematical logic (in particular, Church's theorem concerning the undecidability of formal arithmetic, Gödel's theorem concerning the incopleteness of formal arithmetic, Tarski's theorem concerning truth, and Turing's theorem concerning the Halting Problem), and that the line could be described as an analogical mapping. In other words, Cantor's diagonal argument and the proofs of the limitative theorems are structurally the same. Hence they can be represented as instances (or special cases) of the same general scheme.

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“… Compare Awodey and Reck [2002] for a survey of contributions to the (meta-)theory of formal axiomatics in the late 19th and early 20th centuries.55 SeeJané [2006] for more on the axiomatic background of Tarski's paper.56 The terminology of "schematic terms" in this context, while not widespread in mathematical logic, is fairly common in the philosophical literature; see, e.g.,Quine [1986].…”

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Logic in the 1930s: Type Theory and Model Theory

Schiemer¹,

Reck²

2013

Bull. symb. log

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In historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style of Principia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing several early attempts to develop the semantics of axiomatic theories in the 1930s, by two proponents of the type-theoretic tradition (Carnap and Tarski) and two proponents of the first-order tradition (Gödel and Hilbert), we argue that, instead, the move from type theory to first-order logic is better understood as a gradual transformation, and further, that the contributions to semantics made in the type-theoretic tradition should be seen as central to the evolution of model theory.

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Recursive Functions and Metamathematics

Cited by 17 publications

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Troubles with (the concept of) truth in mathematics

Troubles with (the concept of) truth in mathematics

Analogy and diagonal argument

Logic in the 1930s: Type Theory and Model Theory

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