# [FOM] Arithmetic in ancestral logic

Peter Smith ps218 at cam.ac.uk
Wed May 3 19:44:54 EDT 2006

```Suppose we add to the logical apparatus of a first-order language an
operator *, such that for a relational expression R, R* is interpreted as
the ancestral of R.

We know that the theory you get by augmenting first-order PA with such an
operator and the axiom Ax(x = 0 v S*0x), where S is the successor relation,
is categorical; so it semantically entails, in particular, every truth of
the original language of PA. We also know that we can't give a sound and
complete axiomatized deductive system for the logic of the * operator.

But we can give a partial axiomatization. For a start, there will be a
series of introduction rules like Rab |- R*ab, Ex(Rax & Rxb) |- R*ab,
ExEy(Rax & Rxy & Ryb) |- R*ab, and so on; and a schematic elimination rule
that says, in effect, that R*ab implies that if a property is had by a and
if x has it, and Rxy, then y has it, then b has it.

Question: is PA augmented with the *-operator and the new axiom and this
sort of partial deductive system for the operator deductively conservative
over PA for the *-less wffs of PA's original language?
--
Dr Peter Smith: Faculty of Philosophy, University of Cambridge

Gödel's Theorems: http://www.godelbook.net
LaTeX for Logicians: http://www.phil.cam.ac.uk/teaching_staff/Smith/LaTeX/
Idle musings: http://logicmatters.blogspot.com

```

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