1
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.