FWIW, under the usual definitions, the rationals are enumerable and so are a smaller set than the reals. I'd suppose that if people can figure that out with our nifty fleshy brains, then a well-designed computer brain could, too. -Gabe
On Friday, January 24, 2014 1:23:40 AM UTC-6, Brian Tenneson wrote: > > There are undecidable statements (about arithmetic)... There are true > statements lacking proof. There are also false statements about arithmetic > the proof of whose falsehood is impossible; not just impossible for you and > me but for a computer of any capacity or other forms of rational > processing. We'll never have a computer, then, that will work as a > mathematically-omniscient device. By that I mean a computer such that every > question that has a mathematically-oriented theme having an answer > truthfully can be answered by such a device. Calculators demonstrate the > concept but are clearly not mathematically-omniscient: you ask the > calculator what is 2+2 and press a button and "presto" you get an answer. > What I'm talking about would be questions like "is the set of rational > numbers equal in size to the set of real numbers", and get the correct > answer. So we will never have such a computer no matter what its capacities > are, even if computer encompasses the entire human brain. Unfortunately, > that means that even for humans, we will never know everything about math. > Unless something weird would happen and we suddenly had infinite > capacities; that might change the conclusions. > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
