Nonstandard Models of Arithmetic: The Series

Posts 1–10 are available as pdf files, formatted for small and medium screens.

Non-standard Models of Arithmetic 1: John kicks off the series by asking about recursively saturated models, and I say a bit about n-types and the overspill lemma. I mention the arithmetical hierarchy.

Non-standard Models of Arithmetic 2: John mention some references, and sets a goal: to understand Enayat’s paper “Standard Models of Arithmetic”. He describes his dream: to show that “the” standard model is a much more nebulous notion than many seem to believe. He says a bit about the overspill lemma, and Joel David Hamkins gives a quick overview of saturation.

Non-standard Models of Arithmetic 3: A few remarks on the ultrafinitists Alexander Yessenin-Volpin and Edward Nelson; also my grad-school friend who used to argue that 7 might be nonstandard.

Non-standard Models of Arithmetic 4: Some back-and-forth on Enayat’s term “standard model” (or “ZF-standard model”) for the omega of a model of ZF. Philosophy starts to rear its head.

Non-standard Models of Arithmetic 5: Hamlet and Polonius talk math, and I hold forth on my philosophies of mathematics.

Non-standard Models of Arithmetic 6: John weighs in with why he finds “the standard model of Peano arithmetic” a problematic phrase. The Busy Beaver function is mentioned.

Non-standard Models of Arithmetic 7: We start on Enayat’s paper in earnest. Some throat-clearing about Axiom SM, standard models of ZF, inaccessible cardinals, and absoluteness. “As above, so below”: how ZF makes its “gravitational field” felt in PA.

Non-standard Models of Arithmetic 8: A bit about the Paris-Harrington and Goodstein theorems. In preparation, the equivalence (of sorts) between PA and ZF¬∞. The universe Vω of hereditarily finite sets and its correspondence with $\mathbb{N}$. A bit about Ramsey’s theorem (needed for Paris-Harrington). Finally, we touch on the different ways theories can be “equivalent”, thanks to a comment by Jeffrey Ketland.

Non-standard Models of Arithmetic 9: I sketch the proof of the Paris-Harrington theorem.

Non-Standard Models of Arithmetic 10: Ordinal analysis, the function growth hierarchies, and some fragments of PA. Some questions that neither of us knows how to answer.

Non-standard Models of Arithmetic 11: Back to Enayat’s paper: his definition of PAT for a recursive extension T of ZF. This uses the translation of formulas of PA into formulas of ZF, $\varphi\mapsto \varphi^\mathbb{N}$. Craig’s trick and Rosser’s trick.

Non-standard Models of Arithmetic 12: The strength of PAT for various T‘s. PAZF is equivalent to PAZFC+GCH, but PAZFI is strictly stronger than PAZF. (ZFI = ZF + “there exists an inaccessible cardinal”.)

Non-standard Models of Arithmetic 13: Enayat’s “natural” axiomatization of PAT, and his proof that this works. A digression into Tarski’s theorem on the undefinability of truth, and how to work around it. For example, while truth is not definable, we can define truth for statements with at most a fixed number of quantifiers.

Non-standard Models of Arithmetic 14: The previous post showed that PAT implies ΦT, where ΦT is Enayat’s “natural” axiomatization of PAT. Here we show the converse. We also interpret ΦT as saying, “Trust T”.