Abstract
AbstractWe study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $$\omega _2$$
ω
2
and its restrictions to certain cofinalities. Our main result shows that the strengthening $$\mathrm{{MM}}^{++}$$
MM
+
+
of Martin’s Maximum does not decide whether the restriction of the non-stationary ideal on $$\omega _2$$
ω
2
to sets of ordinals of countable cofinality is $$\Delta _1$$
Δ
1
-definable by formulas with parameters in $$\mathrm{{H}}(\omega _3)$$
H
(
ω
3
)
. The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $$\omega _2$$
ω
2
and strong forcing axioms that are compatible with $$\mathrm{{CH}}$$
CH
. Finally, we answer a question of S. Friedman, Wu and Zdomskyy by showing that the $$\Delta _1$$
Δ
1
-definability of the non-stationary ideal on $$\omega _2$$
ω
2
is compatible with arbitrary large values of the continuum function at $$\omega _2$$
ω
2
.
Funder
H2020 Marie Skodowska-Curie Actions
Simons Foundation
Publisher
Springer Science and Business Media LLC