- The Axiom of Choice (Stanford Encyclopedia of Philosophy)
The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set—a transversal or choice set—containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences—many indispensable, some startling—and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it.
- Voyager 1 (Wikipedia)
Voyager 1 is a space probe launched by NASA on September 5, 1977, as part of the Voyager program to study the outer Solar System and the interstellar space beyond the Sun’s heliosphere. It was launched 16 days after its twin, Voyager 2. It communicates through the NASA Deep Space Network (DSN) to receive routine commands and to transmit data to Earth. Real-time distance and velocity data are provided by NASA and JPL. At a distance of 163.1 AU (24.4 billion km; 15.2 billion mi) from Earth as of June 2024, it is the most distant human-made object from Earth. The probe made flybys of Jupiter, Saturn, and Saturn’s largest moon, Titan. NASA had a choice of either doing a Pluto or Titan flyby; exploration of the moon took priority because it was known to have a substantial atmosphere. Voyager 1 studied the weather, magnetic fields, and rings of the two gas giants and was the first probe to provide detailed images of their moons.