Take $\mathbb{H}=\mathbb{R}^\mathbb{N}/\mathcal{U}$, where $\mathcal{U}$ is some ultrafilter. Questions:
- Are there more than one independent infinitessimal in this field. This means $\epsilon_1 > 0$ and $\epsilon_2 > 0$ such that $0 < \epsilon_1 < \epsilon_2 < r$ for all $0 < r \in \mathbb{R}$, and there is no relation (lineair or algebraic) between them ?
- If there are, are there examples ?
Notes:
- This is like asking for an "hyperlarge number comparable to the ordinal $\omega_n$ with $n>1$", I only wish to know if they can be constructed in $\mathbb{H}$.
- I suspect you can proof the existence of more infinitessimals in $\mathbb{H}$ by roughly the same diagonal argument which is used to proof the uncountability of $\mathbb{R}$, but I can't lay my finger on it, and it doesn't lead to a construction (I am not one of the constructionalist. but in this case I wish to know if you can do more than an existence result).
Edit:
I thought of introducing "a second infinitessimal" to $\mathbb{H}=\mathbb{R}^\mathbb{N}/\mathcal{U}$ by looking at $(\mathbb{R}^\mathbb{N}/\mathcal{U})^\mathbb{N}/\mathcal{V}$, with a second ultrafilter, but (aside from the seond ultrafilter), I don't know in wich swamp I am moving then. Any thoughts on this would be appreciated.