2 ms·
Naming a number in mathematics has different rules to assigning names to objects in the physical world. You can "name" a number such as: "a positive number, th
by jiggawatts 4y ago
Naming a number in mathematics has different rules to assigning names to objects in the physical world.
You can "name" a number such as: "a positive number, that when multiplied by itself, the result is 2".
This is a finite statement that requires only a handful of 'bits' to represent, but it exactly and uniquely identifies the square root of two. If written with decimal digits, then the square root of two would require an infinite number of digits, but it can be defined ('named') with a finite number of bits.
It turns out that this is a rare property for real numbers. The vast majority do not have a shorthand name like this, only the infinitely long description exists.
- riversflow 4y agoWhat does naming have to do with AC though? This seems completely tangential to the axiom. It’s about the nature of sets, not the semantics of how you make the selection and reference it. AC says: if there is a set of non-empty sets y with length x, I can construct a new set z of length x by taking a member from each set in y. How does being able to name the constituents z or any member of y matter to the logic of the axiom?
- Smaug123 4y agoThis is all in response to a comment which basically said "why do people think AC is obvious - if you can't even name all elements of the set, why would you expect to be able to choose them". (Which is handwavy, of course, and leads to the wrong intuition about uncountable-but-well-ordered sets, and depending on how you look at it it may even lead you to conclude that you can't ever choose even a single element from an uncountable set.)