In teaching my calculus students about limits and function domination, we ran into the class of functions
Suppose we say that
We can then readily see that
But we can get more complicated total orders with, say
which are isomorphic as total orders to the lexicographic orders on
All of these complicated orders live inside what I'd call "the AP Calc linear order"
where the refinement on
This seems to be a very complicated linear order, as it includes as a suborder things like
My question is the following: is there any combinatorial description or universal construction, i.e. as a colimit, of the isomorphism type of
K2-Think
The "AP Calc linear order" ((\Omega, \preceq)) can be described as a colimit of a directed system of ordered sets, where each component corresponds to the hierarchy of polynomial-logarithmic functions (\Theta_n).
Each (\Theta_n) consists of functions of the form (x^{\alpha_0} (\ln x)^{\alpha_1} \cdots (\ln^{n-1} x)^{\alpha_{n-1}}) with (\vec{\alpha} \in \mathbb{R}^n), and its equivalen...
GPT-5
Short answer: as stated, Ω is not a linear order, and it is far too complicated to admit a nice “universal” combinatorial description. In fact, it already contains, as a suborder, the very complicated poset (P(ω), ⊆*) of subsets of the natural numbers ordered by inclusion modulo finite. So any hope for a simple colimit description in the spirit of your Θn- or Φ-examples cannot work without imposin...
o3 Pro
The relation you have written down is the usual “big-O’’ relation on germs of
functions at
f⪯g ⇔ lim sup_{x→∞}|f(x)/g(x)|<∞.
After passing to the equivalence relation
f∼g ⇔ f⪯g and g⪯f,
the elements of Ω are the positive germs modulo the relation “differ by a
bounded factor’’ and the order is
[f]⪯[g] ⇔ f=O(g).
- (Ω,⪯) is not a linear order
Even afte...
o4 mini
One way to see that there really is a canonical “combinatorial” model for your big order is to observe that nothing mystical is hiding in the class of all continuous germs at infinity – as soon as one quotients out by “having the same size‐of‐growth,” one ends up with exactly the monomial group of the usual log–exp‐transseries. Equivalently one can build (\Omega) as the directed colimit of ...
Claude 3.7 Sonnet
Linear Orders in Asymptotic Comparison of Functions
The question explores a fascinating linear order structure derived from asymptotic function comparison. Let me address the request for a combinatorial description or universal construction of
Understanding the Structure
The ordered set
DeepSeek R1
First, they define a set of functions Θ = {x^α (ln x)^β} for real numbers α and β. They say that g weakly dominates f, written f ⪯ g, if the limit as x approaches infinity of f(x...
o3
Write
F := { all real–valued functions that are defined and continuous on
some interval (λ , ∞) } ,
B := { f∈F : f is bounded on (λ , ∞) for some λ } .
Put an equivalence relation on F by
f ≈ g ⇔ f – g ∈ B ,
and denote the ≈–class of f by [f].
Give the factor–set F/B the order
[f] ≤ [g] ⇔ (∃C∈ℝ)( f(x) ≤ g(x)+C for all large x )....
Gemini 2.5 Pro
This is a fantastic question that touches on some deep and beautiful mathematics connecting analysis, algebra, and logic. The "AP Calc linear order" you've discovered is a gateway to the study of Hardy fields and transseries.
Let's break down the answer.
1. Correction: The Order is Not Linear on Ω
First, a crucial correction. The order
Claude Opus 4
This is a fascinating question about the structure of growth rates of functions, which connects analysis with order theory. Let me provide some insights into the combinatorial and categorical aspects of this linear order.
The Structure of
The linear order
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