Sheperdson showed that if there are recursive nonstandard models of open induction--the theory where the induction axioms are restricted to quantifier free formulas. The construction is algebraic. Let $R$ be the real algebraic numbers. Let $K$ be the field of Puiseux series over R, that is series in $t^{1/n}$ with coefficients from $R$ for each natural number $n$. Then let $M$ be the elements of $K$ of the form $f+n$ where $n$ is an integer, $f=0$ or all exponents in $f$ are negative, and the leading coefficient of $f+n$ is positive. Then $M$ is a model of open induction. To show how weak this theory is $\sqrt 2$ is rational in $M$.

In the other hand, George Wilmers showed that if you consider IE$_1$ the theory where you allow induction over formulas with one bounded existential formula, then there are no computable models improving a theorem of McAloon's for I$\Delta_0$--though I'm not sure it's known if IE$_1$ is actually weaker than I$\Delta_0$.

Sheperdson showed that if there are recursive nonstandard models of open induction--the theory where the induction axioms are restricted to quantifier free formulas. The construction is algebraic. Let $R$ be the real algebraic numbers. Let $K$ be the field of Puiseux series over R, that is series in $t^{1/n}$ with coefficients from $R$ for each natural number $n$. Then let $M$ be the elements of $K$ of the form $f+n$ where $n$ is an integer, $f=0$ or all exponents in $f$ are negative, and the leading coefficient of $f+n$ is positive. Then $M$ is a model of open induction. To show how weak this theory is $\sqrt 2$ is rational in $M$.

In the other hand, George Wilmers showed that if you consider IE$_1$ the theory where you allow induction over formulas with only bounded existential quantifier, then there are no computable models improving a theorem of McAloon's for I$\Delta_0$--though I'm not sure it's known if IE$_1$ is actually weaker than I$\Delta_0$.

Sheperdson showed that if there are recursive nonstandard models of open induction--the theory where the induction axioms are restricted to quantifier free formulas. The construction is algebraic. Let $R$ be the real algebraic numbers. Let $K$ be the field of Puiseux series over R, that is series in $t^{1/n}$ with coefficients from $R$ for each natural number $n$. Then let $M$ be the elements of $K$ of the form $f+n$ where $n$ is an integer, $f=0$ or all exponents in $f$ are negative, and the leading coefficient of $f+n$ is positive. Then $M$ is a model of open induction. To show how weak this theory is $\sqrt 2$ is rational in $M$.

In the other hand, George Wilmers showed that if you consider IE$_1$ the theory where you allow induction over formulas with one bounded existential formula, then there are no computable models improving a theorem of McAloon's for I$\Delta_0$--though I'm know if IE$_1$ is actually weaker than I$\Delta_0$.

Sheperdson showed that if there are recursive nonstandard models of open induction--the theory where the induction axioms are restricted to quantifier free formulas. The construction is algebraic. Let $R$ be the real algebraic numbers. Let $K$ be the field of Puiseux series over R, that is series in $t^{1/n}$ with coefficients from $R$ for each natural number $n$. Then let $M$ be the elements of $K$ of the form $f+n$ where $n$ is an integer, $f=0$ or all exponents in $f$ are negative, and the leading coefficient of $f+n$ is positive. Then $M$ is a model of open induction. To show how weak this theory is $\sqrt 2$ is rational in $M$.

In the other hand, George Wilmers showed that if you consider IE$_1$ the theory where you allow induction over formulas with one bounded existential formula, then there are no computable models improving a theorem of McAloon's for I$\Delta_0$--though I'm not sure it's known if IE$_1$ is actually weaker than I$\Delta_0$.