Math Girls 3 Godel is Incompleteness Theorems 1st Edition by Hiroshi Yuki, Tony Gonzalez 1939326281 9781939326287

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ISBN 10: 1939326281

ISBN 13: 9781939326287

Author: Hiroshi Yuki, Tony Gonzalez

Math Girls 3 Godel is Incompleteness Theorems 1st Table of contents:

1. Liar, Liar

1.1 Who Tells the Truth?

• Mirror, Mirror
• Finding Truth Tellers
• Same Answers
• Answering with Silence

1.2 Logic Problems

• Alice, Boris, and Chris
• Thinking with Tables
• Well-designed Problems

1.3 What Color is Your Hat?

• What Others Don’t Know
• A Problem for the Problem Writer
• Reflections

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2. Playing Peano

2.1 Tetra

• The Peano Axioms
• Infinite Wishes
• Peano’s Axiom PA1
• Peano’s Axiom PA2
• Getting Bigger
• Peano’s Axiom PA3
• How Small is Small?
• Peano’s Axiom PA4

2.2 Miruka

• Peano’s Axiom PA5
• Mathematical Induction

2.3 Infinite Walks

• The Finite and the Infinite
• Dynamic vs. Static

2.4 Yuri

• What’s Addition?
• More Axioms Needed?

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3. Galileo’s Doubts

3.1 Sets

• A Rude Awakening
• Extensional Definitions
• The Empty Set
• Sets of Sets
• Intersections
• Unions
• Inclusion Relations
• Why Are Sets Important?

3.2 Adding Logic

• Intensional Definitions
• Russell’s Paradox
• Set and Logic Operators

3.3 Infinity

• Bijections and Birdcages
• New Concepts

3.4 Heading Home

• Descriptions and Expressions
• At the Bookstore

3.5 Null Answer

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4. Know Your Limits

4.1 At Home

• Yuri
• The Boy’s Proof
• Yuri’s Proof
• Yuri’s Alternate Proof
• My Explanation

4.2 At the Supermarket

• Arriving at Your Destination

4.3 In the Music Room

• Introducing Variables
• Limits
• Sound Makes the Music
• Calculating Limits

4.4 Heading Home

• Moving On

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5. Leibniz’s Dream

5.1 Yuri Implies Not Tetra

• The Meaning of ‘Implies’
• Solving Without Thinking
• The Limits of Reason?

5.2 Tetra Implies Not Yuri

• Test Anxiety

5.3 Miruka Implies Miruka

• In My Classroom
• Formal Systems
• Propositional Formulas
• The Form of ‘Implies’
• Axioms
• Proof Theory
• Inference Rules
• Proofs and Theorems

5.4 Not Me, or Me

• Good Intentions
• The Form of Form
• The Meaning of Meaning
• Implies ‘Implies’?
• Asked Out

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6. Epsilon Delta

6.1 Limits of Sequences

• Crash Course
• In the Lecture Hall
• Understanding Complex Statements
• Reading Absolute Values
• Reading ‘Implies’
• Reading ‘For All’ and ‘Some’

6.2 The Limits of Functions

• Epsilon-delta
• The Meaning of (ε, δ)

6.3 Finals

• Unranked
• Sound of Calm, Voice of Silence

6.4 Continuity

• In the Library
• Continuous Nowhere
• Continuous at Only One Point
• Escape from an Infinite Maze
• Finding the Function
• Realizations

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7. Diagonalization

7.1 Sequences of Sequences

• Countable Sets
• Cantor’s Diagonal Argument
• A Challenge
• Another Challenge

7.2 Systems of Systems

• Completeness and Consistency
• Gödel’s Incompleteness Theorems
• Arithmetic
• Formal Systems of Formal Systems
• Some Vocabulary
• Numerals
• Diagonalization Revisited
• Setting Things Straight

7.3 Search for a Search

• First Date

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8. Between Two Solitudes

8.1 Overlapping Pairs

• What Tetra Noticed
• What I Noticed
• Getting Ready

8.2 At Home

• My Own Math
• Compressed Expressions
• Defining Addition
• In the Zone

8.3 Equivalence Relations

• The Graduation Ceremony
• What Pairs Produce
• Graphing
• From Natural Numbers to Integers
• Revisiting Equivalence Relations
• Quotient Sets

8.4 At the Restaurant

• A Pair of Wings
• Loosening Up

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9. Following Spirals

9.1 Radians

• An Irritated Yuri
• Trigonometric Functions
• sin 45°
• sin 60°
• The Sine Curve

9.2 Teaching Radians

• Radians and Teaching

9.3 Miscellaneous Topics

• Classes Canceled
• Remainders
• The Lighthouse
• On the Beach
• Salving the Wound

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10. Gödel’s Incompleteness Theorems

10.1 At the Narabikura Library

• Entrance
• Chlorine

10.2 Hilbert’s Program

• Hilbert
• Quizzes

10.3 Gödel’s Incompleteness Theorems

• Gödel
• Discussion
• Outline of the Proof

10.4 Spring: Formal System P

• Elementary Symbols
• Numerals and Individuals
• Elementary Formulas
• Axioms
• Inference Rules

10.5 Lunch Break

• Metamathematics
• Math on Math

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Hiroshi Yuki,Tony Gonzalez,Incompleteness,Math,Girls