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Added 2025-04-02 10:31:37 +0000 UTCHow do you make infinite choices? - The Axiom of Choice
Comments
Great points! You're totally right - there's a lot more nuance here. Beyond just accepting or rejecting the Axiom of Choice entirely, mathematicians explored various intermediate forms like the Axiom of Dependent Choice and the Axiom of Countable Choice. Also, good catch regarding equivalence: The video perhaps shows more clearly that the Axiom of Choice implies the Well-Ordering Principle, but these statements are actually equivalent in both directions. If every set can be well-ordered, then every set (and every subset!) has a first element. So, given any collection of nonempty sets, you can simply pick the first element from each set which exactly matches what the Axiom of Choice claims - that you can always make a choice. Your comment highlights precisely why the analogy with geometry and the parallel postulate is so appropriate: just as geometry can branch into Euclidean, hyperbolic, or elliptic forms depending on the parallel postulate, set theory similarly branches into different frameworks depending on which choice axioms we accept!
Veritasium
2025-04-07 14:42:47 +0000 UTCNot goofy at all. The short answer is no. If a number is irrational in base 10, it is irrational in every base. That's because rationality isn't about how a number looks in a particular numeral system - it's about whether it can be expressed as a fraction of two integers. Rational numbers always have expansions that either terminate or repeat, no matter what base you write them in. Irrational numbers, like √2, have expansions that never terminate or repeat, again regardless of base. For example, in base 10 √2 ≈ 1.41421356... (no repeating pattern) and in base 2 (binary) we have √2 ≈ 1.01101010000010011110... (also no repeating pattern). On the other hand, the rational number 1/3 in base 10 is 1/3 = 0.3333... (repeating) and in base 2 (binary) we have 1/3 = 0.01010101... (also a repeating pattern). I hope this helps!
Veritasium
2025-04-07 13:50:42 +0000 UTCGoofy question: Can irrational numbers is base 10 become rational numbers in some other base?
RobF228
2025-04-06 19:22:47 +0000 UTCOne way to strengthen the analogy with choices of geometry is to note that there are more options than "no axiom of choice" and "axiom of choice". For example, Solvay built a measure theory on the reals in which all sets are measurable, but he need the Axiom of Dependent Choice which is stronger than bare ZF, but weaker than (implied by, but does not imply) the Axiom of Choice. Also, you argued that the Axiom of choice proves the well ordering principle for the reals. Then you state that they are equivalent - which is true, but a stronger statement than what was argued.
Ben Galehouse
2025-04-05 12:57:50 +0000 UTC
