Dusko Pavlovic wrote:
Alex Simpson wrote:
Somebody wrote:
that 1/2 + 1/4 +...+ 1/2^k > 1-e. in other words, that there is k s.t. 1/2^k < e. this is *equivalent* to markov's principle.
The property quoted is in fact a trivial consequence of intuitionistic arithmetic alone. It has nothing to do with Markov's principle.
for a real number e, i am pretty sure that the above is equivalent with markov's principle. it must be in books, but i think you can't miss the proof if you try.
I don't remember the original context, so I don't know who's right, but the answer depends on what sort of real number e could be. It can't be 0, for example, so what can it be? * If e > 0, then work with 1/e and use the Archimedean property; Bishop would recognise and accept this proof. * But if you only know that e <= 0 is false, then you need Markov's principle to deduce e > 0. -- Toby