### Logic and Computation Seminar

Monday, January 20, 2020 - 3:30pm

#### Dan Turetsky

Victoria University of Wellington

Location

University of Pennsylvania

4C8

Preservation properties are a tool for separating the reverse mathematical strength of various statements.  As an example, if \$I\$ is a Turing ideal and \$X\$ is a set outside \$I\$, then there is an ideal \$J\$ containing \$I\$ but omitting \$X\$ and which models WKL\$_0\$.  The same holds with RT\$^2_2\$ in place of WKL\$_0\$, but this fails for RT\$^3_2\$, thus showing that WKL\$_0\$ and RT\$^2_2\$ do not prove RT\$^3_2\$.

In fact, for both WKL\$_0\$ and RT\$^2_2\$, the above holds not just for a single set \$X\$, but for countably many sets simultaneously.  In both cases, the proofs for one set and for countably many sets are more or less the same.  It turns out there's a reason for this: any reverse mathematical principal (of the appropriate form) which can be satisfied while avoiding a single set can be satisfied while avoiding countably many.

This is an example of a relationship between preservation properties.  We investigate similar relationships between various preservation properties.