Circles, 2017 University of Southern California
Circles, Gabriel Leiner
Gabriel Leiner
Interstructure Lattices And Types Of Peano Arithmetic, 2017 The Graduate Center, City University of New York
Interstructure Lattices And Types Of Peano Arithmetic, Athar AbdulQuader
All Graduate Works by Year: Dissertations, Theses, and Capstone Projects
The collection of elementary substructures of a model of PA forms a lattice, and is referred to as the substructure lattice of the model. In this thesis, we study substructure and interstructure lattices of models of PA. We apply techniques used in studying these lattices to other problems in the model theory of PA.
In Chapter 2, we study a problem that had its origin in Simpson, who used arithmetic forcing to show that every countable model of PA has an expansion to PA^{∗} that is pointwise definable. Enayat later showed that there are 2^{ℵ0} models with the ...
The Common Invariant Subspace Problem And Tarski’S Theorem, 2017 Nicolaus Copernicus University of Toruń
The Common Invariant Subspace Problem And Tarski’S Theorem, Grzegorz Pastuszak
Electronic Journal of Linear Algebra
This article presents a computable criterion for the existence of a common invariant subspace of $n\times n$ complex matrices $A_{1}, \dots ,A_{s}$ of a fixed dimension $1\leq d\leq n$. The approach taken in the paper is modeltheoretic. Namely, the criterion is based on a constructive proof of the renowned Tarski's theorem on quantifier elimination in the theory $\ACF$ of algebraically closed fields. This means that for an arbitrary formula $\varphi$ of the language of fields, a quantifierfree formula $\varphi'$ such that $\varphi\lra\varphi'$ in $\ACF$ is given explicitly. The construction of $\varphi'$ is ...
Joint Laver Diamonds And Grounded Forcing Axioms, 2017 The Graduate Center, City University of New York
Joint Laver Diamonds And Grounded Forcing Axioms, Miha Habič
All Graduate Works by Year: Dissertations, Theses, and Capstone Projects
In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for κ is joint if for any sequence of targets there is a single elementary embedding j with critical point κ such that each Laver diamond guesses its respective target via j. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for κ yields a joint sequence of length κ, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly ...
Choice Of Choice: Paradoxical Results Surrounding Of The Axiom Of Choice, 2017 Union College  Schenectady, NY
Choice Of Choice: Paradoxical Results Surrounding Of The Axiom Of Choice, Connor Hurley
Honors Theses and Student Projects
When people think of mathematics they think "right or wrong," "empirically correct" or "empirically incorrect." Formalized logically valid arguments are one important step to achieving this definitive answer; however, what about the underlying assumptions to the argument? In the early 20th century, mathematicians set out to formalize these assumptions, which in mathematics are known as axioms. The most common of these axiomatic systems was the ZermeloFraenkel axioms. The standard axioms in this system were accepted by mathematicians as obvious, and deemed by some to be sufficiently powerful to prove all the intuitive theorems already known to mathematicians. However, this system ...
On Tarski's Axiomatic Foundations Of The Calculus Of Relations, 2017 Hungarian Academy of Sciences
On Tarski's Axiomatic Foundations Of The Calculus Of Relations, Hajnal Andréka, Steven Givant, Peter Jipsen, István Németi
Mathematics, Physics, and Computer Science Faculty Articles and Research
It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the righthand distributive law for relative multiplication with its lefthand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown ...
Does Logic Help Us Beat Monty Hall?, 2017 Cedarville University
Does Logic Help Us Beat Monty Hall?, Adam J. Hammett, Nathan A. Harold, Tucker R. Rhodes
The Research and Scholarship Symposium
The classical Monty Hall problem entails that a hypothetical game show contestant be presented three doors and told that behind one door is a car and behind the other two are far less appealing prizes, like goats. The contestant then picks a door, and the host (Monty) is to open a different door which contains one of the bad prizes. At this point in the game, the contestant is given the option of keeping the door she chose or changing her selection to the remaining door (since one has already been opened by Monty), after which Monty opens the chosen ...
From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, 2017 BarIlan University
From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, Mikhail Katz, Luie Polev
Journal of Humanistic Mathematics
In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting “the definition” to the students as a monolithic absolute. We hope that our experiences could be useful to other instructors wishing to follow this method of instruction. A poll run at the conclusion of the course indicates that students tend to favor infinitesimal definitions over epsilondelta ones.
The Proscriptive Principle And Logics Of Analytic Implication, 2017 The Graduate Center, City University of New York
The Proscriptive Principle And Logics Of Analytic Implication, Thomas M. Ferguson
All Graduate Works by Year: Dissertations, Theses, and Capstone Projects
The analogy between inference and mereological containment goes at least back to Aristotle, whose discussion in the Prior Analytics motivates the validity of the syllogism by way of talk of parts and wholes. On this picture, the application of syllogistic is merely the analysis of concepts, a term that presupposes—through the root ἀνά + λύω —a mereological background.
In the 1930s, such considerations led William T. Parry to attempt to codify this notion of logical containment in his system of analytic implication AI. Parry’s original system AI was later expanded to the system PAI. The hallmark of Parry’s ...
Sudoku Variants On The Torus, 2017 Harvey Mudd College
Sudoku Variants On The Torus, Kira A. Wyld
HMC Senior Theses
This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.
