Editing Openai/6912d046-4ed4-8010-bee4-7b68d354c856 (section)

=== A) The math idea (“one out of infinity” on a circle) ===
* Picking a specific point is fine. Choosing (0,1)(0,1)(0,1) (or any designated point) on the unit circle is perfectly well-defined. There’s no ambiguity about “which” point you mean.
* But the “size” of a single point is still zero. In standard measure theory, the circle carries normalized arc-length measure μ\muμ with μ(whole circle)=1\mu(\text{whole circle})=1μ(whole circle)=1 and μ({any single point})=0\mu(\{\text{any single point}\})=0μ({any single point})=0. So if you color every point white (1) except one designated point black (0), any integral/average over the circle cannot “see” that single flipped point, because it contributes zero measure. Concretely, ∫S1f(θ) dμ=1\int_{S^1} f(\theta)\,d\mu = 1∫S1f(θ)dμ=1 even if f(θ0)=0f(\theta_0)=0f(θ0)=0 at exactly one θ0\theta_0θ0. This is why, inside standard real analysis, a literal numeric value like “1/∞1/\infty1/∞” is not a real number distinct from 000.
* Two ways to make your intuition precise (without breaking math): 1. Distributions / measures. You can represent “one flipped point” by adding (or subtracting) a Dirac delta δθ0\delta_{\theta_0}δθ0. For example, “all ones” is the constant function 111. “All ones except a single special point” can be modeled as 1−α δθ01 - \alpha\,\delta_{\theta_0}1−αδθ0 at the level of distributions. This doesn’t turn into a small real number, but it is a rigorous object that encodes a distinguished point with infinite “spike-like” sensitivity. 2. Nonstandard analysis (hyperreals). In the hyperreal field you get genuine infinitesimals and infinite hyperintegers HHH. Then “1/H1/H1/H” is a real, positive infinitesimal distinct from 000. You can model “one out of HHH equal parts” and push HHH to an infinite hyperinteger. This matches your “one out of infinity” intuition, but it lives in a richer number system than the usual reals.
* Infinite coin flips vs. your circle: - For a countably infinite sequence of i.i.d. fair flips, the probability of “all heads” is 000 (not impossible in the logical sense, but measure 000). Same for “exactly one tail.” That mirrors the “single black point” having measure 000. - Your circle picture is an uncountable index set of “flips.” Fully independent fair flips at uncountably many points run into measurability problems; in practice we work with functions on the circle and an underlying measure. The key takeaway: a single-point difference is measure-zero—so any observable defined by integrating/averaging can’t detect it.

Bottom line on the claim: You absolutely can “mark a single point” on a circle rigorously. But in standard real-valued measurement, its “weight” relative to the whole is 000, not a distinct real like “1/∞1/\infty1/∞.” If you want “1/∞1/\infty1/∞” to be a bona fide number, use hyperreals; if you want “one special point” to be mathematically real without changing number systems, use distributions/Dirac delta or a topology that makes point-evaluation an allowed observable.