"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?

 

[jedishrfu]

My understanding is that adding or strengthening an axiom produces these objects.

They are in a sense odd edge cases that pop up when applying the axioms.

The best example to my mind is the progression from real numbers to complex numbers to quaternion numbers to octonions.

As each axiom defining the real number properties is adjusted the more exotic number algebras appear with octonions as the end of the line.

First you give up, ordered to get complex numbers, then commutativity to get quaternions, then associativity to get octonions at the end.

 

[haruspex]

My understanding is that adding or strengthening an axiom produces these objects.

…

First you give up, ordered to get complex numbers

Isn't that dropping an axiom? I don’t see how it is 'strengthening' one.
But then, going from real to complex is adding a closure axiom, the existence of all n polynomial roots.

 

[jedishrfu]

I guess it's a matter of perspective. Historically, we discovered real numbers first and derived the others by deleting an axiom.

But couldn't you view it as starting with octonions of which the reals, complex numbers and quaternions are a subset and by strengthening or adding an axiom you get a subset and a set of what's not in the subset.