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2. Ultrafilters


    1. Add remark

      Problem 2.1.

      [David Ross] Does the choice of specific ultrafilter have an affect on the resulting model of nonstandard analysis? Can special properties of ultrafilters (e.g. idempotence, $P$-points, selective) be utilized for proving combinatorial results?

        •     An element $\alpha\in{^*}\mathbb{N}$ is idempotent if, for any $A\subseteq\mathbb{N}$, whenever $\alpha\in{^*}A$ then $\alpha+a\in{^*}A$ for some $a\in A$.

          Add remark

          Problem 2.2.

          [Mauro Di Nasso] Give a model theoretic proof of the existence of idempotent points in ${^*}\mathbb{N}$.

            •     An ultrafilter $\mathcal{U}$ on $\mathbb{N}$ is Hausdorff if for any $f,g\colon \mathbb{N}\longrightarrow\mathbb{N}$, if $f(\mathcal{U})=g(\mathcal{U})$ then $\{n\in\mathbb{N}:f(n)=g(n)\}\in\mathcal{U}$.

              Assuming the Continuum Hypothesis, Hausdorff ultrafilters on $\mathbb{N}$ exist (see, e.g., [MR2207497]).

              Add remark

              Problem 2.3.

              [Mauro Di Nasso] Give a ZFC proof of the existence of a Hausdorff ultrafilter.

                  Cite this as: AimPL: Nonstandard methods in combinatorial number theory, available at http://aimpl.org/nscombinatorial.