We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in $\mathsf {ZF}$ , and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in $\mathsf {ZF}+\mathsf {DC}$ and $\mathsf {ZFC}$ . Our results confirm $\mathsf {ZF} + \mathsf {DC}$ as a natural foundation for a significant portion of “classical mathematics” and provide support to the idea of this theory being also a natural foundation for a large part of set theory.
The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of $ZF$ between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of $ZF$. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.
We construct a model M of ZF which lies between L and L[c] for a Cohen real c and does not have the form L(x) for any set x. This is loosely based on the unwritten work done in a Bristol workshop about Woodin's HOD Conjecture in 2011. The construction given here allows for a finer analysis of the needed assumptions on the ground models, thus taking us one step closer to understanding models of ZF, and the HOD Conjecture and its relatives. This model also provides a positive answer to a question of Grigorieff about intermediate models of ZF, and we use it to show the failure of Kinna-Wagner Principles in ZF. Bristol whose aim was to see whether or not Woodin's HOD Conjecture implies his Axiom of Choice Conjecture. The participants were (hence forth known as The Bristol Group).They began the workshop by trying to understand how would one refute the hypothesis which they were set to prove. This led them to the construction of a model M which lies between L and L[c], where c is a Cohen real over L, such that Date
Jech proved that every partially ordered set can be embedded into the cardinals of some model of ZF. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of ZF + DC<κ for any regular κ. We use this theorem to show that for all κ, the assumption of DCκ does not entail that there are no decreasing chains of cardinals. We also show how to extend the result to and embed into the cardinals a proper class which is definable over the ground model. We use this extension to give a large-cardinals-free proof of independence of the weak choice principle known as WISC from DCκ.
We show that it is equiconsistent with ZF that Fodor's lemma fails everywhere, and furthermore that the club filter on every regular cardinal is not even σ-complete. Moreover, these failures can be controlled in a very precise manner.Date: June 2, 2017. 2010 Mathematics Subject Classification. Primary 03E40; Secondary 03E05, 03E25, 03E35. Key words and phrases. symmetric extensions, Fodor's lemma, closed and unbounded filter, iterated symmetric extensions, the axiom of choice.
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