November 21
Saeideh Bahrami,
Institute for Research in Fundamental Sciences
$I$-small submodels of countable models of arithmetic
There has been a long tradition in the model theory of arithmetic of attributing the combinatorial properties of cardinal numbers in set theory to initial segments. Considering that the most basic use of cardinal numbers is to assign cardinality to sets, we can adapt a similar notion in models of arithmetic in the following way: for a given initial segment $I$ of any model $\mathcal M$ of a fragment of arithmetic, say I$\Sigma_1$, a subset $X $ of $\mathcal M$ is called I-small if there exists a coded bijection $f$ in $\mathcal M$ such that the range of the restriction of $f$ to $I$ is equal to $X$. It turns out that for a given countable nonstandard model $\mathcal M$ of I$\Sigma_1$, when I is a strong cut, any $I$-small $\Sigma_1$-elementary submodel of $\mathcal M$ contains $I$, and inherits some good properties of $I$. In this talk, we are going to review such properties through self-embeddings of $\mathcal M$.