Undecidability Theorem Research Articles

Gödel’s Undecidability Theorems and the Search for a Theory of Everything

I investigate the question whether Gödel’s undecidability theorems play a crucial role in the search for a unified theory of physics. I conclude that unless the structure of space-time is fundamentally discrete we can never decide whether a given theory is the final one or not. This is relevant for both canonical quantum gravity and string theory. Slightly elaborated version of a Prize winning essay awarded by the Kurt Gödel Circle of Friends Berlin with the support of the University of Wuppertal, first published in https://kurtgoedel.de/kurt-goedel-award-2023/

This Story Is a Lie by Tom Pollock

Reviewed by: This Story Is a Lie by Tom Pollock Karen Coats Pollock, Tom This Story Is a Lie. Soho Teen, 2018 [336p] ISBN 978-1-61695-911-1 $18.99 Reviewed from galleys R Gr. 9-12 British seventeen-year-old Peter Blankman looks for secret codes and patterns in everything, hoping to hold at bay the panic attacks that plague him. His neuroscientist mother is reassuring, but his chief ally has always been his twin sister, Bel, who is as fearless as Peter is fearful. Just prior to the start of an award ceremony to honor his mother, Peter flees the increasingly claustrophobic room only to hear her scream from behind him a few minutes later; she has been stabbed and Bel is missing. Peter and his mother are immediately whisked away to a secret facility, and a plot beyond even Peter's wildest fears begins to unfold. Packed with as much witty inner monologue as nonstop action, this heady mystery invites readers to consider everything from neuro-engineering to the indeterminate meaning of life. Peter's existential crises are those of a hyperintelligent, obsessional math geek as he tries to find the core of himself through a mathematical proof, only to have his world undone by Gödel's undecidability theorem. Readers who are not mathematically inclined, however, will still be swept along by hyperviolent and breathless escape sequences straight out of the latest cinematic spy thriller. Gaspworthy twists come out of nowhere, later to be explained by Peter's "recursions" into memories that, in geek speak, "encrypt" and ultimately "invert" the circumstances of his life that he has taken for granted as true, transforming his deep-seated paranoia into his saving virtue. Readers won't soon forget this very smart, thoroughly original science fiction action-adventure. Copyright © 2018 The Board of Trustees of the University of Illinois

Letter Games: a metamathematical taster

Metamathematics is the mathematical study of mathematics itself. Two of its most famous theorems were proved by Kurt Gödel in 1931. In a simplified form, Gödel's first incompleteness theorem states that no reasonable mathematical system can prove all the truths of mathematics. Gödel's second incompleteness theorem (also simplified) in turn states that no reasonable mathematical system can prove its own consistency. Another famous undecidability theorem is that the Continuum Hypothesis is neither provable nor refutable in standard set theory. Many of us logicians were first attracted to the field as students because we had heard something of these results. All research mathematicians know something of them too, and have at least a rough sense of why ‘we can't prove everything we want to prove’.

The undecidability theorem for the Horn-like fragment of linear logic (Revisited)

There has been an increased interest in the decision problems for linear logic and its fragments. Here, we give a fully self-contained, easy-to-follow, but fully detailed, direct and constructive proof of the undecidability of a very simple Horn-like fragment of linear logic, which is accessible to a wide range of people. Namely, we show that there is a direct correspondence between terminated computations of a Minsky machine M and cut-free linear logic derivations for a Horn-like sequent of the form \begin{equation*} \bang{\Phi_M},\ l_1 \vdash l_0, \end{equation*} where ΦM consists only of Horn-like implications of the following simple forms \begin{equation*} (l \llto l'),\ \ ((l\otimes r) \llto l'),\ \ (l\llto (r\otimes l')),\ \ and \ \ (l\llto (l'\oplus l'')), \end{equation*} where l1, l0, l, l′, l″ and r stand for literals.Neither negation, nor &, nor constants, nor embedded implications/bangs are used here.Furthermore, our particular correspondence constructed above provides decidability for each of the Horn-like fragments whenever we confine ourselves to any two forms of the above Horn-like implications, along with the complexity bounds that come from the proof.

