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I just learned that Robinson Arithmetic proves all true literal sentences of basic arithmetic. My question is, are there weaker systems that also achieve this result?

In particular:

  1. If we drop the axiom $x \neq 0 \rightarrow \exists y(x=S(y))$ from Robinson arithmetic, does the result still hold?
  2. If we drop the induction schema from PRA and disregard all the symbols that we added to represent the primitive recursive functions, does the result also hold in this context?
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goblin GONE
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1 Answers1

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What is meant by "prove all true literal sentences of basic arithmetic"?

Robinson arithmetic certainly doesn't prove all true sentences of the language of basic arithmetic, so "literal" is doing a lot of work here. One standard meaning of "literal" is atom-or-negated-atom.

Atomic wffs of the language of basic arithmetic are equations of the form $\tau_1 = \tau_2$ where $\tau_j$ is a closed term, built up from numerals and applications of successor, addition and multiplication functions. So the question becomes:

Is there a weaker-than-Robinson arithmetic which proves all true wffs of the forms $\tau_1 = \tau_2$ and $\tau_1 \neq \tau_2$, for arbitrary closed terms $\tau_j$?

Yes. Take the quantifier free theory sometimes known as Baby Arithmetic which has as axioms every instance of

  1. $0 \neq Sn$
  2. $Sm = Sn \to m = n$
  3. $m + 0 = m$
  4. $m +Sn = S(m + n)$
  5. $m \times 0 = 0$
  6. $m \times Sn = m \times n +m$

And take the logic to be propositional logic, plus Leibniz's law, and the reflexivity of identity. Then it is easily shown that this proves every true literal, i.e. every true equation or in equation between terms.

In fact Baby Logic is a negation-complete theory for sentences in its restricted, quantifier free, language. (This is spelt out in my Gödel book, §§10.1, 10.2 in the second edition.)

Peter Smith
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