What is Arithmetic Continuum I recently tried understanding Surreal Numbers on a more meaningful level. Along the way I found this answer to a related question. The accepted answer and the paper it suggests contain the term "arithmetic continuum" and "absolute arithmetic continuum".
To my surprise, I could not easily find a webpage telling what the two things are. In special, what is the arithmetic continuum. The paper says that "the real number system should be regarded as constituting an arithmetic continuum modulo the Archimedean axiom", but I still could not figure out a definition for it.

EDIT
An important aspect of why I hesitate in trusting what I think the term means and looking for a formal definition is my level of knowledge in mathematics. Specifically, I am insecure of my understanding of set theory and number theory, there is so much I do not know.
From the beginning of the text, Cantor and Dedeking, and most mathematicians of present times, believed the gap between discretness and continuity was bridged by the creation/discovery of real numbers.
Now, for the part the came the closest, to my view, of a definition.

while the
Cantor-Dedekind theory succeeds in bridging the gap between the domains
of arithmetic and of standard Euclidean geometry, it only reveals a glimpse
of a far richer theory of continua

I believed the term "arithmetic continuum" refers to, specifically, a bridge between arithmetic and Euclidean Geometry, and this made sense for me. In this sense, I thought, there might be other mathematical systems for which there are still gaps to Euclidean Geometry or arithmetic. Therefore, there might be an unnamed "x continuum" to bridge some other such gap.
From the name "absolute arithmetic continuum", I had the impression that it was a bridge between any mathematical system and arithmetic. However, soon in the text, I changed my mind due to theorem 1.

Theorem 1. Whereas $\mathbb{R}$ is (up to iso-
morphism) the unique homogeneous universal Archimedean ordered field,
No is (up to isomorphism) the unique homogeneous universal ordered field

I followed reading the thesis for some time, but it quickly became hard, because I am new to this. In fact, I was not aware of that $\mathbb{R}$ is an universal ordered field, and cannot say I know exactly what that is, for example. Although I gave a break reading the paper I was still interested in knowing what the names meant. Was my understanding in the beggining of the text on the right track, or does the terms have another meaning? What would this other meaning be, or what would be the important aspect that differs an arithmetic to a non-arithmetic continuum?
 A: Disclaimer: I have not seen the phrase "arithmetic continuum" outside of Ehrlich's paper The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small (preprint), but I have seen the adjective "arithmetic" and the noun "continuum" and read sections of the paper to get the meaning.
Feel free to skip the entire "Context" section if you just want to see how I arrived at my best guess as to Ehrlich's intent based on the paper. And skip to the very end if you just want the guess.

Context
"Arithmetic"
The adjective "arithmetic" has a few different meanings (e.g. a set is arithmetic if it's definable in Peano Arithmetic), but they all boil down to "related to arithmetic". That is, somehow connected to basic operations like addition, multiplication, and maybe subtraction and division.
"arithmetic" is sometimes distinguished from/paired up with "geometric" as in "arithmetic mean" vs. "geometric mean".

bridge between arithmetic and Euclidean Geometry

If someone asked me what that bridge was, I'd say the cartesian plane, not anything like the surreals.
And when I think of a bridge between arithmetic something and geometric something, I am reminded of the arithmetic-geometric mean. The word "arithmetic" on its own doesn't suggest that to me.
"Continuum"
The meanings of "continuum" are a bit trickier to pin down. In general, continuum is connected to things like "continuous" as opposed to discrete.
For example, a model of physics might say that there are a continuum of values for a measurement like the wavelength of a laser light. It comes up in the phrase continuum mechanics where calculus is used and the individual particles/atoms of bodies are ignored. You may also have heard of "the space-time continuum".
In the famous continuum hypothesis, "continuum" refers to the reals or perhaps its cardinality.
Generalizing from the reals, we get linear continua which are sets with a linear order that shares the Dedekind-completeness and denseness properties of the reals. Equivalently, one whose order topology shares the connectedness of the reals.
A different generalization from closed intervals of reals is the general concept of a continuum in topology. It's a nonempty compact connected metric space, or perhaps refers more generally to any compact connected Hausdorff space. This is what the tag [continuum-theory] refers to.
Note that the surreals are not connected under the order topology (e.g. there is a gap sometimes denoted "$\infty$" between the infinite surreals greater than every integer and all other surreals), so none of these definitions of "continuum" would seem to apply.

