Summary Abstract Mathematics Without Choice Division by Three arxiv.org
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One Line
The article examines "mathematics without choice" and its impact on division by three, including alternative interpretations of multiplication.
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Key Points
- Conway and Doyle have claimed to be able to divide by three in cardinal arithmetic without the axiom of choice.
- The principle of "shoe division" states that if |A × n| = |B × n|, then |A| = |B|.
- The principle of "sock division" states that if |A × n| = |B × n| and each set Xa in the collection {Xa}a∈A has size n, then |A| = |B|.
- Sock division by 3 is not provable without the axiom of choice.
- Sock division by n is not provable for any n > 1.
- Sock division by n implies that multiplication by n is equal to repeated addition of n.
- Sock division by n implies that strong and weak divisibility by n are equivalent.
- The relationship between sock division and divisibility in set theory without choice is an interesting topic for further research.
Summaries
20 word summary
This article explores "mathematics without choice" and its application to dividing by three. The authors introduce alternative interpretations of multiplication.
90 word summary
This article explores "mathematics without choice" and its application to dividing by three in cardinal arithmetic. The authors attempt to divide by three using Conway and Doyle's method but fail. They introduce "shoe division" and "sock division" as alternative interpretations of multiplication. The article discusses the difficulty of making choices without assuming arbitrariness and proposes an alternative statement of division by three called "sock division." The authors demonstrate that sock division is not provable without the axiom of choice and raise questions about its implications in set theory without choice.
154 word summary
This article examines the concept of "mathematics without choice" and its application to dividing by three in cardinal arithmetic without the axiom of choice. The authors attempt to replicate Conway and Doyle's claim of being able to divide by three, but fail. They introduce the concepts of "shoe division" and "sock division," where shoe division refers to the standard interpretation of multiplication as the area of a rectangle, and sock division refers to interpreting multiplication as repeated addition. The article discusses Bertrand Russell's example of a millionaire buying shoes and socks daily, highlighting the difficulty of making choices without assuming they can be made arbitrarily. The authors propose an alternative statement of division by three called "sock division," and demonstrate that it is not provable without the axiom of choice. The article also explores the relationship between sock division and multiplication, raising questions about the implications of sock division in set theory without choice.
468 word summary
This article delves into the concept of "mathematics without choice" and its application to dividing by three in cardinal arithmetic without the axiom of choice. The authors attempt to replicate Conway and Doyle's claim of being able to divide by three, but ultimately fail. They introduce the concepts of "shoe division" and "sock division," where shoe division refers to the standard interpretation of multiplication as the area of a rectangle, and sock division refers to interpreting multiplication as repeated addition.
The article highlights Bertrand Russell's example of a millionaire who buys a pair of shoes and socks every day, resulting in an infinite collection of pairs. The challenge arises when asked how to choose one shoe or sock from each pair, as there is no obvious way to distinguish them. This example illustrates the difficulty of making choices without assuming they can be made arbitrarily.
The authors propose an alternative statement of division by three called "sock division," which involves collections of unordered sets rather than ordered sets. They question whether this alternative statement is provable without the axiom of choice and explore its connection to Russell's example. The article also mentions previous work on division by four and equivariant division.
The authors demonstrate that sock division by three is not provable without the axiom of choice by constructing a proof that relies on a choice function for a sequence of disjoint sets of size two. They conclude that sock division by two or any other natural number greater than one is also not provable without the axiom of choice.
The article further examines the relationship between sock division and multiplication, presenting two interpretations: multiplication as the area of a rectangle or as repeated addition. Without the axiom of choice, these interpretations may not be equivalent. The authors argue that sock division implies that multiplication is equal to repeated addition.
The article poses several questions regarding the power and implications of sock division. It asks if sock division implies that all "sock bundles" are trivializable and whether the equivalence of strong and weak divisibility implies sock division. These questions aim to shed light on set theory without the axiom of choice.
The authors acknowledge Peter Doyle for coining the term "sock division" and express gratitude to various individuals for their contributions and inspiration.
In summary, this article explores mathematics without choice and investigates the possibility of dividing by three in cardinal arithmetic without the axiom of choice. It introduces shoe division and sock division, discusses Russell's example, and proposes an alternative statement of division by three. The authors demonstrate that sock division by three is not provable without the axiom of choice and delve into the relationship between sock division and multiplication. They raise questions about the power and implications of sock division in set theory without choice.
472 word summary
This article discusses the concept of "mathematics without choice" and explores the idea of dividing by three in cardinal arithmetic without the axiom of choice. The authors attempt to replicate Conway and Doyle's claim that they can divide by three, but ultimately fail. The article introduces the concept of "shoe division" and "sock division," where shoe division refers to the standard interpretation of multiplication as the area of a rectangle, and sock division refers to interpreting multiplication as repeated addition.
The article highlights an example by Bertrand Russell, where a millionaire buys a pair of shoes and socks every day and amasses infinitely many pairs of each. When asked how to choose one shoe or sock from each pair, there is no obvious way to distinguish them. This example illustrates the issue of making choices without assuming they can be made any-which-way.
The authors propose an alternative statement of division by three, called "sock division," which involves collections of unordered sets instead of ordered sets. They question whether this alternative statement is provable without the axiom of choice and discuss its relationship to Russell's example. The article also mentions previous work on division by four and equivariant division.
The authors show that sock division by three is not provable without the axiom of choice, using a proof that relies on constructing a choice function for a sequence of disjoint sets of size two. They conclude that sock division by two or any other natural number greater than one is also not provable without the axiom of choice.
The article then explores the relationship between sock division and multiplication. It discusses two different interpretations of multiplication: as the area of a rectangle or as repeated addition. Without the axiom of choice, these two interpretations may not be equivalent. The authors argue that sock division implies that multiplication is equal to repeated addition.
The article raises several questions about the power and implications of sock division. It asks whether sock division implies that all -sock bundles are trivializable and whether the equivalence of strong and weak divisibility by implies sock division. These questions would provide insight into set theory without choice.
The authors acknowledge Peter Doyle, who coined the term "sock division," and thank various individuals for their contributions and inspiration.
In summary, this article explores the concept of mathematics without choice and investigates the possibility of dividing by three in cardinal arithmetic without the axiom of choice. It introduces the concepts of shoe division and sock division, discusses an example by Bertrand Russell, and proposes an alternative statement of division by three. The authors show that sock division by three is not provable without the axiom of choice and discuss the relationship between sock division and multiplication. They raise questions about the power and implications of sock division in set theory without choice.