"Alien objects", Stillwell's "Reverse Mathematics"

[Hill]

TL;DR

What is an example of "alien objects" allowed by the first four Peano axioms?

Stillwell's "Reverse Mathematics" says on p.41,

Unfortunately, there is no example of such model. Where can I find it?

For the reference, here are the axioms 1-4:

 

[jedishrfu]

When Stillwell says there are alien objects, he means the objects that mysteriously appear when you strengthen an axiom.

Applying some initial conditions, you can show that the object exists, but you can’t compute it, or list its members, or even prove that some object you generated is a part of the set.

 

[Hill]

When Stillwell says there are alien objects, he means the objects that mysteriously appear when you strengthen an axiom.

Applying some initial conditions, you can show that the object exists, but you can’t compute it, or list its members, or even prove that some object you generated is a part of the set.

Thank you.
He refers to "concocting a model." Where can I learn about such model?

 

[jedishrfu]

Try looking for arithmetic systems without the induction axiom but based on the other three.

 

[Hill]

It appears to be quite simple:

(https://en.wikipedia.org/wiki/Robinson_arithmetic)

 

[haruspex]

When Stillwell says there are alien objects, he means the objects that mysteriously appear when you strengthen an axiom.

Do you mean, when you omit an axiom?