Finite axiomatizability in Łukasiewicz logic

We classify every finitely axiomatizable theory in infinite-valued propositional Łukasiewicz logic by an abstract simplicial complex (V,Σ) equipped with a weight function ω:V→{1,2,…}. Using the Włodarczyk–Morelli solution of the weak Oda conjecture for toric varieties, we then construct a Turing computable one–one correspondence between (Alexander) equivalence classes of weighted abstract simplicial complexes, and equivalence classes of finitely axiomatizable theories, two theories being equivalent if their Lindenbaum algebras are isomorphic. We discuss the relationship between our classification and Markov’s undecidability theorem for PL-homeomorphism of rational polyhedra.

On Markov's Undecidability Theorem for Integer Matrices

We study a problem considered originally by A. Markov in 1947: Given two finitely generated matrix semigroups, determine whether or not they contain a common element. This problem was proved undecidable by Markov for 4 x 4 matrices, even in a very restrict form, and for 3 x 3 matrices by Krom in 1981. Here we give a new proof in the 3 x 3 case which gives undecidability in an almost as restricted form as the result of Markov.

The idealized quantum two-slit gedanken experiment revisited—Criticism and reinterpretation

Abstract An idealized two-slit experiment is envisaged in which the hypothetical experimental set-up is constructed in such a way as to resemble a toy model giving information about the structure of quantum space–time itself. Thus starting from a very simple equation which may be interpreted as a physical realization of Godel’s undecidability theorem, we proceed to show that space–time is very likely to be akin to a fuzzy Kahler-like manifold on the quantum level. This remarkable manifold transforms gradually into a classical space–time as we decrease the resolution in a way reversibly analogous to the processes of recovering classical space–time from the Riemannian space of general relativity. The paper’s main philosophy is to emphasize that the quintessence of the two-slit experiment as well as Feynman’s path integral could be given a different interpretation by altering our classical concept of space–time geometry and topology. In turn this would be in keeping with the development in theoretical physics since special and subsequently general relativity. In the final analysis it would seem that we have two different yet, from a positivistic philosophy viewpoint, completely equivalent alternatives to view quantum physics. Either we insist on what we see in our daily experiences, namely, a smooth four-dimensional space–time, and then accept, whether we like it or not, things such as probability waves and complex probabilities. Alternatively, we could see behind the facade of classical space–time a far more elaborate and highly complex fuzzy space–time with infinite hierarchical dimensions such as the so-called Fuzzy K3 or E –Infinity space–time and as a reward for this imaginative picture we can return to real probabilities without a phase and an almost classical picture with the concept of a particle’s path restored. We say almost classical because non-linear dynamics and deterministic chaos have long shown the central role of randomness in classical mechanics and this is reinforced once more in our model which is directly related not to Newtonian motion, but rather to a diffusion-like random walk similar to that used with great skill by Einstein and later on by Nagasawa and particularly the English-Canadian physicist Garnet Ord.

A new solution for the two-slit experiment

A completely new mathematical solution of the two-slit experiment is given. The corresponding physical model suggests, for the first time, that the wave-particle duality is simply an expression of the non-classical topology and geometry of quantum spacetime when projected into our 3 + 1 Euclidean space . However if we disregard the topological cause of the projected behaviour and concentrate only on what is the case, then the observed wave-particle behaviour may be regarded as a physical realization of Gödel’s undecidability theorem. The essence of the present solution may be summarized in the following statement: If we could perform the two-slit experiment in a laboratory of a size comparable to that of an elementary particle, the wave-particle duality would cease to exist. It is remarkable that we do not assume quantum mechanics but end up confirming its results.

