# In Non-standard analysis, is the number of natural numbers a hyperreal number?

In Non-standard analysis, is the number of natural numbers a hyperreal number? In other words, if $$H$$ is the hyperreal infinite unit, does the sum $$\sum\limits_{n=1}^H 1$$ yield the number of natural numbers? Or, put some other way, is there some hyperreal $$H$$ which would yield the number of natural numbers?

My anwser is a vague "no", because of the nature of the question. The vaguest expressions are written in italic.

The "number of natural numbers" is usually conceived as the cardinal number $$\aleph_0$$. The most common convention is that this corresponds as a set to the set $$\mathbb{N}$$ of positive integers.

This will not be naturally identifiable with a non-standard natural number $$H$$, of which there is no smallest. In particular the set of standard positive integers is not itself a non-standard positive integer since it is not an object of the universe in IST.

(By the way, we have $$\forall h \in \mathbb{N}, h=\sum \limits_{k =1}^h 1$$ so $$\sum \limits_{n=1}^H 1$$ is equal to $$H$$ for any non-standard positive integer, or any hyperatural number).

In generic systems of hyperreal numbers given by ultrapowers of $$\mathbb{R}$$ to the power $$\mathbb{N}$$, you can if you want decide that some hypernatural number such as the equivalence class of the sequence $$(0,1,2,...)$$ corresponds to $$\aleph_0$$, but this will most likely be arbitrary. Moreover, in such systems, there may not be anything that looks like any uncountable cardinal number. Two remarks to illustrate this:

1. I believe it is consistent with ZFC that every order embedding $$(\omega_1,\in) \longrightarrow ({}^*\mathbb{N},{}^*<)$$ is cofinal.
2. For any hypernatural number $$H \in {}^*\mathbb{N}$$, the set $$\{N \in {}^*\mathbb{N} : N has cardinality continuum.

Nevertheless, there are systems of hyperreal numbers where one can identify such sums with ordinal numbers such as $$\omega \equiv \aleph_0 \equiv \mathbb{N}$$. Although this identification still suffers from the issue 2. above, it provides a more solid link between certain so-called euclidean numbers and certain ordinals.