Sliders SDF 60th Birthday Celebration Natasha Dobrinen University of Denver Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 1 / 27
Remeniscences Thanks Sy for giving me a job at KGRC. (2004-2007) Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 2 / 27
Remeniscences Thanks Sy for giving me a job at KGRC. (2004-2007) It was first time I worked around more than one other set theorist. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 2 / 27
Remeniscences Thanks Sy for giving me a job at KGRC. (2004-2007) It was first time I worked around more than one other set theorist. Lovely to do research all day! Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 2 / 27
So, what are sliders? Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 3 / 27
Pulled Pork Sliders Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 4 / 27
Chicken Sliders Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 5 / 27
Hamburger Sliders Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 6 / 27
Sliders come in many forms Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 7 / 27
Sliders come in many forms Yet, all sliders of the same form are indistinguishable from each other. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 7 / 27
Sliders come in many forms Yet, all sliders of the same form are indistinguishable from each other. In mathematics, sliders are formally known as indiscernibles . Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 7 / 27
History During a working lunch at an AIMS Conference in December 2004, Sy introduced me to indiscernibles. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 8 / 27
History During a working lunch at an AIMS Conference in December 2004, Sy introduced me to indiscernibles. (He also introduced me to the working lunch.) Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 8 / 27
History During a working lunch at an AIMS Conference in December 2004, Sy introduced me to indiscernibles. (He also introduced me to the working lunch.) The indiscernibles were pleasant, though undistinguished lunch guests. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 8 / 27
History During a working lunch at an AIMS Conference in December 2004, Sy introduced me to indiscernibles. (He also introduced me to the working lunch.) The indiscernibles were pleasant, though undistinguished lunch guests. Since that initial introduction, indiscernibles keep sliding into key positions in my work. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 8 / 27
Sy and I were interested in the following: Problem. Given models V ⊆ W of ZFC, when does having a new subset of κ in W \ V make ( P κ + ( λ )) W \ ( P κ + ( λ )) V stationary in W ? i.e. When is the ground model co-stationary ? Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 9 / 27
Sy and I were interested in the following: Problem. Given models V ⊆ W of ZFC, when does having a new subset of κ in W \ V make ( P κ + ( λ )) W \ ( P κ + ( λ )) V stationary in W ? i.e. When is the ground model co-stationary ? P κ ( λ ) = { x ⊆ λ : | x | < κ } . Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 9 / 27
Sy and I were interested in the following: Problem. Given models V ⊆ W of ZFC, when does having a new subset of κ in W \ V make ( P κ + ( λ )) W \ ( P κ + ( λ )) V stationary in W ? i.e. When is the ground model co-stationary ? P κ ( λ ) = { x ⊆ λ : | x | < κ } . C ⊆ P κ ( λ ) is club if it is closed under < κ -unions and ⊆ -cofinal in P κ ( λ ) . Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 9 / 27
Sy and I were interested in the following: Problem. Given models V ⊆ W of ZFC, when does having a new subset of κ in W \ V make ( P κ + ( λ )) W \ ( P κ + ( λ )) V stationary in W ? i.e. When is the ground model co-stationary ? P κ ( λ ) = { x ⊆ λ : | x | < κ } . C ⊆ P κ ( λ ) is club if it is closed under < κ -unions and ⊆ -cofinal in P κ ( λ ) . S ⊆ P κ ( λ ) is stationary if S meets every club set. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 9 / 27
Background [Abraham/Shelah 1983]: ccc forcings adding a new subset of ℵ 0 [Gitik 1985]: models V ⊆ W where W has a new subset of ℵ 0 make the ground model co-stationary for P κ ( λ ), for all cardinals ℵ 1 < κ < λ in the larger model. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 10 / 27
Background [Abraham/Shelah 1983]: ccc forcings adding a new subset of ℵ 0 [Gitik 1985]: models V ⊆ W where W has a new subset of ℵ 0 make the ground model co-stationary for P κ ( λ ), for all cardinals ℵ 1 < κ < λ in the larger model. What if the larger model has a new subset of ℵ 1 but no new subsets of ℵ 0 ? Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 10 / 27
Background [Abraham/Shelah 1983]: ccc forcings adding a new subset of ℵ 0 [Gitik 1985]: models V ⊆ W where W has a new subset of ℵ 0 make the ground model co-stationary for P κ ( λ ), for all cardinals ℵ 1 < κ < λ in the larger model. What if the larger model has a new subset of ℵ 1 but no new subsets of ℵ 0 ? Sy knew that Erd˝ os cardinals would be necessary if we add no new ω -sequences, because of a covering theorem of [Magidor 1990]. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 10 / 27
Background [Abraham/Shelah 1983]: ccc forcings adding a new subset of ℵ 0 [Gitik 1985]: models V ⊆ W where W has a new subset of ℵ 0 make the ground model co-stationary for P κ ( λ ), for all cardinals ℵ 1 < κ < λ in the larger model. What if the larger model has a new subset of ℵ 1 but no new subsets of ℵ 0 ? Sy knew that Erd˝ os cardinals would be necessary if we add no new ω -sequences, because of a covering theorem of [Magidor 1990]. Erd˝ os cardinals involve indiscernibles. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 10 / 27
Background [Abraham/Shelah 1983]: ccc forcings adding a new subset of ℵ 0 [Gitik 1985]: models V ⊆ W where W has a new subset of ℵ 0 make the ground model co-stationary for P κ ( λ ), for all cardinals ℵ 1 < κ < λ in the larger model. What if the larger model has a new subset of ℵ 1 but no new subsets of ℵ 0 ? Sy knew that Erd˝ os cardinals would be necessary if we add no new ω -sequences, because of a covering theorem of [Magidor 1990]. Erd˝ os cardinals involve indiscernibles. This was the beginning of our work on finding the equiconsistency of co-stationarity of the ground model and broader work in which indiscernibles play an important role. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 10 / 27
Indiscernibles Def. M a structure, X ⊆ M linearly ordered by < . Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 11 / 27
Indiscernibles Def. M a structure, X ⊆ M linearly ordered by < . � X , < � is a set of indiscernibles for M iff for all ϕ ( v 1 , . . . , v n ) in the language of M , for all x 1 < · · · < x n and y 1 < · · · < y n in X , M | = ϕ [ x 1 , . . . , x n ] iff M | = ϕ [ y 1 , . . . , y n ] . Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 11 / 27
α -Erd˝ os cardinals κ is α -Erd˝ os if for each structure M in a countable language with universe κ (endowed with Skolem functions), for each club C ⊆ κ there is a set I ⊆ C of order type α such that Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 12 / 27
α -Erd˝ os cardinals κ is α -Erd˝ os if for each structure M in a countable language with universe κ (endowed with Skolem functions), for each club C ⊆ κ there is a set I ⊆ C of order type α such that I is a set of indiscernibles for M and Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 12 / 27
α -Erd˝ os cardinals κ is α -Erd˝ os if for each structure M in a countable language with universe κ (endowed with Skolem functions), for each club C ⊆ κ there is a set I ⊆ C of order type α such that I is a set of indiscernibles for M and I is remarkable: whenever α 0 < · · · < α n ; β 0 < · · · < β n are from I , α i − 1 < β i , τ is a term, and τ M ( α 0 , . . . , α n ) < α i , then Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 12 / 27
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