3 ms·
> But wait. It’s true that there are only countably many “symbolic descriptions,” and uncountably many real numbers, but it doesn’t follow that there are specif
by rcme 4y ago
> But wait. It’s true that there are only countably many “symbolic descriptions,” and uncountably many real numbers, but it doesn’t follow that there are specific real numbers which cannot be described using a finite collection of symbols, does it?
It does follow because the set of finite collections of symbols is countable.
- tshaddox 4y agoI'm not sure what you mean, because you just repeated the first part of my sentence. The fact that one set is countable and the other is uncountable doesn't mean there are specific, fixed numbers in the uncountable set which cannot be paired with any element in the first set. Consider the usual example of the natural numbers (countable) and the reals (uncountable). It doesn't follow that there is some specific, fixed real number which cannot be part of a bijection of the naturals and the reals. Name any real number and I can give you a bijection of the naturals and the reals which contains your real number.
- rcme 4y agoI mean what you say is trivially true. Give me a number r in R. I will call that number h. That’s not really interesting.
- tshaddox 4y agoIt’s interesting only as a refutation of the original quote: > consider the set of numbers that cannot be described using any finite collection of symbols. Such numbers must exist because there are only a countably infinite number of numbers that can be described using a finite collection of symbols, but there are an uncountably infinite number of real numbers.
- rcme 4y agoYou’re misunderstanding the quote. It’s not saying that you can’t name any individual number. It’s saying that any scheme you can think of to name numbers involving a finite number of symbols will have numbers that can’t be expressed in that scheme.