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Carl's answer is correct.

There is also a to more direct way to achieve the same thing. The Tennenbaum's theorem holds for much weaker theories, e.g. $I\Delta_0$ (even for far weaker theories like $IOpen$ plus some number theoretic principles)principles, but not for $IOpen$, a result due to Shepherdson).
$I\Delta_0+Exp$ I\Delta_0+exp$ is finitely axiomatization due to a result by axiomatizable, see Haim Gaifman and Constantine Dimitracopoulosin their paper , "Fragments of Peano's Arithmetic and the MRDP Theorem" and it Theorem". It is also a sub-theory of $PA$.

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For more on Tennenbaum's theorem and weak arithmetics, have a look at this paperfor what is known about the weak theories and Tennenbaum's theorem:
Shahram Mohsenipour, "Hierarchies of Subsystems of Weak Arithmetic", to appear
in "Set theory, Arithmetic, Philosophy: Essays in Memory of Stanley Tennenbaum" (edited by J. Kennedy and R. Kossak), Cambridge University Press.
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Carl's answer is correct.

There is also more a more direct way to achieve the same thing. The Tennenbaum's theorem holds for much weaker theories, e.g. $I\Delta_0$ (even far weaker theories $IOpen$ plus some number theoretic principles). $I\Delta_0+Exp$ is finitely axiomatization due to a result by Haim Gaifman and Constantine Dimitracopoulos in their paper "Fragments of Peano's Arithmetic and the MRDP Theorem" and it is also a sub-theory of $PA$.

Edit

Take a look at this paper for what is known about the weak theories and Tennenbaum's theorem:
Shahram Mohsenipour, "Hierarchies of Subsystems of Weak Arithmetic",
to appear in Set theory, Arithmetic, Philosophy: Essays in Memory of Stanley Tennenbaum (edited by J. Kennedy and R. Kossak), Cambridge University Press.
show/hide this revision's text 1

Carl's answer is correct.

There is also more a direct way to achieve the same thing. The Tennenbaum's theorem holds for much weaker theories, e.g. $I\Delta_0$ (even far weaker theories $IOpen$ plus some number theoretic principles). $I\Delta_0+Exp$ is finitely axiomatization due to a result by Haim Gaifman and Constantine Dimitracopoulos in their paper "Fragments of Peano's Arithmetic and the MRDP Theorem" and it is also a sub-theory of $PA$.