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Andrej Bauer asked in a comment, "What is a model of open induction?" Let me give the most modern and relevant answer and then explain the term "open induction", which is somewhat archaic.

Let $F$ be an ordered field. An (ordered) subring $R$ of $F$ is called an "integer part" of $F$ if

1. $R$ is discretely ordered: This means the inequality $x\le x < y < x+1$ has no solution in $R$. Equivalently, nothing in $R$ lies between 0 and 1.

2. For all $x\in F$ there is some $r\in R$ such that $r\le x < r+1$.

Item 2 is equivalent to saying that every element of $F$ is a finite distance from some element of $R$, where "finite" means bounded by an element of $\mathbb{Z}$. Remember that $F$ can and usually will be nonarchimedian.

Items 1 and 2 together imply that there is a unique function $\lfloor\cdot\rfloor$ from $F$ to $R$ given by the inequality $\lfloor x \rfloor \le x < 1+\lfloor x \rfloor$. Think of this as an abstract analog of the integer part operator.

A "model of open induction" is an integer part of a real closed field. Mourges and Ressayre proved that every real closed field has an integer part. The term "open induction" comes from a paper of Shepherdson, who started this whole topic by proving that an ordered ring $R$ is an integer part of its real closure if and only if the positive semiring of $R$ satisfies the axioms of Peano Arithmetic, with the induction axioms restricted to quantifier-free (i.e. open) formulas. Shepherdson also gave recursively presentable nonstandard models of open induction, which is interesting because according to Tennenbaum's Theorem there are no such models of Peano Arithmetic.

Since then there have been many successful attempts to build recursive nonstandard models of theories a little stronger than open induction. The champ, to-date, is a recursive nonstandard normal model of open induction, given in a paper by Otero and Berarducci. Here "normal" means integrally closed in its quotient field.

The main unsolved problems concerning open induction, in my opinion, are (1) Is the universal theory of open induction decidable (i.e. is it decidable whether a diophantine equation has a solution in some model of open induction) and (2) Is there a recursive non-standard diophantine correct model of open induction, where "diophantine correct" means "extends to a model of true arithmetic".

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Andrej Bauer asked in a comment, "What is a model of open induction?" Let me give the most modern and relevant answer and then explain the term "open induction", which is somewhat archaic.

Let $F$ be an ordered field. An (ordered) subring $R$ of $F$ is called an "integer part" of $F$ if

1. $R$ is discretely ordered: This means the inequality $x\le y < x+1$ has no solution in $R$. Equivalently, nothing in $R$ lies between 0 and 1.

2. For all $x\in F$ there is some $r\in R$ such that $r\le x < r+1$.

Item 2 is equivalent to saying that every element of $F$ is a finite distance from some element of $R$, where "finite" means bounded by an element of $\mathbb{Z}$. Remember that $F$ can and usually will be nonarchimedian.

Items 1 and 2 together imply that there is a unique function $\lfloor\cdot\rfloor$ from $F$ to $R$ given by the inequality $\lfloor x \rfloor \le x < 1+\lfloor x \rfloor$. Think of this as an abstract analog of the integer part operator.

A "model of open induction" is an integer part of a real closed field. A theorem of Mourges and Ressayre proves proved that every real closed field has an integer part. The term "open induction" comes from a paper of Shepherdson, who started this whole topic by proving that an ordered ring $R$ is an integer part of its real closure if and only if the positive semiring of $R$ satisfies the axioms of Peano Arithmetic, with the induction axioms restricted to quantifier-free (i.e. open) formulas. Shepherdson also gave recursively presentable nonstandard models of open induction, which is interesting because according to Tennenbaum's Theorem there are no such models of Peano Arithmetic.

Since then there have been many successful attempts to build recursive nonstandard models of theories a little stronger than open induction. The champ, to-date, is a recursive nonstandard normal model of open induction, given in a paper by Otero and Berarducci. Here "normal" means integrally closed in its quotient field.

The main unsolved problems concerning open induction, in my opinion, are (1) Is the universal theory of open induction decidable (i.e. is it decidable whether a diophantine equation has a solution in some model of open induction) and (2) Is there a recursive non-standard diophantine correct model of open induction, where "diophantine correct" means "extends to a model of true arithmetic".

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Andrej Bauer asked in a comment, "What is a model of open induction?" Let me give the most modern and relevant answer and then explain the term "open induction", which is somewhat archaic.

Let $F$ be an ordered field. An (ordered) subring $R$ of $F$ is called an "integer part" of $F$ if

1. $R$ is discretely ordered: This means the inequality $x\le y < x+1$ has no solution in $R$. Equivalently, nothing in $R$ lies between 0 and 1.

2. For all $x\in F$ there is some $r\in R$ such that $r\le x < r+1$.

Item 2 is equivalent to saying that every element of $F$ is a finite distance from some element of $R$, where "finite" means bounded by an element of $\mathbb{Z}$. Remember that $F$ can and usually will be nonarchimedian.

Items 1 and 2 together imply that there is a unique function $\lfloor\cdot\rfloor$ from $F$ to $R$ given by the inequality $\lfloor x \rfloor \le x < 1+\lfloor x \rfloor$. Think of this as an abstract analog of the integer part operator.

A "model of open induction" is an integer part of a real closed field. A theorem of Mourges and Ressayre proves that every real closed field has an integer part. The term "open induction" comes from a paper of Shepherdson, who started this whole topic by proving that an ordered ring $R$ is an integer part of its real closure if and only if the positive semiring of $R$ satisfies the axioms of Peano Arithmetic, with the induction axioms restricted to quantifier-free (i.e. open) formulas. Shepherdson also gave recursively presentable nonstandard models of open induction, which is interesting because according to Tennenbaum's Theorem there are no such models of Peano Arithmetic.

Since then there have been many successful attempts to build recursive nonstandard models of theories a little stronger than open induction. The champ, to-date, is a recursive nonstandard normal model of open induction, given in a paper by Otero and Berarducci. Here "normal" means integrally closed in its quotient field.

The main unsolved problems concerning open induction, in my opinion, are (1) Is the universal theory of open induction decidable (i.e. is it decidable whether a diophantine equation has a solution in some model of open induction) and (2) Is there a recursive non-standard diophantine correct model of open induction, where "diophantine correct" means "extends to a model of true arithmetic".