Re: Cauchy completeness of Cauchy reals

[Toby Bartels <toby+categories@math.ucr.edu>]

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