26 May 06
Originally posted by DoctorScribblesI can prove that no process exists to pick a real uniformly.
I can prove that no process exists to pick a real uniformly.
But I can't tell if you're claiming that you can or not, complicated by the AC red herring with which you have confused the issue.
Finite time or infinite time?
26 May 06
Originally posted by lucifershammerI claim that no such process exists. (Note that this does not deny the AC.) Any process you have generates numbers from only a subset of that range.
I don't think so. That whatever procedure I come up with is capable of generating every real in [0,1] should be sufficient information.
26 May 06
Originally posted by lucifershammerProcesses that don't terminate obviously don't count, since they never produce a number. That's the very thing I'm claiming - it's impossible to generate a uniform real if you never generate any number at all.
[b]I can prove that no process exists to pick a real uniformly.
Finite time or infinite time?[/b]
26 May 06
1 edit
Originally posted by DoctorScribblesIt would be just as correct to say that an infinite-time process does terminate and returns a number.
Processes that don't terminate obviously don't count, since they never produce a number. That's the very thing I'm claiming - it's impossible to generate a uniform real if you never generate any number at all.
Just like the Adam-sin experiment. After all, for Adam sinning to be inevitable, the experiment would have to terminate with Adam sinning.
26 May 06
5 edits
Originally posted by lucifershammerIt would. It's guaranteed to.
After all, for Adam sinning to be inevitable, the experiment would have to terminate with Adam sinning.
You are confusing not terminating with a process requiring an infinite amount of time.
Some processes are guaranteed to terminate while not having an upper bound on the time required to terminate. Adam's sinning is one such process. Flipping a fair coin until it lands heads is another.
Any process that attempts to generate a uniform real either does not terminate ever or does not actually generate uniformly. For every process that you specify, I can prove which category it lies in. Thus, there is no process to generate uniform reals.
26 May 06
Originally posted by DoctorScribblesIt's not guaranteed to terminate - it just has a well-defined termination procedure. For an infinite sequence of tails, it will take the same time to run as a process that generates a uniform real.
It would. It's guaranteed to.
You are confusing not terminating with a process requiring an infinite amount of time.
Some processes are guaranteed to terminate while not having an upper bound on the time required to terminate. Adam's sinning is one such process. Flipping a fair coin until it lands heads is another.
Any process that atte ...[text shortened]... , I can prove which category it lies in. Thus, there is no process to generate uniform reals.
26 May 06
Originally posted by lucifershammerWrong again, bucko.
It's not guaranteed to terminate - it just has a well-defined termination procedure. For an infinite sequence of tails, it will take the same time to run as a process that generates a uniform real.
You are absolutely guaranteed to eventually get a heads from a fair coin. In fact, you are absolutely guaranteed to eventually get every possible finite-length sequence of outcomes.
You are absolutely guaranteed that any terminating process is one that does not generate uniform reals.
You simply don't know what you're talking about. If you don't believe me, ask any other informed mathematician or computer scientist for a second opinion. Or better yet, try to come up with a counterexample or a proof against my claims. Good luck.
26 May 06
Originally posted by DoctorScribblesAn infinite sequence of tails is sufficient counterexample against your claim.
Wrong again, bucko.
You are absolutely guaranteed to eventually get a heads from a fair coin. In fact, you are absolutely guaranteed to eventually get every possible finite-length sequence of outcomes.
You are absolutely guaranteed that any terminating process is one that does not generate uniform reals.
You simply don't know what you're ...[text shortened]... Or better yet, try to come up with a counterexample or a proof against my claims. Good luck.
26 May 06
Originally posted by lucifershammerNo, it's not, because that is an impossible outcome of the experiment when a fair coin is used. That outcome is only possible when a biased coin -- one with p(tails) = 1 -- is used.
An infinite sequence of tails is sufficient counterexample against your claim.
26 May 06
1 edit
Originally posted by DoctorScribblesAn infinite sequence of tails may have p=0, but it's clearly not impossible unless:
No, it's not, because that is an impossible outcome of the experiment when a fair coin is used. That outcome is only possible when a biased coin -- one with p(tails) = 1 -- is used.
1. You're arguing that infinite sequences are impossible.
or
2. p=0 < = > Impossible (it's the same equivalence you used earlier)
(2) is axiomatic - and it's precisely the one I'm challenging.