# LJ calculus with stoup: A pedagogical approach

We prove a theorem in calculus LJ and in LJT (LJ with stoup) as well. We show that while there are many proofs to one single theorem in LJ, there is exactly one proof in LJT.

https://doi.org/10.31219/osf.io/z6h4q

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 4, Article 268, 2022 [KN]

# Personal Handbook of Logic

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]

# 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]

# GÃ¶del’s incompleteness theorem in a nutshell

[microreview]

We present a microversion of GÃ¶del’s theorem.

https://doi.org/10.31219/osf.io/gphsk

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 45, 2019 [BS]

[microresearch]