scholarly journals Distilling the Requirements of Gödel’s Incompleteness Theorems with a Proof Assistant

2021 ◽  
Author(s):  
Andrei Popescu ◽  
Dmitriy Traytel
Keyword(s):  
Standard Model ◽  
Sufficient Conditions ◽  
Semantic Interpretation ◽  
Proof Assistant ◽  
The Standard Model ◽  
Alternative Approaches ◽  
Incompleteness Theorems ◽  
Gödel’S Incompleteness Theorems

AbstractWe present an abstract development of Gödel’s incompleteness theorems, performed with the help of the Isabelle/HOL proof assistant. We analyze sufficient conditions for the applicability of our theorems to a partially specified logic. In addition to the usual benefits of generality, our abstract perspective enables a comparison between alternative approaches from the literature. These include Rosser’s variation of the first theorem, Jeroslow’s variation of the second theorem, and the Świerczkowski–Paulson semantics-based approach. As part of the validation of our framework, we upgrade Paulson’s Isabelle proof to produce a mechanization of the second theorem that does not assume soundness in the standard model, and in fact does not rely on any notion of model or semantic interpretation.

UNIVERSAL ROSSER PREDICATES

2017 ◽  
Vol 82 (1) ◽  
pp. 292-302 ◽  
Author(s):  
MAKOTO KIKUCHI ◽  
TAISHI KURAHASHI
Keyword(s):  
Standard Model ◽  
Initial Segment ◽  
Nonstandard Model ◽  
Complete Extension ◽  
The Standard Model ◽  
Nonstandard Models ◽  
Incompleteness Theorems ◽  
Image Position ◽  
Gödel’S Incompleteness Theorems

AbstractGödel introduced the original provability predicate in the proofs of Gödel’s incompleteness theorems, and Rosser defined a new one. They are equivalent in the standard model ${\mathbb N}$ of arithmetic or any nonstandard model of ${\rm PA} + {\rm Con_{PA}} $, but the behavior of Rosser’s provability predicate is different from the original one in nonstandard models of ${\rm PA} + \neg {\rm Con_{PA}} $. In this paper, we investigate several properties of the derivability conditions for Rosser provability predicates, and prove the existence of a Rosser provability predicate with which we can define any consistent complete extension of ${\rm PA}$ in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. We call it a universal Rosser predicate. It follows from the theorem that the true arithmetic ${\rm TA}$ can be defined as the set of theorems of ${\rm PA}$ in terms of a universal Rosser predicate in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. By using this theorem, we also give a new proof of a theorem that there is a nonstandard model M of ${\rm PA} + \neg {\rm Con_{PA}} $ such that if N is an initial segment of M which is a model of ${\rm PA} + {\rm Con_{PA}} $ then every theorem of ${\rm PA}$ in N is a theorem of $\rm PA$ in ${\mathbb N}$. In addition, we prove that there is a Rosser provability predicate such that the set of theorems of $\rm PA$ in terms of the Rosser provability predicate is inconsistent in any nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $.

scholarly journals A MACHINE-ASSISTED PROOF OF GÖDEL’S INCOMPLETENESS THEOREMS FOR THE THEORY OF HEREDITARILY FINITE SETS

2014 ◽  
Vol 7 (3) ◽  
pp. 484-498 ◽  
Author(s):  
LAWRENCE C. PAULSON
Keyword(s):  
Set Theory ◽  
Incompleteness Theorem ◽  
Proof Assistant ◽  
Mechanical Verification ◽  
Finite Set ◽  
Technical Issues ◽  
Incompleteness Theorems ◽  
Isabelle Proof Assistant ◽  
Finite Set Theory ◽  
Gödel’S Incompleteness Theorems

AbstractA formalization of Gödel’s incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows Świerczkowski (2003), who gave a detailed proof using hereditarily finite set theory. The adoption of this theory is generally beneficial, but it poses certain technical issues that do not arise for Peano arithmetic. The formalization itself should be useful to logicians, particularly concerning the second incompleteness theorem, where existing proofs are lacking in detail.

Indirect measurements in micro-space with the EM

1995 ◽  
Vol 53 ◽  
pp. 598-599
Author(s):  
Sterling P. Newberry
Keyword(s):  
Electron Microscope ◽  
Standard Model ◽  
Particle Physics ◽  
Information Storage ◽  
Storage Space ◽  
Indirect Measurements ◽  
Storage And Retrieval ◽  
Adequate Information ◽  
Compelling Reason ◽  
The Standard Model

At the 1958 meeting of our society, then known as EMSA, the author introduced the concept of microspace and suggested its use to provide adequate information storage space and the use of electron microscope techniques to provide storage and retrieval access. At this current meeting of MSA, he wishes to suggest an additional use of the power of the electron microscope.The author has been contemplating this new use for some time and would have suggested it in the EMSA fiftieth year commemorative volume, but for page limitations. There is compelling reason to put forth this suggestion today because problems have arisen in the “Standard Model” of particle physics and funds are being greatly reduced just as we need higher energy machines to resolve these problems. Therefore, any techniques which complement or augment what we can accomplish during this austerity period with the machines at hand is worth exploring.

An Interview with the Physicist that Her Head in the Universe and Both Feet on the Earth

2019 ◽  
Author(s):  
Adib Rifqi Setiawan
Keyword(s):  
Standard Model ◽  
Particle Physics ◽  
Space Time ◽  
Additional Dimension ◽  
The Earth ◽  
The Standard Model ◽  
The Universe

Put simply, Lisa Randall’s job is to figure out how the universe works, and what it’s made of. Her contributions to theoretical particle physics include two models of space-time that bear her name. The first Randall–Sundrum model addressed a problem with the Standard Model of the universe, and the second concerned the possibility of a warped additional dimension of space. In this work, we caught up with Randall to talk about why she chose a career in physics, where she finds inspiration, and what advice she’d offer budding physicists. This article has been edited for clarity. My favourite quote in this interview is, “Figure out what you enjoy, what your talents are, and what you’re most curious to learn about.” If you insterest in her work, you can contact her on Twitter @lirarandall.

An Interview with the Physicist that Her Head in the Universe and Both Feet on the Earth

2019 ◽  
Author(s):  
Adib Rifqi Setiawan
Keyword(s):  
Standard Model ◽  
Particle Physics ◽  
Space Time ◽  
Additional Dimension ◽  
The Earth ◽  
The Standard Model ◽  
The Universe

Put simply, Lisa Randall’s job is to figure out how the universe works, and what it’s made of. Her contributions to theoretical particle physics include two models of space-time that bear her name. The first Randall–Sundrum model addressed a problem with the Standard Model of the universe, and the second concerned the possibility of a warped additional dimension of space. In this work, we caught up with Randall to talk about why she chose a career in physics, where she finds inspiration, and what advice she’d offer budding physicists. This article has been edited for clarity. My favourite quote in this interview is, “Figure out what you enjoy, what your talents are, and what you’re most curious to learn about.” If you insterest in her work, you can contact her on Twitter @lirarandall.

Enhanced Attribute-Based Authenticated Key Agreement Protocol in the Standard Model

2014 ◽  
Vol 36 (10) ◽  
pp. 2156-2167
Author(s):  
Qiang LI ◽  
Deng-Guo FENG ◽  
Li-Wu ZHANG ◽  
Zhi-Gang GAO
Keyword(s):  
Standard Model ◽  
Key Agreement ◽  
Authenticated Key Agreement ◽  
Key Agreement Protocol ◽  
The Standard Model
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