Yes. Godel showed that the continuum hypothesis (CH, which says aleph_1 = |R| = pow(aleph_0)) is consistent with ZFC -- therefore, it cannot be DISproved. Cohen in 1965 or so showed that CH is not decidable within ZFC: that it cannot be proved, either.
[ZFC is set of axioms used as the basis for a (the?) standard formulation of set theory: you start with these 9 axioms and rules of logic and all of your set theory follows.]
Therefore CH is an axiom that is independent of ZFC -- you can choose to adjoin it to your axioms or not, and you get a different set theory depending on whether you use it or not. Some might say that you choose whether to 'believe' CH -- more accurately, you choose to use CH depending on what kind of model of the real world you want your theory to be.
[btw, one can also prove that the 8 axioms of ZF are independent of C (the axiom of choice). So as above, you can choose whether to adjoin it to your set of axioms and make ZFC or just stick with ZF.]
In more usual crimes like physical vandalism or arson, laws are needed to prohibit them because there's no other way to stop these crimes. (There's no such thing as totally spray-paint resistant walls, for example.) Laws are meant to stop crime by punishing it. They are not perfect.
In recent years, the same philosophy has been applied to information crimes like hacking. The difference is that there is such a thing as a hack-proof web site. If the goal is to stop hacking, the best way to do it is to make your web site hack-proof, not rely on the incredibly inefficient legal system as a deterrent. (inefficient: how much does it cost for the judge, court staff, courtroom, lawyers, etc. to prosecute a single case?)
As society changes, legal philosophies need to change too. (c.f. the FSF.:-)
As a side note, 15 months in prison? For a 19 year old who was able to put some files on a disk in Washington because the web site designers didn't do their jobs correctly? How many lives did he put at risk? Give me a break.
Security by obscurity is doomed to fail, as we have seen over and over again, with the DVD encryption being the most recent example. Sure, the designers of the CSS spec assumed that no one would ever reverse engineer a DVD encoder. Right, good luck -- it's only a matter of time. (It amazes me that there are still people that have not learned this lesson. It is a FUNDAMENTAL notion of cryptography.)
You MUST design security into the protocol assuming that everyone already knows the protocol.
In this specific case, in order to prevent people from claiming they have 5 million mp3's, you might want to try some sort of trust system where people who download mp3s and get what they want rate the person they got them from. This would be similar to the Ebay ranking system or moderation on slashdot.
What if it's a fact. Perhaps the encoding and compression of some music actually makes it "better" (remember that better in this case is defined as "less annoying for the majority of the population". Possible, but this test doesn't allow you to say that. Notice that none of the error bars clear the 0 line in the positive direction, while there are clearly those that do so in the negative direction. Yes, these are 95% confidence intervals, but keep in mind that n=16 after a selection process that threw out about the same amount of other listeners. So you can say that some codecs actually enhance audio, but this test doesn't back you up. (It doesn't convincingly refute you either, though.:)
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Yes. Godel showed that the continuum hypothesis (CH, which says aleph_1 = |R| = pow(aleph_0)) is consistent with ZFC -- therefore, it cannot be DISproved. Cohen in 1965 or so showed that CH is not decidable within ZFC: that it cannot be proved, either.
[ZFC is set of axioms used as the basis for a (the?) standard formulation of set theory: you start with these 9 axioms and rules of logic and all of your set theory follows.]
Therefore CH is an axiom that is independent of ZFC -- you can choose to adjoin it to your axioms or not, and you get a different set theory depending on whether you use it or not. Some might say that you choose whether to 'believe' CH -- more accurately, you choose to use CH depending on what kind of model of the real world you want your theory to be.
[btw, one can also prove that the 8 axioms of ZF are independent of C (the axiom of choice). So as above, you can choose whether to adjoin it to your set of axioms and make ZFC or just stick with ZF.]
In more usual crimes like physical vandalism or arson, laws are needed to prohibit them because there's no other way to stop these crimes. (There's no such thing as totally spray-paint resistant walls, for example.) Laws are meant to stop crime by punishing it. They are not perfect.
:-)
In recent years, the same philosophy has been applied to information crimes like hacking. The difference is that there is such a thing
as a hack-proof web site. If the goal is to stop hacking, the best way to do it is to make your web site hack-proof, not rely on the incredibly inefficient legal system as a deterrent. (inefficient: how much does it cost for the judge, court staff, courtroom, lawyers, etc. to prosecute a single case?)
As society changes, legal philosophies need to change too. (c.f. the FSF.
As a side note, 15 months in prison? For a 19 year old who was able to put some files on a disk in Washington because the web site designers didn't do their jobs correctly? How many lives did he put at risk? Give me a break.
Security by obscurity is doomed to fail, as we have seen over and over again, with the DVD encryption being the most recent example. Sure, the designers of the CSS spec assumed that no one would ever reverse engineer a DVD encoder. Right, good luck -- it's only a matter of time.
(It amazes me that there are still people that have not learned this lesson. It is a FUNDAMENTAL notion of cryptography.)
You MUST design security into the protocol assuming that everyone already knows the protocol.
In this specific case, in order to prevent people from claiming they have 5 million mp3's, you might want to try some sort of trust system where people who download mp3s and get what they want rate the person they got them from. This would be similar to the Ebay ranking system or moderation on slashdot.
What if it's a fact. Perhaps the encoding and compression of some music actually makes it "better" (remember that better in this case is defined as "less annoying for the majority of the population". Possible, but this test doesn't allow you to say that. Notice that none of the error bars clear the 0 line in the positive direction, while there are clearly those that do so in the negative direction. Yes, these are 95% confidence intervals, but keep in mind that n=16 after a selection process that threw out about the same amount of other listeners. So you can say that some codecs actually enhance audio, but this test doesn't back you up. (It doesn't convincingly refute you either, though. :)