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ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
Projective absoluteness under Sacks forcing
"... Abstract. We show that Σ 1 3absoluteness under Sacks forcing is equivalent to the Sacks measurability of every ∆ 1 2 set of reals. We also show that Sacks forcing is the weakest forcing notion among all of the preorders which always add a new real with respect to Σ 1 3 forcing absoluteness. 1. ..."
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Abstract. We show that Σ 1 3absoluteness under Sacks forcing is equivalent to the Sacks measurability of every ∆ 1 2 set of reals. We also show that Sacks forcing is the weakest forcing notion among all of the preorders which always add a new real with respect to Σ 1 3 forcing absoluteness. 1.
Chain models, trees of singular cardinality and dynamic EFgames ∗
"... Let κ be a singular cardinal. Carol Karp’s notion of a chain model of size κ is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf(κ). With a notion of satisfaction and (chain)isomorphism such models give an infinitary logic largely mimi ..."
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Let κ be a singular cardinal. Carol Karp’s notion of a chain model of size κ is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf(κ). With a notion of satisfaction and (chain)isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EFgame which gauges when two chain models are chainisomorphic. To this game is associated a tree which is a tree of size κ with no κbranches (even no cf(κ)branches). The measure of how nonisomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κbranches under this notion and prove that when cf(κ) = ω this collection is rather regular; in particular it has universality number exactly κ +. Such trees are then used to develop a descriptive set theory of the space cf(κ) κ. The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive settheoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analogue of the notion of Scott watershed from the Scott analysis of countable models.