Abstract
In this chapter, we will show that there is a very tight connection between ultrafilters (introduced in Chap. 1) and nonstandard methods (introduced in Chap. 2). Indeed, any element (or “point”) in a nonstandard extension gives rise to an ultrafilter and, conversely, any ultrafilter can be obtained in this way. Such an observation makes precise the assertion that ultrafilter proofs can be seen and formulated as nonstandard arguments. The converse is often true but not always, as two different points in a nonstandard extension can give rise to the same ultrafilter.
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Notes
- 1.
Recall from Definition 1.18 that \(A\in k\mathcal {U}\Leftrightarrow A/k=\{n\in \mathbb {N}\mid nk\in A\}\in \mathcal {U}\).
- 2.
This topology is usually named “S-topology” in the literature of nonstandard analysis, where the “S” stands for “standard”.
References
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). Hyperfinite Generators of Ultrafilters. In: Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory. Lecture Notes in Mathematics, vol 2239. Springer, Cham. https://doi.org/10.1007/978-3-030-17956-4_3
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