Hyperreal numbers is a formal way to introduce infinitesimal by extending the real numbers R. Hyperreals do satisfy arithmetic properties of addition, multiplication and division when restricted to the finite hyperreals(which we will define later). Hyperreals is an alternative way to do analysis without the usual epsilons and deltas which is why some mathematicians advocate for their use in introductory analysis courses. Hyperreals themselves are difficult to construct but the arithmetic and the properties make the waffly arguments about infinitesimals into proper mathematical arguments. The construction of hyperreals itself is actually pretty non-trivial. It requires the use of ultrafilters and in particular, non-principled ultraflters which requires the axiom of choice.

1 Ultrafilters

Loosely speaking a filter can be thought of as the “collection of large subsets of X.”)

Definition 1 (Filter). Let X be a set. F ⊂ P(X) is called a filter 1. X ∈ F 2. φ 6∈ F 3. A, B ∈ F =⇒ A ∩ B ∈ F 4. A ∈ F and A ⊂ B =⇒ B ∈ F

• (Trivial Filter) So of course, we may have the trivial filter which is F = {X}. • (Principal Filter) A principal filter is of the form F = {A : x ∈ A} where x is some fixed element in X. • (Cofinite Filter) The cofinite filter is of the form F = {A : X \ A is finite} This is also widely referred to as the Fr´echet filter

Remark 2. Filters satisfy the F.I.P. (finite interesection property) A1, ..., An ∈ F n then ∩i=1Ai 6= φ Lemma 3 (Finite Intersection Property). Given any susbset S ⊂ P (X) with the finite intersection property, there is a minimal filter F ⊃ S

We close downwards by finite intersection. We then close upwards by including all supersets. On top of the assumptions that we have mentioned if the following condition holds, we call such a filter an ultrafilter.

1 5. (Dichotomy) A ⊂ X =⇒ A ∈ F or X \F, but not both. (Alternatively, A ∈ F ⇐⇒ X \ A 6∈ F)

The cofinite filter is not an ultrafilter because given subset A that is infinite with an infinite complement X \ A , neither A nor X \ A belongs to F. On the other hand, the principal filter is an ultrafilter. The question is does there exist a non-principal ultrafilter? Assuming Axiom of Choice(which is equivalent to Zorn’s Lemma), we can construct a non-principal ultrafilter. (Zorn’s Lemma) If a non-empty poset (X, <), every chain in X has an upper bound in X, then the set contains a maximal element. (i.e. a maximal element m is an element such that there are no x ∈ X such that m < x.

Lemma 4 (Ultrafilter Lemma). Every filter is contained in an ultrafilter.

Proof. Apply Zorn’s Lemma to the poset fo filters ordered by inclusion.

• We check that the union of two filters F ∩ F 0 is a filter. So by Zorn we have a maximal filter.

• We check such a maximal filter is indeed an ultrafilter. Let A ⊂ X. If A 6∈ F and X \ A 6∈ F. Then, F ∩ A satisfies the Finite Intersection Property. Thus, F ∩ A can be extended to a filter F 0 which would contradict our assump- tion of maximality. Hence, F is an ultrafilter.

With the ultrafilter lemma, we can then, extend the cofinite filter into a non-principal ultrafilter. The ultrafilter lemma tells us is that an ultrafilter is a filter than satisfies any of the following.

• It satisfies the Dichotomous Property.

• It is a maximal filter.

• It is a prime filter. (i.e. If A ∩ B ∈ F, then A ∈ F or B ∈ F

2 To conclude this part on filters and ultrafilters, we briefly mention that filters and ultrafilters are an important part of many parts of mathematical constructions. One application is to generalize the convergence of sequences in metric spaces to more general topological spaces. One can say that convergence of sequences in the usual context of metric spaces, ultimately boils down to the fact that there are only finite many indices that do not satisfy d(ai, a) <  for a given  > 0. A way to repharse this is to say that the indices that satisfy d(ai, a) <  is in the cofinite filter. But we can replace this with another filter say with the non-principal ultrafilter that we constructed. Just like the reals were constructed from the rationals by Cauchy sequences and identifying two seqeuences whose index satisfy ai − bi <  under a “cofinite filter”, the construction of the hyperreals will follow a similar suite by constructing a real sequence quotiented out by an ultrafilter.

2 Construction of Hyperreals

A = RN has a componentwise addition and multiplication. This constructs a com- mutative ring with a unital element (1, 1, ...) Clearly, this is not a field, or even a domain for that matter as (1, 0, 1, 0, ..) × (0, 1, 0, 1, ...) = (0, 0, 0, 0, ...). We want to quotient out by these troublesome elements so that it does become a field. Let U be a non-principal filter on N.

For two sequences a = (an) and b = (bn), denote where the two indices agree as [[a = b]]. i.e. [[a = b]] = {i ∈ N : ai = bi}

We declare a relation =U given by a =U b if and only if [[a = b]] ∈ U

Claim: =U is an equivalence relation.

• Reflexive: Because N ∈ U (Property 1) • Symmetric: Because [[a = b]] = [[b = a]].

• Transitive: This is the (slightly) non-trivial part. Let [[a = b]], [[b = c]] ∈ U. Then, [[a = b]] ∪ [[b = c]] ∈ U (Property 3) and [[a = b]] ∪ [[b = c]] ⊂ [[a = b]] so [[a = b]] ∈ U (Property 4).

Claim 2: A/U is a field. Commutativity, Associativity, Distributivity are inherited from A = RN.

3 • Identity: We need to show that the additive and multiplicative identitie are unique. If a + e =U a for then, [[a + e = a]] = {i ∈ N : ai + ei = ai} ∈ U =⇒ {i ∈ N : ei = 0} ∈ U. So e =U . (Similarly, for the multiplicative identity.)

• Additive Inverse: Simply define −a = (−an). • Multiplicative Inverse:This is a bit troublesome because some terms in the se- quence may be 0. Let a be a non-zero element. Then, I = {i ∈ N : ai = 0} 6∈ U. Now define a sequence b = (bn): ( ai i ∈ U −1 −1 bi = We can define our multiplicative inverse on b = (b ) 1 i 6∈ U n

Definition 5. ∗R = RN/U is the field of hyperreal numbers.

Before, we can talk about infinitesimals and infinite numbers, we need to endow this set of hyperreal numbers ∗R with a partial order. We do this in the same way as we did before.

Definition 6. Given two sequences a = (an) and b = (bn), we say a 6 b if and ony if {i ∈ N : ai 6 bi} ∈ U

Actually this defines a total order.

• Check for transitivity. a 6 b, b 6 c. Then, {i ∈ N : ai 6 bi}, {i ∈ N : bi 6 ci} ∈ U and the intersection of these is in the ultrafilter. But the intersection is contained in {i ∈ N : ai 6 ci}, hence it is in U so ai 6 ci.

• Suppose a 6 b. Then, {i :∈ N : ai > bi} ∈ U. But {i :∈ N : ai > bi} ⊂ {i :∈ N : ai > bi} ∈ U and so b 6 a. So at total order.

(Check: If a 6 b and b 6 a then a =U b.) We have a total order field. This finally gives us a definition of infinitesimals and infinite numbers. Firstly, note there is a canonical inclusion of the reals in the hyperreals given by R ,→∗ R r 7→ (r, r, ...) Definition 7. A hyperreal x is said to be

• infinitesimal if x 6 n for every n ∈ N.

4 • an infinite number if n 6 x for every n ∈ N.

This means that the fied is “non-archimedean”. (Meaning there are elements that are “bigger” than the natural numbers.)

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