The StarTrek transporter or 'teleporter' is logically possible and physically possible. It has also been implemented with photons I believe, across the river Danube near Vienna. There will be no problem with shortage of physical and computational resources in the future. Whether you believe it will ever be done for large and systems complex systems largely depends on your view of 'identity'. In quantum mechanics (unlike classical), two systems/objects are identical if they are in the same 'quantum state'. So the 'you' on the other end of the transporter would be (identical to) the 'you' on this end. I recall watching a film called 'The Prestige', where Nikola Tesla played by David Bowie was supposed to have invented some such device. The 'Banach-Tarski' paradox discovered in 1925, demonstrates mathematically that in principle you could 'duplicate everything and everyone'.
I expect the rest of your posting to be roughly of the same quality, and indeed, the same will probably be true of all of your postings. Let me say here that the Banach Tarski paradox has nothing whatsoever to do with quantum teleportation, and the copying of quantum states is specifically ruled out by the no cloning theorem.
There's no list as such. If you want to talk about results of quantum mechanics then you should really use the rules of quantum mechanics to derive those results. The Banach Tarski paradox is certainly an interesting subject, but it comes from set theory which doesn't have all that much to do with QM.
The BTP is not 'discovered' in the physics sense, it's a mathematical result. As Prom says, it has nothing to do with the discussion at hand, the only link is it is used to replicate things. Unfortunately for you the 'things' are not physical things. Firstly, as Prom says, quantum mechanics has a no cloning theorem so immediately and without having to go to abstract logic the whole 'clone someone using a transporter' is invalidated. Secondly, the manner in which the BTP works is to split an object into a number of partitions. These partitions require an infinite divisibility of the object, so you can keep cutting bits in half or remove arbitrarily small regions. This doesn't apply to real world objects which are built of particles. Thirdly (or rather secondly and a half) the BTP requires you use the axiom of choice, as you need to be able to stipulate how you select complicated clouds of uncountably infinite points, which can then be reassembled. Obviously the manner in which you cut up a sphere to make 2 spheres is not going to be akin to cutting an apple with a knife, else it wouldn't be called a 'paradox'. Instead it's extremely complicated, so complicated that you have to invoke additional axioms of logic to justify it! The AoC is independent of ZF logic, you can either use it or its negation in ZF and still get a valid logical system. If the BTP could be proven to apply to the real world it would mean Nature uses the AoC. I think a lot of logicians would be appalled by that.
This logician would not be appalled! In my masters thesis I proved that the Axiom of Extensionality (the base axiom, not some 'additional' one) was independent within axiomatic set theory. How would you like a (non-extensional) mathematics as a basis for your commerce, statistics, quantum mechanics, etc? I don't believe that 'nature' 'uses' any mathematical theorem, or indeed any other construct from the human mind. (Except trivially counting the human mind as part of nature). Those mental constructs (inc. 'space' and 'time') are devised (by us) so that we (human beings) can make sense of 'reality'. And everyone of those constructs has a limited range of validity, even 'truth' is relative (Tarski, again). Anyway sorry to get away from the subject. Many thanks for saving us much wasted time. Au revoir (we'll be back within 2.8115 Gs).
Most consider the invoking of the AoC less elegant than proof by some other method. To find out Nature uses the AoC might be seen as the ultimate inelegant solution. And reality is generally much more elegant than that. And that has any relevance because.....? If anything the fact you're familiar with the specifics of logic makes your previous post about its relevance to a transporter all the more puzzling. If you're sufficiently competent at this level of mathematics and mathematical physics then you should see how the requirements of the AoC in the BTP to allow for infinite division of a continuum cannot be applied to a quantum mechanical system, which by definition has quanta which cannot be divided further. The BTP requires a continuum, it doesn't work with atoms. Besides, the mathematical 'universe' which the BTP exists within is not necessarily the same as the one quantum mechanics resides within. As you should be aware from your work in logic, there's many different logical constructs examined and results in one do not necessarily have any relevance to others. And even if QM is constructed within the realm of ZFC the fact remains no real physical object has the properties of the objects considered in the BTP. Other than the entirely qualitative "Split stuff up, put it back together" the BTP has nothing to do with this thread. If you're really someone familiar with logic you should be able to see that. I don't really care what kind of mathematics it is, provided it works. You don't seem to be understanding the concept here. Clearly there are rules by which the universe works. Physics is an attempt to discover those rules. As relativity shows, clearly there's a set of rules not a million miles away from Riemannian geometry which govern how the universe works at large distances. You can make it more abstract, in that quantum mechanics describes the very small and it's mathematical formalism is that of a rigged Hilbert space. Thus you can say with some justification that Nature 'uses' Hilbert spaces. If you could disassemble and then reassemble an object into 2 objects using the BTP method then it would imply the universe 'uses' the AoC. Of course we know it doesn't because the universe doesn't have objects in it which are mathematical idealisations, real objects are coarse and lumpy. That might be the case in pure mathematics, where you get to define the rules of the game but not in reality. There is an objective external thing from our minds which we are attempting to understand. For example, Euclidean geometry is all well and good in mathematics but we know it doesn't apply exactly to reality. Likewise electromagnetism, which is just a U(1) gauge theory. The no cloning theorem comes from a Hilbert space construct and shows that you cannot replicate states in the space using a particular method, which corresponds to physically implementable procedures. The BTP is an entirely separate concept which has no bearing on the Hilbert space setup. If you're a logic person I can understand you wanting to phrase things in terms of what you're familiar with but in this case it simply doesn't have any relevance.
