Borel computation of names in template iterations

TL;DR

This paper demonstrates that in certain template iterations, any real name can be computed from a Borel function in the ground model using only countably many generic reals.

Contribution

It introduces a method to compute names for reals in template iterations via Borel functions evaluated at countably many generics.

Findings

Names for reals can be computed from Borel functions in the ground model.

The method applies to suitable iterations along a template.

It simplifies the analysis of real names in forcing extensions.

Abstract

We prove that, for a suitable iteration along a template, we can compute any name for a real from a Borel function coded in the ground model evaluated at only countably many of the generic reals.