**Friedman's Theorem: from standard systems to fixed points**

Saeideh Bahrami

Institute for Research in Fundamental Sciences (IPM)

**Abstract:** In 1934, Skolem introduced nonstandard models of arithmetic; i.e. those models which are not isomorphic to \((\omega,0,1,+,\times,<)\), in which \(\omega\) is the set of natural numbers. However, the flourishing of the subject was in 1970s, when studies on nonstandard models of Peano arithmetic, \({\sf PA}\), got deeper and many striking results were achieved; one of the most important works in this area is Friedman's 1973 result on *initial segments* and his rediscovering of *standard system* of a model. Standard systems first appeared in Scott's work (1962) on the family of sets which are known today as Scott's sets.

In particular, Friedman proved: *every countable nonstandard model of \({\sf PA}\) carries a proper initial self-embedding; i.e. an isomorphism between **\(\mathcal{M}\)** and a proper initial segment of **\(\mathcal{M}\)**. *There have been many generalizations and refinements of Friedman's Theorem in the recent decades, which have unveiled many characteristics of the structure of countable nonstandard models of arithmetic

In this talk, we will investigate the notion of standard system of a nonstandard model of arithmetic, and then will focus on initial self-embeddings of countable nonstandard models. Finally, we present a short survey of our work on the structure of fixed point sets of self-embeddings of models of arithmetic in [1] and [2]

[1] S. Bahrami, Tanaka's Theorem Revisited, submitted, https://arxiv.org/abs/1811.08514.

[2] S. Bahrami and A. Enayat, Fixed Points of Self-embeddings of Models of Arithmetic, *Ann.Pure & Appl. Logic* 169, 2018, pp. 487-513

** مکان****: ****تهران** **خیابان انقلاب، دانشگاه تهران،**

** دانشکدهی ریاضی، آمار و علوم کامپیوتر، تالار دکتر هشترودی**

**زمان : دوشنبه، 24 تیر1398، ساعت 4 بعد از ظهر**