Star mapping in Non-standard analysis

Question

I'm trying to understand the star mapping in non-standard analysis in particular for the Hyperreals. I know that $*: \mathbb R\to \mathbb{^* R}$ is a mapping such that $^*(x)=^*x$ where $^*x= (x,x,x,x,...)$ which is in $\mathbb{^* R}$. In other words $^*x$ is the sequence that is in our ultrafilter and thus in the hyperreals. What I'm confused about is how this mapping works exactly for example if we consider the natural numbers $\mathbb N$, and apply the star mapping we would get $\mathbb{^*N}$. But what does $\mathbb{^*N}$ contain; all of the constant sequences with the $*$ mapping applied to them? Would it contain anything else, i'm just quiet confused.

Answer 1

The idea of constructing a nonstandard model of the natural numbers using sequences originates with Skolem. Skolem used definable sequences only. As you mentioned, the constant sequence represents a standard natural number. The other numbers in ${}^\ast\mathbb{N}$ are all infinite. Thus, any sequence tending to infinity will represent a nonstandard number.

Answer 2

In general $^*X\supseteq\{^*x;x\in X\}$ for every set X. The equality holds if and only if $X$ is finite (because for $X$ infinite there is always a sequence not in equivalence class of any constant sequence).

So $^*\mathbb{N}$ contains some (actually many) elements other than $^*n$ for $n\in\mathbb{N}$. Such elements are usually called nonstandard natural numbers.

Because $\mathbb{N}$ is an integer part of $\mathbb{R}$, the same thing holds about their $^*$-images. That is $^*\mathbb{N}$ is an integer part of $^*\mathbb{R}$. From the same reasons $^*\mathbb{N}$ is a subsemiring of $^*\mathbb{R}$, it satisfies mathematical induction, and so on ...

What exactly lies in $^*\mathbb{N}$ depends on the details of how $^*$ was constructed (more different approaches exist). But for virtually all purposes this is unimportant. Equally unimportant is whether $^*x$ is the sequence $(x,x,x,\ldots)$ or some other object. What matters are the properties of $^*$ (nonstandard principles).