Rules and parameter-free schemata in arithmetic

Lev Beklemishev (logic seminar)


We present a simple proof-theoretic method to obtain conservation results between the following types of fragments of arithmetic:
(a) schemata with parameters;
(b) schemata without parameters;
(c) fragments axiomatized by inference rules.

Our main concern are the above forms of the traditional induction and collection principles.

It is shown that many curious properties of fragments of type (b) can be easily explained by their tight relationship with (c). For example, we obtain some new insights into the hierarchy by the *number of instances* of such schemata.