Let $T_0$ be the set theory axiomatized by $ZFC^-$ (that is $ZFC$ without powerset) + every set is countable + $\mathbb{V}=\mathbb{L}$.
Question 1: Suppose $\phi$ is a sentence of set theory. Must there be a large cardinal axiom $A$ such that $\phi$ is decided by $T_0$ +"there are transitive set models of $A$ of arbitrarily high ordinal height?"
Update: no.
Question 2: Is there a set $T$ of $\Pi_2$ sentences, such that $ZFC^- \cup T$ is complete?
Update: no, essentially. See my answer below.
Questions 3 and 4 after comments.
Comments.
- Question 1 is related to, but different from several previous questions on Mathoverflow, e.g. Nice Algebraic Statements Independent from ZF + V=L (constructibility) or On statements independent of ZFC + V=L or Natural statements independent from true $\Pi^0_2$ sentences. Regarding the first two---Hamkins gives a long list of examples of things independent of $\mathbb{V}= \mathbb{L}$, but it seems that my schema takes care of all of them. (Of course in those questions $ZFC$ was assumed.)
- Moreover, Question 1 was partly motivated by Dorais' answer to the second question above, which references a question of Shelah from The Future of Set Theory.
- I'm leaving "large cardinal axiom" undefined here (so Question 1 can't be formalized), but obviously we should exclude inconsistent axioms, or things like $ZFC + $ "there are no transitive set models of ZFC."
If we pick a definition of ``large cardinal axiom", then "every set is countable"+ $\mathbb{V}=\mathbb{L}$ +"there are transitive set models of $A$ of arbitrarily high ordinal height" (for each large cardinal axiom $A$) is a set of $\Pi_2$ sentences. So a negative answer to Question 2 implies a negative answer to Question 1.
We can ignore (recursive) large cardinal axiom schemas $\Gamma$ because we can just replace them by $ZFC_0$+"there is a transitive set model of $\Gamma$" where $ZFC_0$ is some large enough finite fragment of $ZFC$. In particular we don't have to write "$ZFC + A$".
$T_0$ + this axiom schema (loosely speaking) is of personal significance to me: in fact I believe it to be true. (Namely, given whatever universe of sets $\mathbb{V}$ in which we are working, it seems reasonable to suppose there is a larger universe of sets $\mathbb{W} \models \mathbb{V}=\mathbb{L}$ in which $\mathbb{V}$ is countable, or such that $\mathbb{W}$ is a model of a given large cardinal axiom $A$. Ergo,...)
Let $\mathcal{L}_{\mbox{set}}$ be the language of set theory $\{\in\}$ and let $\mathcal{L}_1$ be $\mathcal{L}_{\mbox{set}} \cup \{P\}$, $P$ a new unary relation symbol. Let $T_1$ be $T_0$ + the axioms asserting that $P \subseteq \mbox{ON}$ is stationary (for $\in$-definable classes) and for every $\alpha \in P$, $(\mathbb{V}_\alpha, \in) \preceq (\mathbb{V}, \in)$. We insist that large cardinal axioms $A$ be sentences of $\mathcal{L}_{\mbox{set}}$.
Question 3: Suppose $\phi$ is a sentence of set theory (i.e. of $\mathcal{L}_{\mbox{set}}$). Must there be a large cardinal axiom $A$ such that $\phi$ is decided by $T_1$ +"there are transitive set models of $A$ of arbitrarily high ordinal height?"
Question 4: Is there a set $T$ of $\Pi_2$ sentences of set theory, such that $T_1 \cup T$ decides every sentence of set theory?
