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]

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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]

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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]

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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]

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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]

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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

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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]

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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]

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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]

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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]

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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]