3 ms·

> But you can also think of any real number I don't think that's true. We can certainly think of certain numbers which we gave a name to and have defined it in

> But you can also think of any real number I don't think that's true. We can certainly think of certain numbers which we gave a name to and have defined it in some way. But there are a _lot_ of real numbers. We could think about it this way: we can only describe (and therefore think of) numbers using a finite number of symbols out of a finite alphabet. That makes it only countable.

kzrdude 4y ago
Can we think of any computable number? And then some uncomputable ones if we give them a name/concept. It seems quite wild
- karatinversion 4y ago
  What does it take to think of a number? We can certainly refer to particular uncomputable numbers - choose an enumeration of Turing machines, and let x be the real number whose binary expansion has 1 at the i-th place if the i-th Turing machine halts, and 0 otherwise.
- deleted 4y ago
  [deleted]
- jerf 4y ago
  There are computable numbers whose minimum expression requires more bits than the universe can hold, using any possible encoding the universe has sufficient bits to define. Infinity is much bigger than any finite number. This includes even "all possible numbers reachable in all possible finite encodings of a number within a given size", as inconceivably large as that is. No matter how large that set is, somewhere, there is some maximum number that is the largest possible number you can specify under the conditions I've given here, and that number is 0% of the way along the number line, which means that 100% of all "computable numbers" are larger than than that number. There are some uncomputable numbers that have been given names. We even have the first few digits of some of them. See for instance: https://en.wikipedia.org/wiki/Chaitin%27s_constant https://en.wikipedia.org/wiki/Chaitin%27s_constant
lupire 4y ago
But, unless you have predefined a language, you can't know which countable set is expressible, so the union of all possible sets may be uncountable.
- dllthomas 4y ago
  But I think all languages is countable, so it should be a subset of the product of two countable sets, which should be countable?

kzrdude 4y ago

Can we think of any computable number? And then some uncomputable ones if we give them a name/concept. It seems quite wild

karatinversion 4y ago
What does it take to think of a number? We can certainly refer to particular uncomputable numbers - choose an enumeration of Turing machines, and let x be the real number whose binary expansion has 1 at the i-th place if the i-th Turing machine halts, and 0 otherwise.
deleted 4y ago
[deleted]
jerf 4y ago
There are computable numbers whose minimum expression requires more bits than the universe can hold, using any possible encoding the universe has sufficient bits to define. Infinity is much bigger than any finite number. This includes even "all possible numbers reachable in all possible finite encodings of a number within a given size", as inconceivably large as that is. No matter how large that set is, somewhere, there is some maximum number that is the largest possible number you can specify under the conditions I've given here, and that number is 0% of the way along the number line, which means that 100% of all "computable numbers" are larger than than that number. There are some uncomputable numbers that have been given names. We even have the first few digits of some of them. See for instance: https://en.wikipedia.org/wiki/Chaitin%27s_constant https://en.wikipedia.org/wiki/Chaitin%27s_constant

lupire 4y ago

But, unless you have predefined a language, you can't know which countable set is expressible, so the union of all possible sets may be uncountable.

dllthomas 4y ago
But I think all languages is countable, so it should be a subset of the product of two countable sets, which should be countable?