>>The question has already been answered by Joel Hamkins. Here I want to elaborate on Joel's answer (see Proposition A below) so as to point out a nontrivial variant of the question (see Question A below).

**Proposition A.** *Suppose* $M \models I\Delta_0$.  *If* $M$  *can be end extended to some* $M' \models PA$, *then*:

**(a)** $M \models B\Sigma_1$, *and*

**(b)** *For each standard natural number* $n$, $M \models$ Con($I\Sigma_n$).

**Explanation.** The scheme $B \Sigma_1$ is known as the $\Sigma_1$-collection scheme, it is well-known that if a model $M$ of $I\Delta_0$ can be end extended to a model $M'$ of $I\Delta_0$, then $M$ satisfies $B \Sigma_1$ (see Kaye's textbook on models of PA). It is also well-known that $PA$ proves the consistency of each of its finite subtheories, and that for each standard natural number $n$, the fragment $I\Sigma_n$ of $PA$ is finitely axiomatizable (also detailed in Kaye's text). Note that statements of the form Con($ I\Sigma_n$) are $\Pi_1$-statements and therefore their truth is inherited by initial segments closed under addition and multiplication.  

In light of the above, the "right" question to ask is:

**Question A** *Is the converse of Proposition A true?*

**(1)** The answer to Question A is in the positive, if the assumption that $M$ is a model of $I\Delta_0$ is strengthened to the assumption that $M$ is a model of $I\Sigma_1$. This follows from the nontrivial fact that every model of $I\Sigma_1$ (even uncountable ones) has an expansion to a model of $WKL_0$ (a result due originally due to Peter Hájek, and recently revisited in [this paper][1] of Tin Lok Wong). Note that the compactness theorem of first order logic holds in $WKL_0$, and that if $M$ is a nonstandard model of $I\Sigma_1$ satisfying condition (b) of Proposition (a), then Con($I\Sigma_c$) holds for some nonstandard c in $M$ by an application of the $\Pi_1$-overspill principle (provable in $I\Sigma_1$), so an expansion of $M$ to a model of $WKL_0$ can be used to build an end extension of $M$ that satisfies $PA$.

**(2)** Perhaps the answer to Question A is also in the positive if we add the requirement that $M$ is *countable*, and exponentiation is a total function in $M$ (by using ideas in [this paper][2] of Wong and myself). I will report back if I make a progress in this direction.

**(3)** I suspect that (1) and (2) describe the current "state of the art" in relation to Question A.

  


  [1]: http://www.logic.univie.ac.at/~wongt9/papers/wong-wklact.pdf
  [2]: http://www.logic.univie.ac.at/~wongt9/papers/ew-wkl0star.pdf/www.logic.univie.ac.at/