Stereographic projection is a way of mapping ℝ — which is infinitely long — onto a circle, which has a finite length (circumference).

In a sense, the circle is larger than ℝ. That is, you can map injectively every point from ℝ onto the circle, and then there’s still a point left over at the top of the circle. That point gets associated to ∞.

This mapping also generates the Cauchy distribution from statistics.

An infinite thing is smaller than a finite thing. Weird, right? Well on the other hand, you could also map ℝ to the unit circle and exchange “x inches ∈ ℝ” for “x degrees ∈ circle”, like wrapping an infinite string around the circle, and it would of course wrap around many times.

text{proj}: mathbb{R} &to text{circle} subset mathbb{C} \ x &mapsto e^{i x}

So either thing could be considered bigger. This is why the axiom of choice is messed up.

About isomorphismes

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