The Diagonalization of Metaphysics

Lately, there has been increasing discussion of the possibility of discourse between two areas of philosophy formerly considered to be diametrically opposed: so-called Anglo-American (or analytical) philosophy, exemplified by such philosophers as Quine, and so-called Continental philosophy, exemplified by such philosophers as Heidegger. It is of course well known that analytical philosophy seeks to give philosophy the rigor of a science, while Continental philosophy tends to be skeptical about the very notion of rigor. Given this, it is perhaps not surprising that the impetus to reconcile the two kinds of philosophy comes from within the analytical camp rather than the Continental. What I want to do in the present paper is to further this reconciliation, and to demonstrate that both Continental philosophers, such as Heidegger and Derrida, and analytical philosophers such as Quine, make a similar kind of move-a move that can be seen to be a form of the so-called of mathematics. It is my hope that a byproduct of the approach I take will be a certain demystification of the thought of Derrida. At the very least, the essay will be able to be thought of as an account of an unexpected similarity in the thought of two very different traditions. Cantor's Diagonal Argument Given that I hope to show the to be a kind of bridge between the thought of such philosophers as Derrida and Quine, it is clearly necessary for me to spend some time discussing the argument. Cantor's is to be found at the heart of results as apparently diverse as Godel's undecidability theorems and the Pythagorean demonstration that the of a square is incommensurable with one of its sides. With regard to the latter, a square having a side measuring (for example) one inch has a whose length can never be specified to a finite number of decimal places (although its length can be given approximately: 1.4142 inches). Before going any further, however, I remark that in this essay my use of the term diagonal argument will be somewhat looser than is found in some discussions. Interested readers should consult books such as (Webb, 1980) for a mathematical definition of diagonalization. For my purposes, what is crucial in definitions such as Webb's is the idea of the derivation from the binary of something that is irreducible to the binary. A square is, in a sense, a binary figure (a square is two dimensional), but its cannot be reduced to the same kind of measurement used to express the length of the square's sides. One could say that associated with the binary is something that transcends it; and it is this aspect on which I shall be concentrating in my discussion of Derrida and Heidegger. So what is Cantor's argument? In essence, it forms part of Cantor's demonstration that the real numbers-numbers like 0.5 and -3.62 and 563.8678543, which can be expressed as arbitrarily long decimals-are not countable, even if one had forever to count them. First, Cantor assumed a listing of real numbers, which he set up as an array: Here, the subscripts refer to the rows and the columns: thus is the digit found in row 1, column 3. For example, if the number in the top row of the array were 8.273, a^sub 13^ would be 7. Next, Cantor considered the digits forming the number a^sub 11^, a ^sub 22^, a^sub 33^, a^sub 44^ .... Suppose, he said, there is defined a new number, each digit of which differs from the corresponding digit of the number by 1. Because such a number will differ in at least one digit from any number in the array, it can never appear as a row in the array. Consequently, if each row in the array is considered to define the decimal expansion of a real number, it is always possible to define a real number lying outside the array. The real numbers are thus not countable. The Diagonal Argument in Godel: Some Kantian Considerations As was mentioned, such argumentation underlies the derivation of other mathematical results, such as Godel's first undecidability theorem. …

An undecidability theorem for lattices over group rings

Let G be a finite group, T( Z[ G]) denote the theory of Z[ G]-lattices (i.e. finitely generated Z-torsionfree Z[ G]-modules). It is shown that T( Z[ G]) is undecidable when there are a prime p and a p-subgroup S of G such that S is cyclic of order p 4, or p is odd and S is non-cyclic of order p 2, or p = 2 and S is a non-cyclic abelian group of order 8 (notice that, in these cases, Z[ S]-lattices are a class of wild representation type). More precisely, first we prove that T( Z[ S]) is undecidable because it interprets the word problem for finite groups; then we lift undecidability from T( Z[ S]) to T( Z[ G]).

