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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".

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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 $\mathsf Q$.

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  • $\begingroup$ 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? $\endgroup$ Commented May 18, 2019 at 20:37
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    $\begingroup$ @JasonHemann - what about the "sequence" P,Q,R ? $\endgroup$ Commented May 19, 2019 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. $\endgroup$ Commented May 20, 2019 at 23:22
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So far as I know, there are two interpretations why Raphael Robinson uses $\mathsf{Q}$ represents Robinson Arithmetic.

  1. By John Baez, $\mathsf{Q}$ is the name of a character in Star Trek who could instantly judge whether any statement was provable in this system, or not. While Star Trek came out in 1966, and Robinson Arithmetic was first considered by Raphael Robinson in his 1950's paper:

Robinson, R., (1950) 1952. An Essentially Undecidable Axiom System. In: Graves, L. M., Smith, P. A., Hille, E. and Zariski, O. ed. Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U. S. A. August 30-September 6, 1950, Volume I. Rhode Island: American Mathematical Society, 729-730.

and the symbol $\mathsf{Q}$ already appeared on page 51 of his 1953's book:

Tarski, A., Mostowski, A. and Robinson, R., 1953. Undecidable Theories. Amsterdam: North-Holland Publishing Company. Reprinted by Dover Publishing Company, Inc. in 2010.

So the interpretation is not suitable.

  1. In my opinion, the suitable interpretation for Robinson's using $\mathsf{Q}$ to represent Robinson Arithmetic is that $\mathsf{Q}$ is after $\mathsf{P}$ in order since in his 1953's book Robinson uses $\mathsf{P}$ to represent Peano Arithmetic, and Robinson Arithmetic is the subsequent theory they consider.
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