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Woodin's program of refuting CH, as summarized in 1, continues the following assertions (roughly as in Propositions 7, 13, and 20 of that paper):

  • In any model of $\text{ZFC}$, the theory of $(H(\omega), \in)$, equivalently, of $(\omega, +, \cdot)$, is invariant under forcing. Moreover, $\text{ZFC}$ settles every "interesting" first-order sentence about $(H(\omega), \in)$.
  • In any model of $\text{ZFC}+\exists \text{ a proper class of Woodin cardinals}$, the theory of $(H(\omega_1),\in)$, equivalently, of $(\omega, P(\omega),+,\cdot,\in)$ is invariant under forcing. Moreover, $\text{ZFC}+\exists \text{ infinitely many Woodin cardinals}$ settles any "interesting" first-order sentence about $(H(\omega_1), \in)$.

In the second case, the consistency strength of the theory is known to be optimal. In the first case I do not know if full $\text{ZFC}$ is required to settle all "interesting" first-order arithmetic statements. The strongest theory for which I know such an independent "interesting" statement is $\Pi^1_1\text{-CA}_0$, as presented in 2. Thus I ask:

Question 1: Is there an "interesting" first-order arithmetic statement which is true but unprovable in full second-order arithmetic $\text{Z}_2$? If so, what is the weakest "natural" theory $T$ which proves this statement? (e.g. third-order arithmetic, finite-order arithmetic, Zermelo set theory etc)

Meanwhile, the second-order statement $\text{PD}$ already implies all "interesting" second-order consequences of $\text{ZFC}+\exists \text{ infinitely many Woodin cardinals}$. So I also ask:

Question 2: Is $\text{Z}_2$ a conservative extension of a "natural" theory $T_0$ in the language of first-order arithmetic, at least for $\Pi_1$ or $\Pi_2$ formulas? If not, what is the strongest subsystem of $\text{Z}_2$ whose first-order consequences, at least of low complexity, we can axiomatize?

1 Patrick Dehornoy, Recent progress on the Continuum Hypothesis (after Woodin) (author's site, pdf)

2 Stephen G. Simpson, Nonprovability of certain combinatorial properties of finite trees (journal site, pdf of German version)

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    $\begingroup$ I have no idea what you mean by “interesting”; this is entirely subjective. Concerning Question 2, since $Z_2$ is essentially reflexive, you can just axiomatize $T_0$ with the (arithmetic) reflection schema $T_0=Q+\bigcup_{n\in\mathbb N}\mathrm{Rfn_{Z_2\restriction n}}$ (or the uniform reflection schema $\mathrm{RFN}_{Z_2\restriction n}$ if you wish). $\endgroup$ Commented Jan 9 at 8:48
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    $\begingroup$ The notions of "interesting" and "natural" are philosophical concepts, rather than precise mathematical notions, and to my way of thinking it has little meaning to ask precise technical questions involving these notions. They are fine as a motivation for us to articulate precise formulations of what they might mean, but we are not entitled to use these notions as though they have already been precisely articulated. To ask a meaningful mathematical question, one has to do the work of saying what it means to be interesting or natural in order to ask a question involving these ideas. $\endgroup$ Commented Jan 9 at 9:47
  • $\begingroup$ See Harvey Friedman's work. $\endgroup$ Commented Jan 9 at 10:16
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    $\begingroup$ This "interesting" seems to mean "that cannot be refuted by going to a generic extension", as in footnote 42 and the "empirical completeness" in Dehornoy's paper. $\endgroup$ Commented Jul 31 at 19:50
  • $\begingroup$ @MattF. No. Arithmetical statements are forcing invariant. $\endgroup$ Commented Jul 31 at 21:47

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