field-theorynonstandard-analysisorder-theoryordered-fields
This answer shows that the Dedekind completion of the set of hyperreal numbers, endowed with the usual definitions of addition and multiplication of Dedekind cuts, is not an ordered field. But my question is, is it possible to embed this set with its addition and multiplication operations into some ordered field?
I imagine you’d need to add an awful lot of numbers which are greater than all the natural numbers but less than the infinite hyperreal numbers.
Here is a partial answer.
For any infinite cardinal $\kappa$, we write $\mathrm{ded}(\kappa)$ for the supremum, over all linear orders $X$ of size $\kappa$, of the cardinality of the set of Dedekind cuts in $X$. We always have $\kappa<\mathrm{ded}(\kappa)\leq 2^\kappa$.
Now suppose $S$ is a saturated dense linear order of cardinality $\kappa$. I claim that $S$ has the maximal number of Dedekind cuts, i.e., the number of Dedekind cuts in $S$ is $\mathrm{ded}(\kappa)$.
Since $\mathrm{ded}(\kappa)$ is defined as a supremum, and $\kappa<\mathrm{ded}(\kappa)$, it suffices to show that for any $\lambda$ with $\kappa<\lambda\leq \mathrm{ded}(\kappa)$, if there is a linear order $X$ with $\lambda$-many Dedekind cuts, then $S$ has at least $\lambda$-many Dedekind cuts.
Now since $S$ is a saturated dense linear order and $|X|\leq |S|$, there is an order embedding $e\colon X\to S$. For each Dedekind cut $(L,R)$ in $X$, $(e(L),e(R))$ generates a Dedekind cut in $S$, unless it is "filled", i.e., there are elements of $S$ between $e(L)$ and $e(R)$. But since $|S| = \kappa$, at most $\kappa$-many of the images of the cuts in $X$ are filled in $S$. Since $\lambda>\kappa$, $\lambda$-many cuts in $X$ have images that are cuts in $S$. This completes the proof that $S$ has $\mathrm{ded}(\kappa)$-many Dedekind cuts.
How is this relevant to the hyperreals? Let $^*\mathbb{R}$ be an ultrapower of $\mathbb{R}$ by a non-principal ultrafilter on $\omega$. Then $^*\mathbb{R}$ is $\aleph_1$-saturated. If we assume CH, $|^*\mathbb{R}| = \aleph_1$, so the underlying linear order of $^*\mathbb{R}$ is $\aleph_1$-saturated and dense. By the result above, the number of Dedekind cuts in $^*\mathbb{R}$ is $\mathrm{ded}(\aleph_1)>\aleph_1$.
Thus, assuming CH, $|\overline{^*\mathbb{R}}|>\aleph_1$. And if we also assume $2^{\aleph_1} = \aleph_2$, then $|\overline{^*\mathbb{R}}|\leq 2^{|\mathbb{R}|} = \aleph_2$, so we obtain that $|\overline{^*\mathbb{R}}| = \aleph_2 = 2^{|\mathbb{R}|}$.
I suspect it is consistent that CH holds but $\mathrm{ded}(\aleph_1)<2^{\aleph_1}$ (can anyone confirm?). If so, then there is a model of ZFC in which $2^{\aleph_0} = \aleph_1 < \mathrm{ded}(\aleph_1) < 2^{2^{\aleph_0}}$, so $|\mathbb{R}| <|\overline{^*\mathbb{R}}| < 2^{|\mathbb{R}|}$.
I have no idea what happens if CH fails. It seems likely that without CH, the cardinality of $\overline{^*\mathbb{R}}$ depends on the ultrafilter you use to take the ultrapower.