Meaning
Searching for the phrase
In the main body of the paper('s preprint), so not counting the title,  a reference title, the abstract, the introduction or a part title, the phrase "arithmetic continuum" appears only three times, so we can examine them all:

It is this together with Theorem 1 and a number of closely related results (see [Ehrlich 1992, forthcoming 1]) that naturally suggest that $\mathbf{No}$ may be regarded as an absolute arithmetic continuum (modulo $\mathrm{NBG}$)
...
Whereas Theorems 1 and 3 may be said to characterize $\mathbf{No}$ as an absolute arithmetic continuum, Theorem 13 may be said to characterize $\mathbf{No}$ as an $s$-hierarchical absolute arithmetic continuum.

The first quote appears shortly after Theorem 3 and references Theorem 1, and the second quote references Theorems 1 and 3, so we should definitely examine those theorems and the definitions they depend on.
Theorems and definitions

Theorem 1
(Ehrlich 1988; 1989; 1989a; 1992). Whereas $\mathbb R$ is (up to isomorphism) the unique homogeneous universal Archimedean ordered field,
$\mathbf{No}$ is (up to isomorphism) the unique homogeneous universal ordered field. ${\!}^{4}$
...
${\!}^{4}\!$ For the purpose of this paper, an ordered field (Archimedean ordered field) $A$ is said to be homogeneous universal if it is universal—every ordered field (Archimedean ordered field) whose universe is a class of $\mathrm{NBG}$ can be embedded in $A$—and it is homogeneous—every isomorphism between between subfields of $A$ whose universes are sets can be extended to an automorphism of $A$.
...
the notation "$L<R$" indicates that every member of $L$ precedes every member of $R$...
Definition 1 (Ehrlich 1987). An ordered class $\langle A,<\rangle$ will be said to be an absolute linear continuum if for all subsets $L$ and $R$ of $A$ where $L<R$ there is a $y\in A$ such that $L<\{y\}<R.$
...
Theorem 2 (Ehrlich 1988). $\langle\mathbf{No},<\rangle$ is (up to isomorphism) the unique absolute linear continuum.
...
Theorem 3 (Ehrlich 1988). $\mathbf{No}$ is (up to isomorphism) the unique real-closed ordered field that is an absolute linear continuum.
An ordered field is real-closed if and only if it admits no extension to a more inclusive ordered field that results from supplementing the field with solutions to polynomial equations with coefficients in the field...$\mathbf{No}$ not only exhibits all possible algebraic and set-theoretically defined order-theoretic gradations consistent with its structure as an ordered field, it is to within isomorphism the unique such structure that does. It is this together with Theorem 1...absolute arithmetic continuum...

From the above, an "absolute linear continuum" has a beefed-up version of version of the denseness property that the usual "linear continua" have. And $\mathbf{No}$ is called an "absolute arithmetic continuum" because it also has as many solutions to polynomials (which is more related to arithmetic than linear orders) as an ordered field can, because it's a real-closed field.
"(absolute) arithmetic continuum"
While it isn't spelled out explicitly in this paper, I imagine "arithmetic continuum" could mean a real-closed field whose order structure is that of a linear continuum. ($\mathbb R$ is the unique such thing up to isomorphism.)
And an "absolute arithmetic continuum" is almost certainly a class-sized real-closed field in NBG whose order structure $\langle A,<\rangle$ is that of an "absolute linear continuum", meaning that if $L$ and $R$ are sets and every element of $L$ is less than every element of $R$, then we can find $y\in A$ such that $\ell<y$ and $y<r$ for all $\ell\in L$ and $r\in R$. ($\mathbf{No}$ is the unique such thing up to isomorphism.)
