This logician would not be appalled!

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.

In my masters thesis I proved that the Axiom of Extensionality (the base axiom, not some 'additional' one) was independent within axiomatic set theory.

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.

How would you like a (non-extensional) mathematics as a basis for your commerce, statistics, quantum mechanics, etc?

I don't really care what kind of mathematics it is, provided it works.

I don't believe that 'nature' 'uses' any mathematical theorem, or indeed any other construct from the human mind.

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.

And everyone of those constructs has a limited range of validity, even 'truth' is relative (Tarski, again).

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.