Four Years With Russell, Gödel, And Erdős: An Undergraduate's Reflection On His Mathematical Education, 2017 Claremont McKenna College
Four Years With Russell, Gödel, And Erdős: An Undergraduate's Reflection On His Mathematical Education, Michael H. Boggess
CMC Senior Theses
Senior Thesis at CMC is often described institutionally as the capstone of one’s undergraduate education. As such, I wanted my own to accurately capture and reflect how I’ve grown as a student and mathematician these past four years. What follows is my attempt to distill lessons I learned in mathematics outside the curriculum, written for incoming undergraduates and anyone with just a little bit of mathematical curiosity. In it, I attempt to dispel some common preconceptions about mathematics, namely that it’s uninteresting, formulaic, acultural, or completely objective, in favor of a dynamic historical and cultural perspective, with ...
Exploring Mathematical Strategies For Finding Hidden Features In MultiDimensional Big Datasets, 2016 University of Houston
Exploring Mathematical Strategies For Finding Hidden Features In MultiDimensional Big Datasets, Tri Duong, Fang Ren, Apurva Mehta
STAR (STEM Teacher and Researcher) Presentations
With advances in technology in brighter sources and larger and faster detectors, the amount of data generated at national user facilities such as SLAC is increasing exponentially. Humans have a superb ability to recognize patterns in complex and noisy data and therefore, data is still curated and analyzed by humans. However, a human brain is unable to keep up with the accelerated pace of data generation, and as a consequence, the rate of new discoveries hasn't kept pace with the rate of data creation. Therefore, new procedures to quickly assess and analyze the data are needed. Machine learning approaches ...
Mathematical Practice And Human Cognition, 2016 Indiana University  Purdue University Fort Wayne
Mathematical Practice And Human Cognition, Bernd Buldt
Philosophy Faculty Presentations
Frank Quinn (of JaffeQuinn fame, see [1]) worked out the basics of his own account of mathematical practice, an account that is informed by an analysis of contemporary mathematics and its pedagogy (see [2]). Taking this account as our starting point, we can characterize the current mathematical practice to acquire and work with new concepts as a cognitive adaptation strategy that, first, emerged to meet the challenges posed by the growing abstractness of its objects and which, second, proceeds according to the following threepronged approach:

(i) sever as many ties to ordinary language as possible and limit ordinary language explanations ...
On Fixed Points, Diagonalization, And SelfReference, 2016 Indiana University  Purdue University Fort Wayne
On Fixed Points, Diagonalization, And SelfReference, Bernd Buldt
Philosophy Faculty Presentations
We clarify the respective role fixed points, diagonalization, and self reference play in proofs of G ̈odel’s first incompleteness theorem. We first show that the usual fixedpoint construction can be reduced to a double diagonalization; this is done to address widely held views such as that fixedpoint are “paradoxical” (Priest), or work by “black magic” (Soare), or that their construction is “intuitively unclear” (Kotlarski). We then discuss three notions of selfreference; this can be seen an extension of a recent study by Halbach and Visser and is meant to show that we do not (yet?) have a robust theory ...
Mathematical Practice And Human Cognition, 2016 Indiana University  Purdue University Fort Wayne
Mathematical Practice And Human Cognition, Bernd Buldt
Philosophy Faculty Presentations
Frank Quinn (of JaffeQuinn fame, see [1]) worked out the basics of his own account of mathematical practice, an account that is informed by an analysis of contemporary mathematics and its pedagogy (see [2]). Taking this account as our starting point, we can characterize the current mathematical practice to acquire and work with new concepts as a cognitive adaptation strategy that, first, emerged to meet the challenges posed by the growing abstractness of its objects and which, second, proceeds according to the following threepronged approach:

(i) sever as many ties to ordinary language as possible and limit ordinary language explanations ...
Sentential Logic, 2016 CSU, San Bernardino
Sentential Logic, Tony Roy
Excerpted from the longer Roy, Symbolic Logic, including chapter 1 and just the first parts of chapters 2  7.
From the preface: There is, I think, a gap between what many students learn in their first course in formal logic, and what they are expected to know for their second. While courses in mathematical logic with metalogical components often cast only the barest glance at mathematical induction or even the very idea of reasoning from definitions, a first course may also leave these untreated, and fail explicitly to lay down the definitions upon which the second course is based. The ...
Symbolic Logic, 2016 CSU, San Bernardino
Symbolic Logic, Tony Roy
From the preface: There is, I think, a gap between what many students learn in their first course in formal logic, and what they are expected to know for their second. While courses in mathematical logic with metalogical components often cast only the barest glance at mathematical induction or even the very idea of reasoning from definitions, a first course may also leave these untreated, and fail explicitly to lay down the definitions upon which the second course is based. The aim of this text is to integrate material from these courses and, in particular, to make serious mathematical logic ...
Kolmogorov’S Axioms For Probabilities With Values In Hyperbolic Numbers, 2016 Chapman University
Kolmogorov’S Axioms For Probabilities With Values In Hyperbolic Numbers, Daniel Alpay, M. E. LunaElizarrarás, Michael Shapiro
Mathematics, Physics, and Computer Science Faculty Articles and Research
We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov’s system of axioms. We show that this new measure verifies the usual properties of a probability; in particular, we treat the conditional hyperbolic probability and we prove the hyperbolic analogues of the multiplication theorem, of the law of total probability and of Bayes’ theorem. Our probability may take values which are zero–divisors and we discuss carefully this peculiarity.
On Fixed Points, Diagonalization, And SelfReference, 2016 Indiana University  Purdue University Fort Wayne
On Fixed Points, Diagonalization, And SelfReference, Bernd Buldt
Philosophy Faculty Publications
We clarify the respective roles fixed points, diagonalization, and self reference play in proofs of Gödel’s first incompleteness theorem.
The Logic Of Uncertain Justifications, 2016 Kenyon College
The Logic Of Uncertain Justifications, Robert Milnikel
Robert Milnikel
No abstract provided.