Below, “logic” means “regular logic containing first-order logic” in the sense of Ebbinghaus/Flum/Thomas. Setup For a logic $\mathcal{L}$, let $\mathcal{PA}(\mathcal{L})$ be the set of $\mathcal{L}$-sentences in the language of arithmetic consisting of: the ordered semiring axioms, and for each $\mathcal{L}$-formula $\varphi(x,y_1,…,y_n)$ the induction instance $$\forall y_1,…,y_n[[\varphi(0,y_1,…,y_n)\wedge\forall x(\varphi(x,y_1,…,y_n)\rightarrow\varphi(x+1,y_1,…,y_n))]\rightarrow\forall x\varphi(x,y_1,…,y_n)].$$ (Note that even if the new logic […]

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## Is there a natural intermediate version of PA?

- Post author By Q+A Expert
- Post date March 30, 2020
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- Tags ..., 'Y_1', "logic" means "regular logic containing first-order logic" in the sense of Ebbinghaus/Flum/Thomas. Setup For a logic $\mathcal{L}$, \omega})$ pins down $\mathbb{N}$ up to isomorphism (consider $\forall x(\bigvee_{i\in\omega}x=1+...+1\mbox{ ($i$ times)}$). So both second-or, \omega}$ is not PA-intermediate, $\mathcal{PA}(FOL)$ is just the usual (first-order) PA, and $\mathcal{PA}(SOL)$ characterizes the standard model $\mathbb{N}$ up to isomorphism. Also, and for each $\mathcal{L}$-formula $\varphi(x, below, I'll interpret "natural" as "has appeared in at least two different papers whose respective authorsets are incomparable.") Partial progress, induction applied to "has finitely many predecessors" gets the job done. (If we run the analogous construction with ZFC in place of PA, let $\mathcal{PA}(\mathcal{L})$ be the set of $\mathcal{L}$-sentences in the language of arithmetic consisting of: the ordered semiring axio, on the other hand, since in fact $\mathcal{PA}(\mathcal{L}_{\omega_1, the latter is stronger semantically: no countable nonstandard model of $PA$ satisfies $\mathcal{PA}(FOL[Q])$ (to prove conservativity over $P, then we can whip up an $\mathcal{L}$-sentence $\varphi$ and an appropriate tuple of formulas $\Theta$ such that $\varphi$ is satisfiable and, there is a surprising amount of variety among logics without the downward Lowenheim-Skolem property even in relatively concrete contexts - e., things seem more interesting ....) Question Say that a logic $\mathcal{L}$ is PA-intermediate if we have $PA<\mathcal{PA}(\mathcal{L})<Th_{, though, we always have $\mathbb{N}\models_\mathcal{L}\mathcal{PA}(\mathcal{L})$. Note that it's not quite true that $\mathcal{PA}(SOL)$ "is" second-, we still have a scheme as opposed to a single sentence. However, we wind up with a conservative extension of $PA$. That said, y_n))]\rightarrow\forall x\varphi(x, y_n)].$$ (Note that even if the new logic $\mathcal{L}$ has additional types of variable - e.g. "set" variables - the corresponding inductio, y_n)\rightarrow\varphi(x+1, y_n)\wedge\forall x(\varphi(x, y_n)$ the induction instance $$\forall y_1, y_n[[\varphi(0, y_n$ for the parameters.) For example