#### Title

On automorphisms of models of Peano arithmetic

#### Date of Completion

January 2004

#### Keywords

Mathematics

#### Degree

Ph.D.

#### Abstract

When studying the automorphism group Aut(*M*) of a model * M*, one is interested to what extent *M* is recoverable from Aut(*M*). We show that if *M* is a countable arithmetically saturated of Peano Arithmetic, then Aut(*M*) can recognize if a maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. ^ We use that fact to show that if *M*_{1}, * M*_{2} are countable arithmetically saturated models of Peano Arithmetic such that Aut(*M*_{1}) ≅ Aut(* M*_{2}), then (ω, Rep(Th(*M*_{1}))) ⊨RT^{n} 2 iff (ω, Rep(Th(*M*_{2})) ⊨RT^{n} 2 . Here: RT^{n}2 is Ramsey's Theorem stating that every 2-coloring of [ω]* ^{ n}* has an infinite homogeneous set; and if

*T*⊃ PA is a complete and consistent theory, then we define Rep(

*T*) = {ω ∩

*X*:

*X*is a definable set in a prime model of

*T*}. ^ Using this result we show the existence of countable arithmetically saturated models

*M*

_{0},

*M*

_{1},

*M*

_{2},

*M*

_{3}of Peano Arithmetic such that they have the same standard system and Aut(

*M*) ≇ Aut(

_{i}*M*), where

_{j}*i < j*< 4. ^ We also show the similar results for saturated models of Peano Arithmetic of cardinality ℵ1 . ^

#### Recommended Citation

Nurkhaidarov, Ermek S, "On automorphisms of models of Peano arithmetic" (2004). *Doctoral Dissertations*. AAI3144603.

https://digitalcommons.lib.uconn.edu/dissertations/AAI3144603