Quantum Entanglement (A Love Story)

Yes, there are several areas in mathematics where key principles, axioms, or deep theorems hold (or are formulated) in a way that applies cleanly to both ordinary/finite cases (like typical real numbers or normal sequences) and to limiting/infinite cases — including actual infinities, asymptotic regimes, or nonstandard/infinitesimal extensions.Here are some of the clearest and most natural examples:1. Nonstandard analysis (hyperreals and transfer principle)This is probably the most direct match to what you're asking.We build an extension field *ℝ (hyperreals) that contains the usual reals ℝ plus genuine infinitesimals and infinite numbers.
The transfer principle is the key "postulate" here: any first-order statement in the language of real numbers that is true for ℝ is automatically true for *ℝ (and vice versa).
This means almost all the familiar axioms, definitions, and theorems from standard calculus/analysis (limits, continuity, differentiability, integrals, intermediate value theorem, mean value theorem, etc.) hold verbatim in the hyperreals — including for numbers that are infinite or infinitely close to finite ones.
Example: the definition of derivative works exactly the same way whether you use ordinary h → 0 or an actual infinitesimal h ≈ 0 but h ≠ 0.

So the same core "axioms/postulates" of real analysis govern both ordinary finite numbers and ones that are literally infinite (or infinitesimal).2. Asymptotic density / natural density in number theoryMany statements about the distribution of sets of integers (primes, squares, numbers with certain digit properties, etc.) are phrased in terms that make sense for both finite prefixes and the infinite limit.Natural density (when it exists):
lim (n→∞😉 |A ∩ {1,…,n}| / n = d
This is an "infinite" notion, but the finite approximations (the proportion up to n) are what we actually compute.
Many theorems hold with the same form in both worlds:The primes have density 0 (by the prime number theorem, the count up to n is ~ n / log n → 0 as n → ∞😉.
Normal numbers in base 10 have digit density 1/10 for each digit 0–9 (finite initial segments approach this, and the infinite expansion satisfies it exactly).

The finite and limiting behaviors are tied together by the very definition of the limit.

3. Ultralimits and nonstandard methods in discrete settingsUltralimits (using a non-principal ultrafilter) produce a very similar situation to nonstandard analysis, but often in more combinatorial/discrete contexts.You take sequences of finite/ordinary objects (graphs, metric spaces, numbers, etc.) indexed by ℕ.
The ultralimit object behaves as if it's "at infinity".
Many properties transfer: if every finite graph in a sequence satisfies some first-order property, then the ultralimit graph does too (by Łoś's theorem).
This is used heavily in graph theory, ergodic theory, and geometric group theory — the same statement holds for all finite n and for the infinite limit case.

4. Large cardinal axioms / strong axioms of infinity (more set-theoretic)These are less about "normal numbers" and more foundational, but still relevant:The ordinary axiom of infinity (there exists an infinite set, usually used to get ℕ😉 is the baby version.
Stronger large cardinal axioms (measurable cardinals, supercompact cardinals, huge cardinals, etc.) are "postulates about infinity" that imply the existence of extremely large infinite sets with reflection-like properties.
A striking feature is that many of these large cardinal axioms are downward absolute: if they hold in a universe V, they hold in many inner models and forcing extensions — so the "infinite strength" propagates downward to affect statements about ordinary sets and even finite mathematics (via consistency strength hierarchies).

In short — yes, there are several such "postulates" or principles, especially in analysis and set theory. The cleanest everyday example is nonstandard analysis, where the transfer principle makes the same core rules apply equally to ordinary real numbers and to hyperreal numbers that are actually infinite/infinitesimal.The spirit is that mathematics often finds elegant ways to make the same fundamental ideas govern both the finite/ordinary world and various kinds of limiting/infinite worlds at once.