On Sun, Oct 6, 2019, 4:47 AM John Rose <johnr...@polyplexic.com> wrote:
> > Pi doesn't exist, only expressions and approximations of it and you gave > an example. You must expend energy and time to generate a better > approximation, IOW add power. > Do numbers exist? It's a philosophical question. Philosophy is arguing about the meanings of words. What do you mean by "exist"? Pi is a computable number. There exists a Turing machine that inputs n and outputs n digits for any positive integer n. There are 1 to 1 mappings between the sets of natural numbers, integers, rational numbers like 1/2, algebraic numbers like sqrt(2), computable numbers like pi, and describable numbers like Omega, the halting probability. There is no such mapping of any of these sets to the set of real numbers, which has a larger cardinality. I can't give you any examples of undescribable real numbers even though they vastly outnumber the numbers that have finite length descriptions. I can only prove that they exist in the sense that natural numbers exist, as a subset of the real numbers. ------------------------------------------ Artificial General Intelligence List: AGI Permalink: https://agi.topicbox.com/groups/agi/T8eabd59f2f06cc50-Ma99d7c197f9b7745d5f0e4e5 Delivery options: https://agi.topicbox.com/groups/agi/subscription
