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 [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
