# Non-set-theoretic consequences of forcing axioms

... forcing axioms ... are workhorses that regular mathematicians “can actually go out and use in the field, so to speak,” ...

What are some examples of uses and/or consequences of forcing axioms in "regular mathematics" (i.e. not set theory / logic)?

• Even more strongly one could ask, are there examples of using the technique/philosophy of forcing to show something other than that a particular statement is independent of ZFC (or a closely related logical system). Dec 9 '20 at 5:30
• @Jordan This is common in point-set topology. There are also many examples from algebra (group theory, or the theory of modules.) Dec 9 '20 at 6:37
• @SamHopkins One genre of argument is the forcing + absoluteness argument. That is, you force to make P true and then you show that P had to be true in the ground model all along, thus proving P from ZFC (or whatever axioms you started with). This talk by Assaf Rinot and this talk by Matteo Viale both give overviews of the technique and give some examples of how it's been used. Dec 9 '20 at 18:24
• There is already an MO question about Forcing as a tool to prove theorems. Dec 9 '20 at 19:24
• @JordanMitchellBarrett : They are not; I guess I was mainly responding to Sam Hopkins. Dec 9 '20 at 21:02

Indeed there is a vast of applications, for example:

Using Martin's axiom, Shelah showed that there is a non-free Whitehead group. The book  Consequences of Martin's Axiom'' contains many other examples as well

PFA implies the consistency of Kaplasky's conjecture in functional analysis,

See also List of statements independent of ZFC and Martin's axiom and The Proper Forcing Axiom and possibly many more.

• There is a nice topological formulation of the Whitehead problem, which might be useful for those unfamiliar with Ext: Every path connected compact abelian group is a torus, i.e. $(\mathbb{R}/\mathbb{Z})^\lambda$ for a cardinal $\lambda$. The equivalence follows by Pontyagrin duality, a non-free Whitehead group is the Pontyagrin dual of a non-torus path connected compact abelian group. Dec 13 '20 at 3:20

Farah's proof that all automorphism's of the Calkin algebra are inner under ZFC + Open coloring axiom,. The Calkin algebra is the quotient of the algebra of continuous linear operators on a separable Hilbert space by the ideal of compact operators (if I remember correctly this is the only closed ideal). It was known for a long time that you can construct an outer automorphism under ZFC + Continuum Hypothesis, but it wasn't known if CH is necessary for this.

I'm not sure if OCA is considered to be a forcing axiom, but it is a consequence of PFA, which certainly is. My somewhat vague understanding is that OCA is supposed to be a consequence of PFA which is both comprehensible and maybe even useful to non-logicians. In general, CH gives lots of isomorphisms between structures (so lots of automorphisms) and PFA makes things more rigid. Mathematicians like rigidity, so maybe they will like consequences of PFA. The stuff on automatic continuity of homomorphisms between Banach algebras (Kaplansky's conjecture) mentioned by Golshani fits in this picture. CH gives you discontinuous homomorphisms and PFA gives automatic continuity.

Farah's theorem descends from the earlier theorem of Shelah that is it consistent with ZFC that every automorphism of $$\mathcal{P}(\mathbb{N})/\mathrm{Fin}$$ is induced by an automorphism of $$\mathcal{P}(\mathbb{N})$$, here $$\mathcal{P}(\mathbb{N})$$ is the boolean algebra of subsets of $$\mathbb{N}$$ and $$\mathrm{Fin}$$ is the ideal of finite sets. Shelah's original argument is a complicated forcing, it was later proven from ZFC + OCA + Martin's axiom. Analogous results hold for other quotients of $$\mathcal{P}(\mathbb{N})$$ by other ideals, Farah has a nice survey.

Farah has other interesting applications of set theory to operator algebras. Some of this uses forcing axioms, and some of it doesn't. For example Farah and Hirshberg have constructed, under ZFC + a form of diamond, an inductive limit of matrix algebras which is not isomorphic to its opposite. The extra axiom they use is intermediate between CH and V = L, and I don't think it's considered to be a forcing axiom.

It seems that in general set theory has a lot to say about non-separable operator algebras. I think most operator algebraists now mainly work with separable algebras, probably partly to avoid set-theoretic issues, but there are certainly interesting non-separable algebras such as the Calkin algebra, and it is interesting to see which results from the separable case generalize to the non-separable case and which conjectures in the separable case fail in the non-separable case. My impression is that operator algebraists have been pretty open to input from logic. (Logicians are often very interested in applying logic to other areas, other areas have varied levels of interest in being applied to.)