# The full basis theorem does not imply analytic wellordering

## TL;DR

This paper constructs a model of ZFC where every non-empty analytically definable set of reals contains an analytically definable real, yet there is no analytically definable wellordering of the continuum, showing the full basis theorem does not imply analytic wellordering.

## Contribution

It demonstrates that the full basis theorem does not necessarily lead to an analytically definable wellordering of the continuum by constructing a specific model.

## Key findings

- Every non-empty lightface analytically definable set of reals contains an analytically definable real.
- There is no analytically definable wellordering of the continuum in the constructed model.
- The model uses a finite support product of Jensen minimal $oldsymbol{
m extPi^1_2}$ singleton forcing.

## Abstract

We make use of a finite support product of $\omega_1$ clones of the Jensen minimal $\varPi^1_2$ singleton forcing to obtain a model of ZFC in which every non-empty lightface analytically definable set of reals contains a lightface analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the continuum.