On cuts in ultraproducts of linear orders I

Mohammad Golshani, Saharon Shelah
2016 Journal of Mathematical Logic  
For an ultrafilter D on a cardinal κ, we wonder for which pair (θ 1 , θ 2 ) of regular cardinals, we have: for any (θ 1 + θ 2 ) + −saturated dense linear order J, J κ /D has a cut of cofinality (θ 1 , θ 2 ). We deal mainly with the case θ 1 , θ 2 > 2 κ . 3 In fact it is easily seen that κ * = θ 1 . M. GOLSHANI AND S. SHELAH Proof. Suppose not. Let J be a θ + −saturated dense linear order and suppose f α /D : α < θ , g α /D : α < θ witnesses (θ, θ) ∈ C (D). For α < β < θ we have Since 2 κ < θ
more » ... θ is weakly compact, we can find So it suffices to pick some element in (f αn (i), g αn (i)) J , which is possible as J is dense. Since X is unbounded in θ, we have ∀α < θ, f α < D h < D g α , and we get a contradiction. The next theorem generalizes the above result. Theorem 2.10. Suppose that D is an ultrafilter on κ, κ < μ ≤ λ, θ, where μ is a supercompact cardinal and λ, θ are regular. Then (λ, θ) / ∈ C (D). Proof. Suppose not. Let J be a (λ + θ) + −saturated dense linear order, J ⊆ Ord, and let f α /D : α < λ , g γ /D : γ < θ witness a pre-cut in J κ /D, which is a pre-cut in J κ * /D, for each linear order J * ⊇ J. Letf/D = f α /D : α < λ andḡ/D = g γ /D : γ < θ . Let η = (λ + θ), U be a normal measure on P μ (η) and let j : V → M Ult(V, U ) be the corresponding elementary embedding. So we have crit(j) = μ and M η ⊆ M. It follows that j(κ) = κ and j(D) = D. Note that j[J] is also a (λ + θ) + −saturated dense linear order and for f ∈ J κ , j(f ) = j[f ] ∈ j[J] κ , as |f | = κ < η = crit(j). It follows that: " j(f α )/D : α < λ , j(g γ )/D : γ < θ witnesses a pre-cut in ON CUTS IN ULTRAPRODUCTS OF LINEAR ORDERS I 7 ( * ) 1 j[J] κ /D, which is also a pre-cut in J κ * /D, for each linear order J * ⊇ j[J]". Also, as j is an elementary embedding, the following hold in M : "j(J) is a j((λ + θ) + )−saturated dense linear order, j(J) ⊇ j[J],
doi:10.1142/s0219061316500082 fatcat:osc373z3rvcbznrzvz6swu2o7q