Internal Set Theory: $n$ is standard $\implies\;n+1$ is standard

I'm reading a bit about Nelson's version of nonstandard analysis and in the notes it is said that [$n$ is standard]$\implies$[$n+1$ is standard]. Right after that it is mentioned that an inductive proof is impossible (using the fact $0$ is standard as the base) because it would be based on external sets. I have two questions about this:

1. My proof attempt: We're given a standard $n\in\mathbb{N}$ and the internal formula $A(x)$ defined by $[x=n+1]$ (emphasizing dependence on $x$ with $n$ being a parameter). By a version of the transfer principle $\forall^{\text{st}}n[\exists^{\text{st}}xA(x)\iff \exists xA(x)]$, $x=n+1$ can be chosen standard, and by the uniqueness of $n+1$ it necessarily is standard. Is this correct?

2. Since any object that can be uniquely described in internal mathematics is standard, why exactly is it untrue that every natural is standard? After all, no matter what $n\in\mathbb{N}$ i'm given, I can uniquely describe it using the typical $\left\{ \emptyset,\left\{ \emptyset \right\},\dots \right\}$.. What exactly is wrong about this approach of dealing with each natural number separately?

Your argument that $n+1$ is standard if $n$ is seems correct; I would suggest a different approach: show that evaluating a standard function at a standard value produces a standard value. Then this problem would follow by using $f(x) = x+1$.

For your second question, are you sure every natural number can be written as $\{ \varnothing, \{ \varnothing \}, \ldots \}$?

You're making a level slip here: every natural number within an IST-universe can indeed be expressed by an string of symbols $\{\},\varnothing$ one can construct within the IST-universe. But that does not imply it can be expressed by an external string of the same symbols.

(in fact, depending precisely on your choice foundations, you might even have standard natural numbers that can't be expressed by external strings)

• What does that show about the set-theoretic formula? – Mikhail Katz Aug 10 '14 at 13:35
• This is a little confusing because you are not using "external" in the same sense OP used it. What you are referring to apparently is the construction of a model for Nelson's IST starting with the intended model of ZFC. – Mikhail Katz Aug 10 '14 at 13:37
• @user: I intend "external" to mean "of the formal logic in which I've written the axioms of IST" and anything else on the same level (e.g. the associated notion of natural number) – user14972 Aug 10 '14 at 13:39
• @user: I think it was "internal" I was using differently. – user14972 Aug 10 '14 at 13:44
• Thanks for your answer! Two questions: 1) Doesn't the usual recursive set-theoretic definition of $\mathbb{N}$ indeed enable us to write every natural number in that way? 2) What does internal/external string mean? – user153312 Aug 10 '14 at 13:45

The formula $x=n+1$ is internal but the statement $[n \text{ standard}] \implies [n+1 \text{ standard}]$ is not. You haven't provided a proof that does not involve external entities.

• The statement we're trying to prove involves external entities, so it's unavoidable that external entities will be involved. – user14972 Aug 10 '14 at 13:23
• @Hurkyl, thanks for your comment. I am not sure how to respond to the OP's remark about the set-theoretic construction. Do you want to try? – Mikhail Katz Aug 10 '14 at 13:32