A fragment of Asperó–Mota’s finitely proper forcing axiom and entangled sets of reals

Asperó–mota Iteration and the Size of the Continuum

“…This proof is a standard proof in the context of the side condition method (see, e.g., [12, Lemma 4]), and similar to the proof in [10]. Suppose that (then ), , is a predense subset of , and .…”

“…The forcing iteration in this paper deals with Asperó–Mota iterations with symmetric markers, like in [5, 10]. We disclose a technical limitation of this type of iterations; in fact our results show that the length of Asperó–Mota iterations with symmetric markers in the style of [5, 10] must be at most in order to ensure their properness.…”

“…The main ingredient is not only the use of symmetric systems of models but also symmetric systems of models with markers. Asperó–Mota iterations are used in the papers [2–5, 10, 13]. In [5, 10], the iterations require that markers of models in symmetric systems also have symmetry in a suitable sense (Definition 4.1 , , , and ).…”

“…Asperó–Mota iterations are used in the papers [2–5, 10, 13]. In [5, 10], the iterations require that markers of models in symmetric systems also have symmetry in a suitable sense (Definition 4.1 , , , and ).…”

“…The Asperó–Mota iterations used in [10] have length and are proper. The Asperó–Mota iterations used in [5] have length beyond and are claimed to be proper.…”