Rules and parameter-free schemata in arithmetic
Abstract:
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.