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 $\{N \in {}^*\mathbb{N} : N<H\}$ 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.


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