TITLE: Does $\mathrm{PA}^-$ prove all true literal sentences of basic arithmetic?
QUESTION [5 upvotes]: By "the literal sentences of basic arithmetic" let us mean sentences like
$$3+4=7,\;\; 2\cdot 3 = 3\cdot 2, \;\;S(2)=3$$
where for example $3$ is shorthand for $S(S(S(0)).$
Note that some literal sentences of basic arithmetic are true (e.g. $3+4=7$) while others are false (e.g. $2\cdot 3 = 1$).
Now let $\mathrm{PA}^-$ denote Peano arithmetic without induction. Does $\mathrm{PA}^-$ prove all true literal sentences of basic arithmetic?

REPLY [5 votes]: This theory is called Robinson arithmetic and the answer is yes assuming you mean sentences without quantifiers or variables.  There are some very simple quantified sentences that cannot be proved such as commutativity of addition ($\forall x, y: x + y = y + x$), although any instance of that such as $3 + 5 = 5 + 3$ can be.