The parameterfree Comprehension does not imply the full Comprehension in the 2nd order Peano arithmetic

TL;DR

This paper demonstrates that the parameter-free fragment of second-order Peano arithmetic does not necessarily imply the full comprehension schema, using forcing techniques to construct specific models where comprehension fails.

Contribution

It introduces forcing-based methods to construct models of second-order Peano arithmetic where full comprehension does not hold, highlighting limitations of parameter-free systems.

Findings

Existence of models where full comprehension fails

Parameter-free PA2* does not imply full comprehension

Forcing techniques effectively demonstrate comprehension limitations

Abstract

The parameter-free part $\text{PA}_2^\ast$ of $\text{PA}_2$, the 2nd order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an $\omega$-model of $\text{PA}_2^\ast + \text{CA}(\Sigma^1_2)$, in which an example of the full Comprehension schema $\text{CA}$ fails. Using Cohen's forcing, we also define an $\omega$-model of $\text{PA}_2^\ast$, in which not every set has its complement, and hence the full $\text{CA}$ fails in a rather elementary way.