Abstract
If one wants to present the methods of nonstandard analysis in their full generality and with full rigor, then notions and tools from mathematical logic such as “first-order formula” or “elementary extension” are definitely needed. However, we believe that a gentle introduction to the basics of nonstandard methods and their use in combinatorics does not directly require any technical machinery from logic. Only at a later stage, when advanced nonstandard techniques are applied and their use must be put on firm foundations, detailed knowledge of notions from logic will be necessary.
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Notes
- 1.
According to the usual set-theoretic foundational framework, every mathematical object is identified with a set (see Remark A.2 in the Appendix). However, here we will stick to the common perception that considers numbers, ordered pairs, relations, functions, and sets as mathematical objects of distinct nature.
- 2.
Recall that notation f : A → B means that f is a function with domain(f) = A and range(f) ⊆ B.
- 3.
In logic, properties that talks about elements of a given structure are called first-order properties; properties about subsets of the given structure are called second-order; properties about families of subsets of the given structure are called third-order; and so forth.
- 4.
Recall that an ordered field is real closed if every positive element is a square, and every polynomial of odd degree has a root.
- 5.
An element η is the successor of ξ (or ξ is the predecessor of η) if ξ < η and there are no elements ζ with ξ < ζ < η.
- 6.
There are a few exceptions to this statement, but we will never see them in the combinatorial applications presented in this book.
- 7.
In most cases, the converse is also true, namely that if the diagonal embedding is onto, then the ultrafilter is principal. A precise statement of the converse would require a discussion of measurable cardinals, taking us too far afield. It suffices to say that when I is an ultrafilter on a countable set, the converse indeed holds.
- 8.
This is because they are elementarily equivalent ℵ 1-saturated structures of cardinality ℵ 1 in a finite language, and they have size ℵ 1 under the Continuum Hypothesis.
- 9.
The existence of non-principal ultrafilters that are countably complete is equivalent to the existence of the so-called measurable cardinals, a kind of inaccessible cardinals studied in the hierarchy of large cardinals, and whose existence cannot be proved by ZFC. In consequence, if one sticks to the usual principles of mathematics, it is safe to assume that every non-principal ultrafilter is countably incomplete.
- 10.
The subformula “z = {t ∈ x∣φ(t, y 1, …, y n)}” is elementary because it denotes the conjunction of the two formulas:
$$\displaystyle \begin{aligned} ``\forall t\in z\ (t\in x\ \text{and}\ \varphi(t,y_1,\ldots,y_n))\text{''}\ \ \text{and}\ \ ``\forall t\in x\ (\varphi(t,y_1,\ldots,y_n)\Rightarrow t\in z)\text{''}. \end{aligned}$$ - 11.
More formally, one transfers the formula: “\(\forall x,y\in \mathbb {N}\ [(x<y\Rightarrow (\exists A\in \text{Fin}(\mathbb {N})\ \forall z\,(z\in A\leftrightarrow x\le z\le y))]\)”.
- 12.
We remark that the enlarging property is strictly weaker than saturation, in the sense that for every infinite κ there are models of nonstandard analysis where the κ-enlarging property holds but κ-saturation fails.
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). Nonstandard Analysis. 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_2
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