Return to Answer

You can also talk about arithmetically definable real numbers: those for which the Dedekind cut of rationals is of the form: $$\{m/n: \forall x_1 \exists x_2 \ldots \forall x_{k-1} \exists x_k\, p(m,n,x_1,\ldots,x_k)=0\},$$ where the $x$'s range over integers, and $p$ is a polynomial with integer coefficients.

Then on this definition of definability: $e$ and $\pi$ and all the familiar reals are definable. Only countably many numbers are definable. There must be other real numbers which are undefinable. And it all makes sense, and is even probably consistent, in ordinary set theory.

A standard reference for this way of thinking is the system $ACA_0$ in Simpson's Subsystems of Second-Order Arithmetic.

The cost of this metamathematical simplicity is a small change to the mathematics: any definable bounded sequence of reals has a definable least upper bound, but an uncountable definable set of reals may not. Feferman's notes on Predicative Foundations of Analysis show how to develop standard analysis on this basis. If we changed mathematics as taught in universities to be based on predicative analysis, few undergraduates or people outside the math department would notice much difference.

You can also talk about arithmetically definable real numbers: those for which the Dedekind cut of rationals is of the form: $$\{m/n: \forall x_1 \exists x_2 \ldots \forall x_{k-1} \exists x_k\, p(m,n,x_1,\ldots,x_k)=0\},$$ where the $x$'s range over integers, and $p$ is a polynomial with integer coefficients.

Then on this definition of definability: $e$ and $\pi$ and all the familiar reals are definable. Only countably many numbers are definable. There must be other real numbers which are undefinable. And it all makes sense, and is even provably consistent, in ordinary set theory.

A standard reference for this way of thinking is the system $ACA_0$ in Simpson's Subsystems of Second-Order Arithmetic.

The cost of this metamathematical simplicity is a small change to the mathematics: any definable bounded sequence of reals has a definable least upper bound, but an uncountable definable set of reals may not. Feferman's notes on Predicative Foundations of Analysis show how to develop standard analysis on this basis. If we changed mathematics as taught in universities to be based on predicative analysis, few undergraduates or people outside the math department would notice much difference.

You can also talk about arithmetically definable real numbers: those for which the Dedekind cut of rationals is of the form: $$\{m/n: \forall x_1 \exists x_2 \ldots \forall x_{k-1} \exists x_k\, p(m,n,x_1,\ldots,x_k)=0\},$$ where the $x$'s range over integers, and $p$ is a polynomial with integer coefficients.

Then on this definition of definability: $e$ and $\pi$ and all the familiar reals are definable. Only countably many numbers are definable. There must be other real numbers which are undefinable. And it all makes sense in ordinary set theory.

A standard reference for this way of thinking is the system $ACA_0$ in Simpson's Subsystems of Second-Order Arithmetic.

You can also talk about arithmetically definable real numbers: those for which the Dedekind cut of rationals is of the form: $$\{m/n: \forall x_1 \exists x_2 \ldots \forall x_{k-1} \exists x_k\, p(m,n,x_1,\ldots,x_k)=0\},$$ where the $x$'s range over integers, and $p$ is a polynomial with integer coefficients.

Then on this definition of definability: $e$ and $\pi$ and all the familiar reals are definable. Only countably many numbers are definable. There must be other real numbers which are undefinable. And it all makes sense, and is even probably consistent, in ordinary set theory.

A standard reference for this way of thinking is the system $ACA_0$ in Simpson's Subsystems of Second-Order Arithmetic.

The cost of this metamathematical simplicity is a small change to the mathematics: any definable bounded sequence of reals has a definable least upper bound, but an uncountable definable set of reals may not. Feferman's notes on Predicative Foundations of Analysis show how to develop standard analysis on this basis. If we changed mathematics as taught in universities to be based on predicative analysis, few undergraduates or people outside the math department would notice much difference.

You can also talk about arithmetically definable real numbers: those for which the Dedekind cut of rationals is of the form: $$\{p/q: \forall x_1 \exists x_2 \ldots \forall x_{n-1} \exists x_n\, \phi(p,q,x_1,\ldots,x_n)\},$$ where the $x$'s range over integers, $\phi$ is of the form $p_1=0 \vee \cdots \vee p_k=0$, and the $p_i$'s are polynomials in $p,q,x_1,\ldots,x_n$ with integer coefficients.

Then on this definition of definability: $e$ and $\pi$ are all the familiar reals are definable. Only countably many numbers are definable. There must be other real numbers which are undefinable. And it all makes sense in ordinary set theory.

A standard reference for this way of thinking is the system $ACA_0$ in Simpson's Subsystems of Second-Order Arithmetic.

You can also talk about arithmetically definable real numbers: those for which the Dedekind cut of rationals is of the form: $$\{m/n: \forall x_1 \exists x_2 \ldots \forall x_{k-1} \exists x_k\, p(m,n,x_1,\ldots,x_k)=0\},$$ where the $x$'s range over integers, and $p$ is a polynomial with integer coefficients.

Then on this definition of definability: $e$ and $\pi$ and all the familiar reals are definable. Only countably many numbers are definable. There must be other real numbers which are undefinable. And it all makes sense in ordinary set theory.

A standard reference for this way of thinking is the system $ACA_0$ in Simpson's Subsystems of Second-Order Arithmetic.