Abstract
We have seen in Chap. 3 how ultrafilters corresponds to points in a nonstandard extension. We will see in this chapter how one can describe operations between ultrafilters, such as the Fubini product, in terms of the corresponding nonstandard points.
In the most common approach to nonstandard methods, one assumes that the star map goes from the usual “standard” universe to a different (larger) “nonstandard” universe. We will see in this chapter that one can dispense of this distinction assume that there is just one universe which is mapped to itself by the star map. This has fruitful consequences, as it allows one to apply the nonstandard map not just one, but any finite number of times. This yields the notion of iterated nonstandard extension, which will be crucial in interpreting the Fubini product and other ultrafilter operations as operations on the corresponding nonstandard points.
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Notes
- 1.
A construction of such star maps is given in Sect. A.1.4 of the foundational appendix.
- 2.
To avoid ambiguity, some authors call the hyper-extensions \({ }^{\ast }A\in \mathbb {V}\) “internal-standard”.
- 3.
Every map f : A → B yields a natural map \(f^{\mathcal {U}}:A^{\mathbb {N}}/\mathcal {U}\to B^{\mathbb {N}}/\mathcal {U}\) between their ultrapowers, by setting \(f^{\mathcal {U}}([\sigma ])=[f\circ \sigma ]\) for every \(\sigma :\mathbb {N}\to A\).
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). Many Stars: Iterated Nonstandard Extensions. 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_4
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