4 ms·
How do you define real numbers without uncountable sets though? I'm not sure most people are willing to give up the real numbers just because they are a bit uni
by ufo 13y ago
How do you define real numbers without uncountable sets though? I'm not sure most people are willing to give up the real numbers just because they are a bit unintuitive.
- thenerdfiles 13y agoYou have to make a distinction between the intensional and extensional domain. Real numbers are merely infinite decimal expansions, where "infinite" is a convenience term (semantically no different from "really big"). I believe the mantra is "π is a rule, not a thing".
- VladRussian2 13y agoreal numbers are a closure of the set of rational numbers under the limit operation. It just happens that the closure is uncountable. It is pretty standard device in mathematics - having a set and an operation, to produce and explore another set built as a closure of the first set under the operation. Seems pretty intuitive to me :) What is really unintuitive, puzzling is the starting point of it all - natural numbers... 1, 2, 3... why there is a one stone, a one star, ... why this discretization?
- eru 13y agoYou can start lower, if you want to. {} == 0 {{}} == 1 ...
- michaelochurch 13y agoComputable (real) numbers are a countable subset of the reals and represent the largest set we can analytically reason about; however, equality of computable numbers is not itself computable-- you can have computable c1 and c2 where cmp(c1, c2) is nonterminating (which means that c1 == c2 but it cannot be shown to be so in finite time). I'd argue that, for most people's tastes, computable analysis is wonkier than real analysis, even if the latter has uncountable infinities (which is rarely considered a problem).