Mathematics is either inconsistent or incomplete.

What is your philosophical interpretation of Gödel’s incompleteness theorems?

https://youtu.be/jtPgdy80YZ8

#math #mathematics #maths #logic #MathematicalLogic #gödel #kurtgödel #godel #KurtGodel #Philosophy #PhilosophyOfMathematics #PhilosophicalLogic #philosophyoflogic #logic

Thinking about non-well-founded set theory (Aczel 1988) and the Quine atom Ω = {Ω}.

Russell's paradox arises from self-reference in sets. But Aczel showed self-containing sets are consistent if you drop the Foundation Axiom.

Here's what I'm chewing on: Russell's paradox feels like a linguistic trap as much as a logical one. Sets are framed as containers (nouns). A container containing itself → paradox.

But Ω = {Ω} works because we're describing relationship, not containment. The set doesn't "hold" itself like a box — it refers to itself like a pattern.

Has anyone explored whether the noun/verb distinction (container vs. relationship) is doing hidden work in self-reference paradoxes?

Pointers to literature welcome.

#SetTheory #MathematicalLogic #SelfReference #FoundationsOfMathematics

Do we have any Gödel experts in the house?

I'm trying to understand why Gödel used this encoding in the original Gödel numbering. To be clear, these are supposed to be the exponents in the unique factorisation 2ᵃ3ᵇ5ᶜ⋯, not the bases which are also coincidentally prime.

Why did Gödel pick prime exponents too?

And why did is the 0 assigned to the exponent 1 and not 2? Why is that first one not prime?

At first I thought this might be a typo in the inset figure in Wikipedia, but upon consulting the cited reference I saw that the same encoding is used in the original paper. https://en.wikipedia.org/wiki/G%C3%B6del_numbering#G%C3%B6del's_encoding

#GödelNumbers #MathematicalLogic

Logic of Relatives
• https://inquiryintoinquiry.com/2024/08/05/logic-of-relatives-a/

Relations Via Relative Terms —

The logic of relatives is the study of relations as represented in symbolic forms known as rhemes, rhemata, or relative terms.

Introduction —

The logic of relatives, more precisely, the logic of relative terms, is the study of relations as represented in symbolic forms called rhemes, rhemata, or relative terms. The treatment of relations by way of their corresponding relative terms affords a distinctive perspective on the subject, even though all angles of approach must ultimately converge on the same formal subject matter.

The consideration of relative terms has its roots in antiquity but it entered a radically new phase of development with the work of Charles Sanders Peirce, beginning with his paper “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic” (1870).

References —

• Peirce, C.S., “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 1870. Reprinted, Collected Papers CP 3.45–149. Reprinted, Chronological Edition CE 2, 359–429.
• https://www.jstor.org/stable/25058006
• https://archive.org/details/jstor-25058006
• https://books.google.com/books?id=fFnWmf5oLaoC

Resources —

Charles Sanders Peirce
• https://mywikibiz.com/Charles_Sanders_Peirce

Relation Theory
• https://oeis.org/wiki/Relation_theory

Survey of Relation Theory
• https://inquiryintoinquiry.com/2024/03/23/survey-of-relation-theory-8/

Peirce's 1870 Logic of Relatives
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/
• https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview

#Peirce #Logic #LogicOfRelatives #MathematicalLogic
#Mathematics #RelationTheory #Semiotics #SignRelations

Novo semestre começando, revisando alguns conteúdos e formulações para o ensino de lógica matemática, computabilidade e prolog! #logic #mathematicallogic

Fonte: A Concise Introduction to Mathematical Logic, Wolfgang Rautenberg, Third Edition, Springer: 2010.

A question: is transfinite induction constructive?

Sure it would depend on the definition of ordinal, but how?
#mathematicallogic #settheory