The proper forcing axiom for $\aleph_1$-sized posets, $ω_1$-linked symmetrically proper forcing, and the size of the continuum
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916912343023616 |
|---|---|
| author | Asperó, David Golshani, Mohammad |
| author_facet | Asperó, David Golshani, Mohammad |
| contents | We show that the Proper Forcing Axiom for forcing notions of size $\aleph_1$ is consistent with the continuum being arbitrarily large. In fact, assuming $GCH$ holds and $κ\geqω_2$ is a regular cardinal, we prove that there is a proper and $\aleph_2$-c.c.\ forcing giving rise to a model of this forcing axiom together with $2^{\aleph_0}=κ$ and which, in addition, satisfies all statements of the form $\mathcal{H}(\aleph_2)\models \exists yφ(a, y)$, where $a\in \mathcal{H}(\aleph_2)$ and $φ(x, y)$ is a $Σ_0$ formula with the property that for every ground model $M$ of $CH$ with $a\in M$ there is, in $M$, a suitably nice poset -- specifically, a poset $\mathbb{Q}\subseteq\mathcal{H}(κ)^M$ which is $ω_1$-linked and symmetrically proper -- adding some $b$ such that $φ(a, b)$. In particular, $\mathbb{P}$ forces Moore's Measuring principle, Baumgartner's Axiom for $\aleph_1$-dense sets of reals, Todorčević's Open Colouring Axiom for sets of size $\aleph_1$, the Abraham-Rubin-Shelah Open Colouring Axiom, and Todorčević's P-ideal Dichotomy for $\aleph_1$-generated ideals on $ω_1$, among other statements. Hence, all these statements are simultaneously compatible with a large continuum. Finally, we show that a further small variation of our construction yields a model satisfying, in addition to all the earlier conclusions, Martin's Maximum for posets of size $\aleph_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_01395 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The proper forcing axiom for $\aleph_1$-sized posets, $ω_1$-linked symmetrically proper forcing, and the size of the continuum Asperó, David Golshani, Mohammad Logic We show that the Proper Forcing Axiom for forcing notions of size $\aleph_1$ is consistent with the continuum being arbitrarily large. In fact, assuming $GCH$ holds and $κ\geqω_2$ is a regular cardinal, we prove that there is a proper and $\aleph_2$-c.c.\ forcing giving rise to a model of this forcing axiom together with $2^{\aleph_0}=κ$ and which, in addition, satisfies all statements of the form $\mathcal{H}(\aleph_2)\models \exists yφ(a, y)$, where $a\in \mathcal{H}(\aleph_2)$ and $φ(x, y)$ is a $Σ_0$ formula with the property that for every ground model $M$ of $CH$ with $a\in M$ there is, in $M$, a suitably nice poset -- specifically, a poset $\mathbb{Q}\subseteq\mathcal{H}(κ)^M$ which is $ω_1$-linked and symmetrically proper -- adding some $b$ such that $φ(a, b)$. In particular, $\mathbb{P}$ forces Moore's Measuring principle, Baumgartner's Axiom for $\aleph_1$-dense sets of reals, Todorčević's Open Colouring Axiom for sets of size $\aleph_1$, the Abraham-Rubin-Shelah Open Colouring Axiom, and Todorčević's P-ideal Dichotomy for $\aleph_1$-generated ideals on $ω_1$, among other statements. Hence, all these statements are simultaneously compatible with a large continuum. Finally, we show that a further small variation of our construction yields a model satisfying, in addition to all the earlier conclusions, Martin's Maximum for posets of size $\aleph_1$. |
| title | The proper forcing axiom for $\aleph_1$-sized posets, $ω_1$-linked symmetrically proper forcing, and the size of the continuum |
| topic | Logic |
| url | https://arxiv.org/abs/2209.01395 |