A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points

Following F. William Lawvere, we show that many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the same simple scheme. We demonstrate these similarities by showing how this simple scheme encompasses the semantic paradoxes, and how they arise as diagonal arguments and fixed point theorems in logic, computability theory, complexity theory and formal language theory

Review of 'The Outer Limits of Reason' by Noson Yanofsky 403p (2013) (review revised 2019)

I give a detailed review of 'The Outer Limits of Reason' by Noson Yanofsky from a unified perspective of Wittgenstein and evolutionary psychology. I indicate that the difficulty with such issues as paradox in language and math, incompleteness, undecidability, computability, the brain and the universe as computers etc., all arise from the failure to look carefully at our use of language in the appropriate context and hence the failure to separate issues of scientific fact from issues of how language works. I discuss Wittgenstein's views on incompleteness, paraconsistency and undecidability and the work of Wolpert on the limits to computation. To sum it up: The Universe According to Brooklyn---Good Science, Not So Good Philosophy. Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019) and Suicidal Utopian Delusions in the 21st Century 4th ed (2019

What Do Paraconsistent, Undecidable, Random, Computable and Incomplete mean? A Review of Godel's Way: Exploits into an undecidable world by Gregory Chaitin, Francisco A Doria, Newton C.A. da Costa 160p (2012) (review revised 2019)

In ‘Godel’s Way’ three eminent scientists discuss issues such as undecidability, incompleteness, randomness, computability and paraconsistency. I approach these issues from the Wittgensteinian viewpoint that there are two basic issues which have completely different solutions. There are the scientific or empirical issues, which are facts about the world that need to be investigated observationally and philosophical issues as to how language can be used intelligibly (which include certain questions in mathematics and logic), which need to be decided by looking at how we actually use words in particular contexts. When we get clear about which language game we are playing, these topics are seen to be ordinary scientific and mathematical questions like any others. Wittgenstein’s insights have seldom been equaled and never surpassed and are as pertinent today as they were 80 years ago when he dictated the Blue and Brown Books. In spite of its failings—really a series of notes rather than a finished book—this is a unique source of the work of these three famous scholars who have been working at the bleeding edges of physics, math and philosophy for over half a century. Da Costa and Doria are cited by Wolpert (see below or my articles on Wolpert and my review of Yanofsky’s ‘The Outer Limits of Reason’) since they wrote on universal computation, and among his many accomplishments, Da Costa is a pioneer in paraconsistency. Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019

On the Goedel's formula

This article examines the formula G (of Goedel). We demonstrated that the Goedel's number of the formula G is not a finite number if (i) G is comprehended as a self-referential statement or (ii) there is an infinite set S of well-formed formulae such that the elements of S are theorems or antitheorems in T.Comment: 4 pages. Section 3 rewritte

Diagonalization in Formal Mathematics

The use of diagonalization reasonings is transversal to the Mathematical practise. Since Cantor, diagonalization reasonings are used in a great variety of areas that vanish from Topology to Logic. The objective of the present thesis was to study the formal aspects of diagonalization in Logic and more generally in the Mathematical practise. The main goal was to find a formal theory that is behind important diagonalization phenomena in Mathematics. We started by the study of diagonalization in theories of Arithmetic: Diagonalization Lemma and self-reference. In particular, we argued that important properties related to self-reference are not decidable, and we argued that the diagonalization of formulas is substantially different from the diagonalization of terms, more precisely, the Diagonal Lemma cannot prove the Strong Diagonal Lemma. We studied in detail Yablo’s Paradox. By presenting a minimal theory to express Yablo’s Paradox, we argued that Yablo’s Paradox is not a paradox about Arithmetic. From that theory and with the help of some notions of Temporal Logic, we claimed that Yablo’s Paradox is self-referential. After that, we studied several paradoxes — the Liar, Russell’s Paradox, and Curry’s Paradox— and Löb’s Theorem, and we presented a common origin to those paradoxes and theorem: Curry System. Curry Systems were studied in detail and a consistency result for specific conditions was offered. Finally, we presented a general theory of diagonalization, we exemplified the formal use of the theory, and we studied some results of Mathematics using that general theory. All the work that we present on this thesis is original. The fourth chapter gave rise to a paper by the author ([SK17]) and the third chapter will also give rise in a short period of time to a paper. Regarding the other chapters, the author, together with his Advisors, is also preparing a paper