Geometric Set Theory

Sequential topologies and Dedekind finite sets

“…However, I will show that if W is the choiceless Solovay model, then the P ‐extension of W is a model for the theory required by Theorem 1.2. In order to do that, an analysis of its balance as in [5] is necessary. This analysis takes place in $$\mathsf {ZFC}$$.…”

“…This analysis takes place in $$\mathsf {ZFC}$$. Recall [5, § 5.2] that a pair $$\langle Q, \sigma \rangle$$ is balanced if Q is a poset, σ is a Q ‐name, $$Q\Vdash \sigma \in P$$ and for any two mutually generic extensions $$V[H_0], V[H_1]$$ and any filters $$G_0, G_1\subset Q$$ in $$V[H_0], V[H_1]$$ repectively, which are generic over V , and any conditions $$p_0\le \sigma /G_0$$ and $$p_1\le \sigma /G_1$$ in $$V[H_0], V[H_1]$$ respectively, the two conditions $$p_0, p_1$$ are compatible in the poset P in the model $$V[H_0, H_1]$$. The poset P is balanced if for every condition $$p\in P$$ there is a balanced pair …”

“…Proof Suppose towards a contradiction that this fails. Recall that by [5, Theorem 9.1.1] and the fact that the poset P is balanced in $$\mathsf {ZFC}$$, the P ‐extension of the model W contains no new reals and no new ω‐sequences of reals. Thus, it must be the case that there is a condition $$p\in P$$ and an ω‐sequence x for elements of [0, 1] converging to some y such that $$p\Vdash \operatorname{\mathrm{rng}}(\check{x})\subset \dot{A}$$ and $$\check{y}\notin \dot{A}$$.…”

“…In particular, in it every infinite set contains a countably infinite subset, and no uncountable Polish space is sequential. The inaccessible cardinal assumption is used only to make the construction fit under the umbrella of geometric set theory [5]; I do not know if it is necessary. It is also unclear if it is possible to distinguish between sequentiality of various uncountable Polish spaces; one rather egregious example is $$\mathbb {R}$$ and $$\mathbb {R}^2$$.…”

“…The paper uses standard set theoretic notation as in [3]; in matters of geometric set theory it follows [5].…”