Are there examples of theorems proved via proper (i.e. non-conservative) extensions?

Question

This is not a question about set theory specifically, but lets talk about ZFC just for concreteness

Suppose we have a sentence $\phi$ in the language of ZFC, and a proof that either $(\mathrm{ZFC} \vdash \phi) \vee (\mathrm{ZFC} \vdash \neg\phi)$. We believe $\mathrm{ZFC} \vdash \phi$, but actually demonstrating this is very difficult. So after a lot of head-banging, we get crafty: instead of trying to demonstrate $\mathrm{ZFC} \vdash \phi$, we instead properly extend our theory with a new axiom, say $\mathrm{GCH}$. Note this is not a conservative extension. We demonstrate $\mathrm{ZFC}+\mathrm{GCH} \vdash \phi,$ and thereby conclude that $\mathrm{ZFC} \vdash \phi$.

Has this sort of approach ever been used in practice?

Answer 1

Yes, this is a nice idea, and the approach is used in practice. I list four examples below, but there are many others.

1. Any arithmetic statement, or any first order statement about $(\mathbb R,\mathbb N,+,\times,<,0,1)$ provable with $\mathsf{CH}$ is provable without this additional assumption. This is because, even though adding $\mathsf{CH}$ gives us a non-conservative extension, we can force $\mathsf{CH}$ without adding reals. This is useful, since $\mathsf{CH}$ gives us nice additional structure. For a specific way this can be used: Suppose $\Gamma$ is a countable amenable group. By a result of Christensen and Mokobodzki, if $\mathsf{CH}$ holds, then there is a universally measurable right-invariant mean of $\Gamma$, that is, a positive linear functional on $\ell_\infty(\Gamma)$. This is useful in the descriptive set theoretic study of orbit equivalence relations, rigidity theorems, etc. Alexander Kechris and his collaborators have several results proved by taking advantage of precisely this situation. See for instance his recent monograph Global aspects of ergodic group actions.

2. The $\Sigma^1_2$ theory cannot be changed by forcing, so any $\Sigma^1_2$ statement (in particular, any arithmetic statement) proved using any statement that can be forced, can be proved without using the statement. This is Shoenfield's absoluteness theorem. (An example of its use in computability theory is in the study of infinite time Turing machines by Hamkins and Lewis (doi: 10.2307/2586556, jstor): If $\lnot\mathsf{CH}$ holds, then there are incomparable infinite time degrees, since every degree has only countably many predecessors. But the statement that there are incomparable degrees is $\Sigma^1_2$.)

3. In the partition calculus of $\omega_1$, it is a common strategy to assume $\mathsf{MA}$, Martin's axiom, or some of its consequences, in order to obtain certain positive partition relations. One then argues that these relations must hold without the assumption, using what amounts to a well-foundedness argument. This way of reasoning is key in a famous paper by Baumgartner and Hajnal (A proof (involving Martin's axiom) of a partition relation, Fund. Math. 78 (3), (1973), 193–203; eudml.), and has been used as well by Schipperus, Todorcevic, Jones, and others.

4. Another common application of the idea is to use choice to establish a statement and then argue that the statement holds in $\mathsf{ZF}$. For an example, see this MO question.