This is a personal collection of definitions and results from first-order logic.
https://doi.org/10.31219/osf.io/8wck9
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 252, 2021 [JX]
Tag: logic
Two non-isomorphic structures with the same number of elements
[white paper: pedagogical]
For pedagogical purposes, we define a simple language to show that two different structures with the same number of elements in their universes are not isomorphic.
https://doi.org/10.31219/osf.io/ytcru
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 234, 2021 [JE]
The isomorphism between structures is an equivalence relation
[white paper: pedagogical]
We present a pedagogical proof that the function of an isomorphism between two structures is an equivalence relation.
https://doi.org/10.31219/osf.io/7uwnh
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 233, 2021 [JD]
The terms of a language with one constant, one binary function, and one 4-ary function have an odd number of symbols
[white paper: pedagogical]
We show using induction on complexity that all terms of a language with one constant, one binary function, and one 4-ary function have an odd number of symbols.
https://doi.org/10.31219/osf.io/ue32a
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 231, 2021 [JB]
Goldbach Conjecture, Twin Primes Conjecture, and Bounded Gap Theorem in the language of number theory
[white paper: pedagogical]
We write the formulas of the theorem and the conjectures highlighted in the title of this white paper in the language of number theory for pedagogical purpose in first-order logic.
https://doi.org/10.31219/osf.io/u45y3
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 229, 2021 [IZ]
Substitutions and Substitutability
[knowledge base]
SUBSTITUTIONS, SUBSTITUTABILITY, and their underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/k6w7u
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 212, 2021 [II]
Model
[knowledge base]
MODEL (mathematical logic) and its underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/vs6j9
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 209, 2021 [IF]
A unidimensional windmill in the plane with varying pivot points
We show that we can choose a point in the plane such that the resulting windmill process with varying pivot uses each point of the plane infinitely many times.
https://doi.org/10.31219/osf.io/mwkaf
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 2, Article 148, 2020 [FT]
Logical Loop
[microresearch]
We discuss the dubbed term “logical loop” and it’s implication regarding provability in undecidable theorems.
https://doi.org/10.31219/osf.io/c5ezm
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 2, Article 123, 2020 [ET]
Non-classical logic and undecidability
[conjecture]
We elaborate a conjecture by applying the definition of Non-Axiomatic System [1] in Non-Classical Logic.
https://doi.org/10.31219/osf.io/h4f7p
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 2, Article 112, 2020 [EI]
The logic of entanglement in gravity
[microresearch]
We explore the relation between gravity and entanglement using mathematical logic (Modus Ponens).
https://doi.org/10.31219/osf.io/m2dca
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 1, Article 58, 2019 [CF]
Hilbert-style proof Calculus for Propositional Logic in ABC notation
[microreview]
All nine axioms and a single inference rule of logic (Modus Ponens) within the Hilbert axiomatic system are presented using capital letters (ABC) in order to familiarize the beginner student in hers/his first contact with the topic.
https://doi.org/10.31219/osf.io/jd3gp
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 1, Article 57, 2019 [CE]
The Incorrectness Lemma
[original idea]
We address two questions. Is there an Incorrectness Lemma? Is this system complete?
https://doi.org/10.31219/osf.io/f8p7k
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 1, Article 53, 2019 [CA]
A pedagogical approach to convert “if and only if” to “if then”
[microreview]
We show, using the equivalence of some formulas, that (A iff B) ~ ((A v B) => (A ^ B)).
https://doi.org/10.31219/osf.io/xhzc9
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 1, Article 52, 2019 [BZ]
Generalized Pierce law
[microresearch]
An extension of the Pierce law is presented, considering n “if then” connectives.
https://doi.org/10.31219/osf.io/6fbgp
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 1, Article 50, 2019 [BX]
Gödel’s incompleteness theorem in a nutshell
[microreview]
We present a microversion of Gödel’s theorem.
https://doi.org/10.31219/osf.io/gphsk
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 1, Article 45, 2019 [BS]
Vacuously true sentences: Are they non-Boolean and undecidable?
[microresearch]
We propose a discussion on the non-Boolean mathematical object embedded in vacuously true sentences.
https://doi.org/10.31219/osf.io/8r3j2
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 1, Article 35, 2019 [BI]
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