Abstract
We reimplement the creature forcing construction used by Fischer et al. (Arch Math Log 56(7–8):1045–1103, 2017. https://doi.org/10.1007/S00153-017-0553-8. arXiv:1402.0367 [math.LO]) to separate Cichoń’s diagram into five cardinals as a countable support product. Using the fact that it is of countable support, we augment our construction by adding uncountably many additional cardinal characteristics, sometimes referred to as localisation cardinals.
Highlights
Set theory began as a mathematical subject when Georg Cantor discovered the notion of infinite cardinality and proved that the cardinality of the set of real numbers is different from the cardinality of the set of natural numbers א0
Klausner consistently be infinite sets of reals of intermediate cardinality, several cardinal numbers of potentially “intermediate” size were known, and the inability of mathematicians to prove equalities between them already hinted at the vast range of unprovability results that emerged as Cohen’s forcing method was developed and refined
We show how to add a variant of the lim sup creature forcing posets used to separate the localisation cardinals c f,g from [13] to this construction
Summary
Set theory began as a mathematical subject when Georg Cantor discovered the notion of infinite cardinality and proved that the cardinality of the set of real numbers (the continuum 2א0 ) is different from the cardinality of the set of natural numbers א0. A lim inf creature forcing poset Qnm to increase non(M) (and cof(N ) = max(non(M, d)) = non(M) to to κnm The latter two are still simpler than the parts of the original creature forcing construction corresponding to them; we believe they cannot be replaced by countable support products of creature forcing posets. This new representation allows describing the methods and proofs used in a more modular way, which can more be combined with other lim sup creature forcing posets. – Sections 8 and 11 prove (M4), – Section 9 proves (M2), – Section proves (M5), and – Sections and 12 prove (M3). in Sect. 13, we give a brief account of the limitations of the method (and some of our failed attempts to add factors to the construction) and open questions
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