I see Peano arithmetic so often abbreviated as "P" or "PA". Why is Robinson Arithmetic "Q"? Following the obvious pattern, I would have expected R" or "RA".


I think that there is no special reason here.

See Alfred Tarski & Andrzej Mostowski & Raphael Robinson, Undecidable Theories (North Holland, 1971); Ch.2 : Undecidability in Arithmetic (page 51) deals with the :

formalized arithmetic of natural numbers [...] referred to as Theory $\mathsf N$.

We shall be interested in some axiomatic subtheories of $\mathsf N$ referred to as Theories $\mathsf P, \mathsf Q, \mathsf R$.

And see also the footnote on page 39:

A finitely axiomatizable and essentially undecidable subtheory $\overline {\mathsf Q}$ of the arithmetic of natural numbers was first constructed by Mostowski and Tarski in 1939. [...] R.M.Robinson has shown that Theory $\overline {\mathsf Q}$ can be replaced by a weaker theory based upon a simpler axiom system, in fact, by Theory $\mathsf Q$.

  • $\begingroup$ Can you talk more about how that passage from Tarski et al. 71 leads to you the conclusion that there's no real reasoning behind the name? $\endgroup$ – Jason Hemann May 18 at 20:37
  • 1
    $\begingroup$ @JasonHemann - what about the "sequence" P,Q,R ? $\endgroup$ – Mauro ALLEGRANZA May 19 at 7:53
  • $\begingroup$ That's a thought. I'd sure feel more comfortable with something stronger to confirm it, though. I don't have access to that book/page. I haven't traced back the reference to Mostowski & Traski '39, to see what they called their axiom system, if any, or Robinson's original paper on the subject. If Tarski et al. is the origin of that nomenclature, then I'd be sold. But for some reason I'm doubting that it is, so to me, the question still seems open. $\endgroup$ – Jason Hemann May 20 at 23:22

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