Complexity, decidability and undecidability results for domain-independent planning

In this paper, we examine how the complexity of domain-independent planning with STRIPS-style operators depends on the nature of the planning operators. We show conditions under which planning is decidable and undecidable. Our results on this topic solve an open problem posed by Chapman (1987), and clear up some difficulties with his undecidability theorems. For those cases where planning is decidable, we explain how the time complexity varies depending on a wide variety of conditions: • • whether or not function symbols are allowed; • • whether or not delete lists are allowed; • • whether or not negative preconditions are allowed; • • whether or not the predicates are restricted to be propositional (i.e., 0-ary); • • whether the planning operators are given as part of the input to the planning problem, or instead are fixed in advance. • • whether or not the operators can have conditional effects.

Peter B. Andrews. An introduction to mathematical logic and type theory: to truth through proof. Computer science and applied mathematics. Academic Press, Orlando etc. 1986, xv + 304 pp.

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.

Gregory Cherlin, Lou van den Dries, and Angus Macintyre. Decidability and undecidability theorems for PAC-fields. Bulletin of the American Mathematical Society, n.s. vol. 4 (1981), pp. 101–104.

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Brouwer's Anticipation of the Principle of Charity

ion now interacts with causal attention in that several causal sequences are subsumed under a common form or law. Here, according to Brouwer, lies the starting point of exact science. A Viennese digression. The history of ideas is full of striking coincidences of simultaneous discoveries of similar results by one or more researchers. The discovery of the planet Neptune is perhaps the most spectacular instance. The different, but equivalent, formulations of Quantum Mechanics by Heisenberg and Schr6dinger give another well-known example. The Foundations of Mathematics provide many instances around 1930: Tarski and Herbrand proved the Deduction Theorem, Gentzen and Jaskowski discovered Natural Deduction, Church and Turing proved the Undecidability theorem etc.. It is a striking coincidence that, at the very time when Brouwer delivered his 'Mathematik, Wissenschaft und Sprache' in Vienna, Carnap must have been busy preparing his Logische Aujbau der Welt for the press. (The preface is dated May 1928.) Carnap's infinitely more detailed account has many similarities with Brouwer's position. Brouwer constructs his world through, or perhaps better, tells us that his world is given through, among other things, persisting causal sequences. Carnap uses, as the basic elements of his construction system, what he calls 'elementary experiences'. (This is a primitive concept, but might perhaps be understood as a section of the stream of consciousness.) The central relation between these is that of a 'likeness recollection' and from this a world is built. Objects become regularities in the stream of elementary experiences (which strongly reminds one of Brouwer's 'identifications'). The name of Carnap's position-methodological solipsism-describes Brouwer's view in a most appropriate way. The crucial difference between the two is that Carnap is a 'constructionist' or translator. He wants to show that on the basis of his primitives he can, using the logic of Principia, give definitions of the concepts one needs in order to talk about the world. Brouwer, on the other hand, does nothing of the sort. He tells us what his world is like, and not what you would have to do to translate your world into his. Brouwer and his lecture do not get mentioned in the Aujbau, This content downloaded from 207.46.13.101 on Sat, 08 Oct 2016 05:59:01 UTC All use subject to http://about.jstor.org/terms

Is quantum theory universally valid?

Quantum theory has been criticized for not being deterministic and therefore not universal: it cannot completely describe a measurement aimed at verifying its predictions (although any given apparatus can be considered as a quantum system). We investigate the possible alternatives. Theories where the observed world is deterministic but the observer is not (whatever the reason for that) lead to Bell’s nonseparability theorem. If, on the other hand, the observer too is deterministic, the theory is not verifiable. It follows that quantum theory must be the logically preferred option. Its inability to completely describe the measurement process appears to be not a flaw of the theory but a logical necessity which is analogous to Gödel’s undecidability theorem.

Decidability and undecidability theorems for PAC-fields

Decidability and undecidability theorems for PAC-fields