On Fri, Jan 24, 2014 at 2:23 AM, Brian Tenneson <[email protected]> wrote:
> There are undecidable statements (about arithmetic)... There are true > statements lacking proof. > Yes. > There are also false statements about arithmetic the proof of whose > falsehood is impossible; > A proof is a FINITE number of statements establishing the truth or falsehood of something; if Goldbach's Conjecture is untrue then there is a FINITE even number that is NOT the sum of 2 primes. It would only take a finite number of lines to list all the prime numbers smaller than that even number and show that no two of them equal that even number, and that would be a proof that Goldbach's Conjecture is wrong. The real problem would come if Goldbach's Conjecture is true (so we'll never find two primes to show it's wrong) but can not be proven to be true (so we will never find a finite proof to show its correct). John K Clark > 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. > -- 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.
