Why is it said that all surreal numbers with birthdate $<\omega_1$ are isomorphic to a Hardy field?

Question

In this answer I have encountered with the following statement:

Assuming CH, every maximal Hardy field is isomorphic to $(\bf{No}(\omega_1), \partial_{\omega_1})$, where $\bf{No}(\omega_1)$ is the set of all surreal numbers having tree-rank (or birthday) $<\omega_1$

Effectively, in my impression, this claims that there are no countable surreals between the Hardy field germs and $\omega_1$. But in my impression, there are surreal numbers, which are greater than all Hardy germs, but countable.

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Let us construct them and analyze the implications.

First of all, let us establish the principle of Dirac Comb equivalence.

• We assume that any germ of Hardy field, canonically embedded into surreals is equivalent to a divergent integral: $f(\omega)=[f(x)]=f(a)+\int_a^\infty f'(x)dx$ for arbitrary real $a$.

• We assume that if we decompose any divergent (or not) integral into (finite or infinite) number of segments with areas $A_n$ and in each segment replace the function under integral with Dirac Delta, centered at the center of mass of the segment and coefficient equal to the area of the segment, the whole integral remains the same. In other words, $$\int_{a}^{b} f'(x)dx=\int_a^b \sum_{n}A_n\delta\left(x-\frac{\int_{a_n}^{a_{n+1}}tf'(t)dt}{A_n}\right)dx$$ If we divide the integral into segments of area $A_n=1$, the formula takes form $$\int_{a}^{b} f'(x)dx=\int_a^b \sum_{n}\delta\left(x-\int_{a_n}^{a_{n+1}}tf'(t)dt\right)dx.$$ Denoting the center of mass of each segment as $s_n=\int_{a_n}^{a_{n+1}}tf'(t)dt$, we get $\int_{a}^{b} f'(x)dx=\int_a^b \sum_{n}\delta\left(x-s_n\right)dx.$ We call this integral numerosity of the set $\{s_n\}$ and denote $N(\{s_n\})$. We can express any germ $f(\omega)$ as a numerosity of an increasing sequence and often the vice versa as well. (Wolfram language code for numerosity to sequence conversion: S = Log[\[Omega]]; DifferenceDelta[Integrate[Normal[SolveValues[S == k, \[Omega]]], k], k] /. C[1] -> 0 // Last // FullSimplify // Expand; for sequence to numerosity: a[k] := k^2; SolveValues[D[Sum[a[k], k], k] == \[Omega], k] /. C[1] -> 0 // FullSimplify // Expand)

Now, let us define the direct product of two numerosities.

• If copies $A_n$ of a symmetric set $A$ are symmetrically centered around each element $b_n$ of set $B$, then the resulting set of all elements of $\cup A_n$ has numerosity $N(A)*N(B)$. For instance, $N(\mathbb{Z})=2\omega$. So, the numerosity of $N(\mathbb{Z}\pm1/3)=4\omega$.

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Now, consider two divergent integrals: $\int_0^{\infty } \left(\frac{1}{4} \pi ^2 \cot ^2\left(\frac{\pi x}{2}\right)+\frac{\pi ^2}{4}\right) \, dx$ and $\int_0^\infty (2x)dx$. By Dirac Comb decomposition and direct product rule we can see that $\int_0^{\infty } \left(\frac{1}{4} \pi ^2 \cot ^2\left(\frac{\pi x}{2}\right)+\frac{\pi ^2}{4}\right) \, dx=2\omega/2*\omega=\omega*\omega$, but by Hardy field operations we also see that $\int_0^\infty (2x)dx=2\cdot\frac{\omega^2}2=\omega^2$. At the same time, we see that the later integral corresponds to the numerosity of a strict subset of the former one, so by Euclid's principle it should be smaller.

We came to a paradox, which can be interpreted in different ways:

• Euclid's principle is valid for numerosities and imposes a strict total order on them, coinciding with the order of surreals. The direct product of numerosities is a different operation than Hardy field multiplication. There are surreal numbers that correspond to countable numerosities but greater than any element of the Hardy field. For instance, the numerosity $\omega*\omega$ is countable but greater than all Hardy germs. The statement in the linked answer and linked from there papers is wrong.

• Euclid's principle is not applicable (at least) to sets with accumulation points and/or dense ones. Numerosities of a set and its subset can be equal. The direct product of numerosities is the same operation as the Hardy field multiplication of germs. All countable surreal numbers are the germs of the Hardy field, for instance, $N(\mathbb{Q})=2\pi\omega!$. The statement in the linked post is correct.

To me this dilemma is a hard one, but I feel that Euclid's principle is more important. As such, I cannot agree with the linked statement. So, on what the linked statement is based?

Answer 1

As Philip Ehrlich had mentioned in the other post, the initial claim of your question is Corollary B of the following paper

• Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven, Filling maximal gaps in Hardy fields, arxiv:2308.02446.

Not surprisingly, the proof proceeds by a back-and-forth construction, building up the isomorphism by a sequence of countable approximations in an iterative construction of length $\omega_1$.

You had asked what are the reasons for that claim, and so this paper may be the answer to your question.

Regarding some of the other remarks you made, perhaps it helps to say that the field $\text{No}(\omega_1)$ denotes the surreal numbers that are born on a countable ordinal birthday, that is, strictly before $\omega_1$. (I suppose that one could call these the "countable" surreal numbers, but I wouldn't use that terminology myself, because (1) in fact there are automorphisms of the surreal field that do not respect birthdays and which move these surreal numbers to others born at $\omega_1$ or later; (2) the so-called countable surreals do not form a convex set, so something smaller than a countable surreal is not itself countable.)

Meanwhile, yes, there are many surreal numbers born at $\omega_1$ and after, and so there are many additional surreal numbers outside $\text{No}(\omega_1)$. In short, there are uncountable surreals much much larger than anything in $\text{No}(\omega_1)$. And also others much much smaller infinitesimal than anything in $\text{No}(\omega_1)$.