Let me give you two seemingly contradictory answers:
We can prove that there is no maximal ordered field. Based on your post, I think some of this will be new to you, but I want to keep things brief and high level. There’s a theorem in mathematical logic called the “Lowenheim-Skölem Theorem”. This theorem has a trivial corollary that says the following. Suppose we have a mathematical structure that can be defined by “first order sentences”. For example, Groups, Rings, Fields, Ordered Fields, Graphs, etc. Suppose we also can prove that there is an infinite example of our structure (like an infinite group, or $\mathbb{Q}$ as an ordered field). Then there are examples of our structure of all infinite cardinalities.
From the above corollary to Lowenheim-Skölem, there are ordered fields of all infinite cardinalities. Since there is no largest cardinality, there can be no largest ordered field.
Why doesn’t this argument apply to complete ordered fields? “completeness” isn’t a first order property. Hence the Lowenheim-Skölem Theorem does not apply.
The above argument assumes we are defining an ordered field as a set with additional structure. In set theory, there are collections of sets that are “too big” to be sets. We call such collections “proper classes”. For example, there is no set of all sets, but there is a class of all sets. Suppose we allow the underlying collection of objects for our ordered field to be a proper class.
Then we can define the field of Surreal Numbers! This ordered field has the property that every ordered field defined on a set can be embedded inside it.
This can be formalized in set theories other than ZFC, like NBG set theory.
Whether or not there is a “maximal” ordered field is going to depend on what exactly we mean by “maximal ordered field” and our underlying set theory.
Here I’ll try to clear up some of the concerns raised in the comments:
“Joe, so the surreal numbers are not really a set, but a class?” — Yes. The underlying collection of objects on which the surreal field is defined is a proper class.
“Why is completeness a second order property when we have the axiom of power set?” — This is a subtle point, and to really understand what’s going on you’re going to need to learn some mathematical logic. If you’re interested in learning this, Chris Leary’s book is an exceptionally gentle treatment. Enderton’s book is also consider canonical, but it’s harder.
When we say “completeness is a second order property”, we mean as a property of ordered fields. We can only quantify over elements of the field. You are absolutely correct that in set theory, we can “hack this” by declaring there is a power set. But to quantify over the power set of elements of an ordered field is different than quantifying over elements of the field itself. Specifically, when quantifying over elements of the ordered field, there is no sentence we can build out of quantifications, and, or, not, $+,\cdot, <, 0,1$ that corresponds to completeness.
From the perspective of mathematical logic, our ordered field axioms are purely “syntactic”. We then find sets in a meta-theoretic set theory and interpretations of $0,1,+,\cdot,<$ that satisfy those axioms. The axioms we write down exist in their own domain. They can’t appeal to the power set of a model of the axioms.
The point you raise, is, in some sense though, the way we prove that the “real numbers” that we can construct in set theory are complete. It is also how we are able to talk about “complete ordered fields” in set theory.
You are correct that Dedekind completing a non-Archimedean field will not give you a field. You then made the following proposition:
Proposition: Let $\mathbb{F}$ be an ordered field. Let $0^* := \{ x \in \mathbb{F} : x < 0 \}$ denote the zero-cut. If for all Dedekind cuts $A \subseteq \mathbb{F}$ there is an additive inverse $-A$ such that $A + (-A) =0^*$, then $\mathbb{F}$ is Archimedean.
Proof: This proposition is correct. Indeed, it is easy to demonstrate that the contrapositive is true. Suppose that $\mathbb{F}$ is not Archimedean. Consider the following set: $$X := \{x \in \mathbb{F} : \text{ there is a } q \in \mathbb{Q} \text{ such that } x < q \}$$ Since $\mathbb{F}$ is non-Archimedean, $X$ is non-empty and forms a left Dedekind cut. Further, it is easy to see that $X$ has no additive inverse. Suppose that there’s another cut $Y$ such that $X + Y = 0^*$. I leave the details to be filled in by you, but we may following the following outline of a proof:
This is a contradiction. Thus $X$ has no additive inverse.
Best Answer
No: addition is not cancellative, so it cannot embed in any abelian group. For instance, if $C$ is the Dedekind cut consisting of all elements that are less than some integer, then it is easy to see that $C+1=C$.