If you are thinking about applying to graduate school in the future, it is essential that you prepare yourself for some standardized tests. These assessments assess your skills and knowledge across various fields and can help strengthen your application package.

The logic-based exam for graduate students tests your comprehension of logical principles. Additionally, it evaluates your analytical writing and verbal reasoning abilities.

## Basics

The Graduate Record Examination (GRE) is the most widely administered graduate school admission test. It measures your capacity for critical thinking, analytical writing, verbal reasoning and quantitative reasoning skills as well as critical thinking and analytical writing capabilities.

The GRE exam consists of two sections – the General Test and Analytical Writing Section – designed to test your logical reasoning skills. Each section tests specific topics and concepts through multiple-choice questions.

Each test has a distinct format, and the General Test includes four sections: Verbal Reasoning, Quantitative Reasoning, Analytical Writing and Integrated Reasoning. Additionally it includes Data Sufficiency and Problem Solving questions which assess your ability to analyze and resolve an issue.

Before you begin studying for the GRE, be sure to understand logic’s fundamental principles and how it applies to mathematics. Doing so will enable you to get the most out of this exam.

Logic requires an intense amount of study, and graduate students often take multiple courses before taking the logic qualifying exam. Among other topics, they learn about model theory, proof theory, and recursion theory.

An important component of logic is set theory, which describes how formal proofs interact with natural language structure and behavior. This relationship is particularly crucial in computer science and philosophy contexts.

Graduate students wishing to gain proficiency in set theory typically enroll in Math 220ABC, a one-year sequence of courses that presents the fundamental principles and their practical application. Furthermore, they attend colloquia hosted by faculty members of the Logic Group to stay abreast of new research directions within logic.

In addition to logic courses, UCLA graduate students take a variety of other classes as well. These include various math and physics classes as well as some other graduate-level offerings from other departments.

Many grad schools require students to submit the Free Application for Federal Student Aid, or FAFSA, in order to be eligible for financial aid. The earlier you submit your application for aid, the greater your chances are of being able to afford school.

## Set Theory

Sets are one of the fundamental concepts in mathematics and have long been a focus for research by mathematicians. Without an understanding of sets, other mathematical concepts such as relations, functions and sequences would become incomprehensible.

Set theory is the study of the properties of well-defined collections of objects. These can range from mathematical objects such as numbers or functions to non-mathematical ones like rivers or colors in a rainbow.

Set theory is an axiomatically defined subject, where sets are given their basic properties by applying certain axioms. As these concepts are so general, virtually all mathematical objects can be construed as sets – making it a highly relevant area in modern mathematics.

Set theory is an incredibly sophisticated field, though seemingly straightforward at first glance. It has allowed us to formalize almost every aspect of mathematical thought into a formal set theoretic language – an incredible accomplishment and now the standard foundation for modern mathematics.

Set theory was first axiomatized in 1904 and has since been utilized to derive most fundamental results in pure mathematics. Additionally, it has contributed significantly to other fields of mathematics like probability theory and geometry.

Grad students taking a logic-based exam must possess an in-depth knowledge of set theory. This subject requires extensive study and practice to gain the required level of comprehension.

At Berkeley, a comprehensive understanding of set theory is an absolute prerequisite to passing the logic-based exams. This course provides a thorough introduction to sets, membership rules, and their expressive power.

## Recursion Theory

Recursion theory is a branch of mathematical logic and computer science that began in the 1930s with the study of computable functions and Turing degrees. It has since broadened to encompass generalized computability and definability, as well as overlaps with proof theory and effective descriptive set theory.

Recursion theory states that a function f is considered to be recursive if its computation can only be accomplished using recursion. This concept is similar to how loops in computer programs may be considered recursive; the main distinction being that the former must be capable of calling itself when its input arguments match up against one single base case value.

Similar to for and while loop statements in programming languages, recursive functions require at least one base case to prevent them from repeatedly calling themselves.

Let us use an example to demonstrate this point: suppose you want to write a function that calls itself until its output equals five. To accomplish this, pass in input arguments x, y and z until each has been matched against a single base case.

Once this is complete, you can recursively call the function until its final result equals five. However, if an initial value of zero (0 0) isn’t passed to it when called, then failure will occur and an error message will display.

Recursive functions can be divided into two categories: b-recursive and a-recursive. Generally speaking, b-recursive functions are weakly admissible while a-recursive ones are strongly admissible.

The latter category is the strongest and most significant. It is cartesian closed, satisfying several other categories-theoretic properties.

B-recursion theory is intimately connected to Peano arithmetic’s formal logic, and many researchers have worked on it with this in mind. Kleene’s work on arithmetical hierarchy is particularly pertinent here, while Jensen’s ordinal recursion theory serves as the driving force behind this area of research.

## Incompleteness

Incompleteness in logic refers to the fact that certain mathematical systems may not be complete. This discovery, attributed to Kurt Godel in 1931, has become one of the cornerstones of modern logic.

Incompleteness is the property that some mathematical statements cannot be proved or disproven within an axiomatic system, although they may still hold true elsewhere. This does not undermine logic nor lead to any speculative nonsense as some claim; on the contrary, it promotes creative problem-solving and constructive debate.

Some have argued that this poses a problem for the program of logicism proposed by Gottlob Frege and Bertrand Russell. However, this argument ignores the incompleteness theorems which affect both arithmetic and first order logic.

Godel defined a valid logical expression as an ordered first order formula without identity, and an expression is refutable (false) if its negation can be proved true, or satisfying (true) if some interpretation of it holds.

Furthermore, an incomplete arithmetic statement is one that is neither provable nor refutable in Peano arithmetic but true according to the standard model of arithmetic. Thus, even though true arithmetic contains all such statements, it does not satisfy the hypotheses posed by incompleteness theorems.

Since their publication in the 1930s, incompleteness theorems have been an influential part of mathematic philosophy and set theory, as well as arithmetic formalism.

Proofs of the first incompleteness theorem have been achieved using computer verification methods; Natarajan Shankar in 1994, Russell O’Connor in 2003 and John Harrison in 2009. Lawrence Paulson announced a computer-verified proof of the second incompleteness theorem with HOL Light in 2013.

Graduate logic students must take a course in logic and pass a foundations exam for credit. Logic I provides proof procedure and semantics for first-order predicate logic with identity, as well as basic familiarity with standard metalogical results concerning completeness, incompleteness and decidability. This should typically be completed during the first year of graduate study prior to taking the foundations exam; exceptions may be granted by either the chair of the department or graduate adviser.