Gödel’s Second Incompleteness Theorem

1979 ◽  
pp. 72-84
Author(s):  
Richard E. Grandy
Keyword(s):  
Incompleteness Theorem

scholarly journals FINDING THE LIMIT OF INCOMPLETENESS I

2020 ◽  
Vol 26 (3-4) ◽  
pp. 268-286
Author(s):  
YONG CHENG
Keyword(s):  
Model Theory ◽  
Recursion Theory ◽  
Minimal Degree ◽  
Incompleteness Theorem ◽  
Turing Degree ◽  
Turing Reducibility ◽  
Inseparable Pair

AbstractIn this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ 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 $\textsf {G1}$ holds.

INCOMPLETENESS THEOREM AND ITS FRONTIER

2005 ◽  
pp. 434-439
Author(s):  
GAISI TAKEUTI
Keyword(s):  
Incompleteness Theorem

scholarly journals Relevant Arithmetic and Mathematical Pluralism

2021 ◽  
Vol 18 (5) ◽  
pp. 569-596
Author(s):  
Zach Weber
Keyword(s):  
Mathematical Practice ◽  
Worked Examples ◽  
Incompleteness Theorem ◽  
Classical Counterpart ◽  
The One ◽  
Mathematical Pluralism

In The Consistency of Arithmetic and elsewhere, Meyer claims to “repeal” Goedel’s second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpart—a corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few worked examples from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture.

scholarly journals A short remark on Gödel incompleteness theorem and its self-referential paradox from Neutrosophic Logic perspective

2019 ◽  
pp. 21-26
Author(s):  
V. Christianto ◽  
◽  
◽  
F. Smarandache
Keyword(s):  
20Th Century ◽  
History Of Mathematics ◽  
The Self ◽  
Incompleteness Theorem ◽  
Aristotelian Logic ◽  
Binary Logic ◽  
Neutrosophic Logic ◽  
History Of ◽  
Incompleteness Theorems ◽  
Modern Mathematics

It is known from history of mathematics, that Gödel submitted his two incompleteness theorems, which can be considered as one of hallmarks of modern mathematics in 20th century. Here we argue that Gödel incompleteness theorem and its self-referential paradox have not only put Hilbert’s axiomatic program into question, but he also opened up the problem deep inside the then popular Aristotelian Logic. Although there were some attempts to go beyond Aristotelian binary logic, including by Lukasiewicz’s three-valued logic, here we argue that the problem of self-referential paradox can be seen as reconcilable and solvable from Neutrosophic Logic perspective. Motivation of this paper: These authors are motivated to re-describe the self-referential paradox inherent in Godel incompleteness theorem. Contribution: This paper will show how Neutrosophic Logic offers a unique perspective and solution to Godel incompleteness theorem.

scholarly journals Methodological Frames: Paul Bernays, Mathematical Structuralism, and Proof Theory

2020 ◽  
pp. 352-382
Author(s):  
Wilfried Sieg
Keyword(s):  
Proof Theory ◽  
Incompleteness Theorem ◽  
Mathematical Structures ◽  
General Characterization ◽  
Mathematical Structuralism ◽  
Mathematical Experience

Mathematical structuralism is deeply connected with Hilbert and Bernays’s proof theory and its programmatic aim to ensure the consistency of all of mathematics. That aim was to be reached on the basis of finitist mathematics. Gödel’s second incompleteness theorem forced a step from absolute finitist to relative constructivist proof-theoretic reductions. This mathematical step was accompanied by philosophical arguments for the special nature of the grounding constructivist frameworks. Against that background, this chapter examines Bernays’s reflections on proof-theoretic reductions of mathematical structures to methodological frames via projections. However, these reflections are focused on narrowly arithmetic features of frames. Drawing on broadened meta-mathematical experience, this chapter proposes a more general characterization of frames that has ontological and epistemological significance. The characterization is given in terms of accessibility: domains of objects are accessible if their elements are inductively generated, and principles for such domains are accessible if they are grounded in our understanding of the generating processes.

The Director’s Cut

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
David Berlinski
Keyword(s):  
Incompleteness Theorem

One of the twentieth century’s great intellectual achievements is Kurt Gödel’s first incompleteness theorem. Gödel’s work remains difficult, uncanny in its power. David Berlinski writes that the phrase the director’s cut is useful “because it sets the scene: the masterful director; his view; his vision. Mathematics has always been rich in its directors of note.” Gödel and Peano are among them.

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