Steprans, Juris2016-09-202016-09-202015-09-242016-09-20http://hdl.handle.net/10315/32107The goal of this dissertation is to prove two theorems related to a question posed by Felix Hausdorff in 1907 regarding pantachies, which are maximal linearly ordered subsets of the space of real-valued sequences partially ordered by eventual domination. Hausdorff's question was as follows: is there a pantachie containing no gaps of order type the first uncountable cardinal? In Chapter 1, some terminology is defined, and Hausdorff's question about pantachies is explored. Some related work by other mathematicians is examined, both preceding and following Hausdorff's paper. In Chapter 2, relevant definitions and results about forcing, gaps, and saturated linear orders are collected. Chapter 3 contains the complete proof of the first theorem, namely, the consistency of the existence of a saturated Hausdorff pantachie in a model where the continuum hypothesis (CH) fails. Finally, in Chapter 4, a different method is used to prove a stronger result, namely, the consistency of the existence of a saturated Hausdorff pantachie in a model of Martin's Axiom along with the negation of CH. The appendix mentions a few related open questions and some partial answers.enAuthor owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.Theoretical mathematicsElectronic Thesis or Dissertation2016-09-20Set TheoryCardinalCardinal ArithmeticIndependenceGenericGenericityDelta-systemHausdorffFelix HausdorffPantachieMaximal linear orderMaximalLinearLinear orderOrderPartial orderSaturatedSaturated linear orderMaximal saturated linear orderSaturationEventual dominationForcingForcing ExtensionIterated ForcingFinite-support iterated forcingGapsGap-fillingPregapsCutsCut-fillingcontinuumContinuum HypothesisMartin’s AxiomCHMAKunenPartial OrderCountable chain conditionCccSpecial gapsStrong gapsSuslin TreesAronszajn TreesPaul du Bois-ReymondRates of convergenceInfinitesimalHyper-realsUltrapowerUltrafilterIdealDivergenceDivergence OrderingConvergenceAlmost ContainmentRichard LaverInvestigations into Order TypesUniversalUniversal linear orderBaumgartnerJames BaumgartnerKenneth KunenWoodinHugh WoodinProper Forcing AxiomUniform DenseSpecial TreesSpecializerTree-SpecializerTree-Specializing forcingccc-fillableccc-indestructiblePre-caliberPre-caliber aleph 1ModelGround modelCountable transitive modelZFCAntichainMaximal antichainNice partial orderKibediFrancisco KibediStepransJuris Steprans