New methods in forcing iteration and applications

Abstract The Theme. Strong forcing axioms like Martin’s Maximum give a reasonably satisfactory structural analysis of $H(\omega _2)$ . A broad program in modern Set Theory is searching for strong beyond $\omega _1$ In other words, one would to figure out the properties taller initial segments universe. However, classical techniques iterations seem unable bypass obstacles, as resulting forcings have not thus far been enough! with his celebrated work on generalised side conditions, I. Neeman introduced us novel paradigm iterate forcings. particular, he could, among things, reprove consistency Proper Forcing Axiom using an iterated finite supports. 2015, technology virtual models, Veličković built up iteration semi-proper supports, hence reproving Maximum, achievement leading notion model. this thesis, we are interested constructing notions finitely many models conditions preserve three uncountable cardinals. thesis constitutes six chapters and appendices that amount 118 pages, where Section 1 devoted preliminaries, 2 warm-up about scaffolding poset proper forcing. 3, present general theory context sets two types, we, e.g., define “meet” between prove its properties. main results joint Boban Veličković, partly appeared Guessing approachability ideal, J. Math. Log. 21 (2021) Pure Side Conditions. 4, use types (countable large non-transitive ones induced by supercompact cardinal, which call Magidor models) construct our pure conditions. covertly uses third type transitive. We also add decorations clubs generic _2$ contrast Neeman’s method, do single chain, but $\alpha $ -chains, ordinal $V_\alpha \prec V_\lambda Thus, starting from suitable cardinals $\kappa <\lambda , whose described earlier. strongly proper, preserves has $\lambda -Knaster property. relevant quotients helps guessing model principles. construction generalisable ${<}\mu -closed forcing, any given cardinal $\mu ^{<\mu }=\mu <\kappa Iteration Theorem. 5, subclass $\aleph -c.c. obtain axiom at level iterable class closely related Asperó–Mota’s Model Principles. 6 encompasses parts thesis. principle $\mathrm {GMP}^+(\omega _3,\omega _1)$ states ${\theta \geq \omega _3}$ set -sized elementary submodels M $H(\theta )$ union -continuous $\in -chain -guessing, I.C. stationary $\mathcal P_{\omega _3}(H(\theta ))$ consequences demonstrated following diagram. can above way -guessing remain outer transitive same denote $\rm{SGMP}^+(\omega_3,\omega_1)$ diagram, $\mathrm{TP}$ stands tree property; $w\mathrm{KH}$ weak Kurepa Hypothesis; $\mathrm{MP}$ Mitchell property, i.e., ideal trivial modulo nonstationary ideal; $\mathrm{AP}$ {AMTP}(\kappa ^+)$ if $2^\kappa <\aleph _{\kappa ^+}$ then every adds new subset ^+$ ground model, collapses some $\leq 2^{\kappa }$ dotted arrow denotes relative consistency, while others logical implications. Appendices. Appendix includes merely B presents proof Mapping Reflection Principle under {PFA}$ C contains open problems. Finally, thesis’s bibliography consists 42 items. prepared Rahman Mohammadpour E-mail : [email protected] URL https://theses.hal.science/tel-03209264