Measurement problem in the orthodox interpretation Let's look at the measurement problem in the orthodox interpretation of quantum mechanics as an inconsistency between inner and outer treatment of the measurement apparatus. You can always push your boundaries of treating the evolution of your system as unitary further and further. You can say OK, the universe as a whole is evolving unitarily (let's not worry about information loss in a blackhole). So it's up to me to consider the boundary to see the evolution of my system and apparatus together or just my system. And I should be able to work out the reduced density matrix of my system equally in every treatment unambiguously! However, If you treat the apparatus externally, the evolution of the system would be:
$$a|\uparrow\rangle + b|\downarrow\rangle \to |\uparrow\rangle$$
with probability $|a|^2$ or
$$a|\uparrow\rangle + b|\downarrow\rangle \to |\downarrow\rangle$$
with probability $|b|^2$.
Whereas, an internal treatment of the apparatus would give
$$|\uparrow\rangle\otimes|\text{ready}\rangle\to U\bigl(|\uparrow\rangle\otimes|\text{ready}\rangle\bigr) = |\uparrow\rangle\otimes|\text{up}\rangle$$
and
$$|\downarrow\rangle\otimes|\text{ready}\rangle\to U\bigl(|\downarrow\rangle\otimes|\text{ready}\rangle\bigr) = |\uparrow\rangle\otimes|\text{down}\rangle$$
with $U$ a linear operator, $U(a|\psi\rangle + b|\phi\rangle) = aU|\psi\rangle + bU|\phi\rangle$, which evolves
$$\bigl(a|\uparrow\rangle + b|\downarrow\rangle\bigr)\otimes|\text{ready}\rangle$$
to
$$U\bigl[a|\uparrow\rangle\otimes|\text{ready}\rangle + b|\downarrow\rangle\otimes|\text{ready}\rangle\bigr] =a|\uparrow\rangle\otimes|\text{up}\rangle + b|\downarrow\rangle\otimes|\text{down}\rangle$$
However, pushing the boundary after the measuring apparatus gives a difference physics. This could be viewed as a problem with measurement in orthodox quantum mechanics (as opposed to realist or operational strategies to solve the measurement problem) 
But I was thinking it's not really an inconsistency within a theory. It's just an inconsistency between two different choices of the internal-external boundaries! I'm not asking about the role of decoherence theory. It sounds to me like the measurement problem wasn't really a problem in the first place! Am I right about that?
update: It has been pointed out that the question is not clear enough yet. Here is my last attempt: It's believed that for an adequate postulates for quantum mechanics, the inner and outer treatment of measuring apparatus shouldn't affect the physics of the system. Which for the orthodox interpretation of quantum mechanics does. For instance in the Bohm's model this has been resolved by denial of representational completeness. And in Operational interpretation it's bypassed by avoiding talking about physical state of the system. Here the question is Are we really allowed to change the boundaries? Because if you don't believe you can, the problem will never appear in the first place.
I hope that explains what I'm asking. Because I don't think I can make it more clear :-)
 A: We're not allowed to change the boundary, especially not in the direction of making a bigger piece of the world "classical" than what is allowed.
In particular, the whole world - not just electrons but also the apparatus - obeys the laws of quantum physics. At some moment, however, the quantum phases, coherence, and interference become de facto impossible because they're lost in the noise of the environmental degrees of freedom. This process is called "decoherence" and once it happens, it is legitimate to consider the quantum-calculated probabilities to be ordinary classical probabilities.
Decoherence depends on the Hamiltonian and exactly determines where the boundary is. The important point is that if one assumed that the world is classical even at some "more microscopic level" than where it becomes classical - i.e. if one assumed that certain objects can't interfere and exhibit entanglement etc. even though they can - he would end up with incorrect results.
See e.g. this introduction to decoherence:
http://www.karlin.mff.cuni.cz/~motl/entan-interpret.pdf
Niels Bohr was implicitly aware of decoherence but unfortunately, his explanations of his intuition weren't too coherent, either. That's why this insight had to wait for an explicit description until the 1980s.
Instead, the Copenhagen school adopted a pragmatic attitude. It didn't explain or calculate where the classical-quantum boundary was located. However, it placed it safely in the "macroscopic realm" so that no interference, entanglement, or other predictions of quantum mechanics were ever lost.
It's important to realize that one can use the full quantum description even with macroscopic objects - they satisfy quantum mechanics, too. I can treat all other observers and physicists as quantum systems who evolve into linear superpositions etc. and only calculate the probability that I see $A$ or $B$ at the very end - because I know that I won't "perceive" any bizarre linear superposition. It never hurts when you treat all things in the quantum mechanical framework. See Sidney Coleman's talk, Quantum Mechanics In Your Face, 
http://motls.blogspot.com/2010/11/sidney-coleman-quantum-mechanics-in.html
where this quantum treatment of the whole world is used in several pedagogical examples. On the other hand, when one deals with macroscopic enough objects, and traces over environmental degrees of freedom that are really impossible to keep track of, the density matrix becomes almost immediately and almost exactly diagonal in the "preferred classical basis vectors". That allows us to treat the diagonal elements of the density matrix as classical probabilities. Quantum mechanics will still predict probabilities only and the results of individual events will be random. However, if the density matrix is kept diagonal, it allows us to assume - just like in classical physics - that the macroscopic degrees of freedom were in a particular state even before we measured (or saw) what they were. We won't reach any contradictions.
However, once again, if we did make this assumption for any system that can still interfere and evolve quantum mechanically - i.e. if we imagined that any aspect of the wave function "collapsed" before we actually make the measurements - we would reach incorrect and contradictory predictions. So if you don't know where the boundary is, you should better assume that everything is quantum - because everything is quantum. Classical physics is just an approximation and one must appreciate that it is often or usually invalid.
A: The orthodox Copenhagen interpretation is entirely unproblematic as long as your self is placed outside of the box. We can always choose to draw the box placing us outside. This is why it works in practice for practically all experiments and engineering work. Madness comes in when your try placing yourself inside the box. You either have to restrict the box, or contrive to place yourself outside the box one way or other. Many worlders try to describe the universe by an uncollapsed wave function, but only at the price of imaginatively abstracting themselves away from the universe, pretending that they're not really inside the universe, or that they don't exist.
A: I think you've just rediscovered the Many-Worlds interpretation. All that Many-Worlds really says is that when a system in a superposition state is measured, the state of the measurement apparatus (and associated observers, etc.) becomes entangled with the state of the system . The system continues to be in a superposition of all possible states, it's just a somewhat more complicated superposition, including the measurement apparatus and all that. 
Contrary to what a lot of popular treatments lead people to believe, Many-Worlds does not involve the creation of an entire separate universe worth of matter for each of the possible measurement outcomes. There is just a single wavefunction describing the single universe worth of matter that we have, which becomes more and more complicated as time goes on. We only see a single branch because decoherence in the form of unmeasured interactions with a larger environment destroys our ability to detect any influence of these different branches on one another. This makes the different branches effectively separate "universes," because the piece of an observer in one branch is entangled with the state of the system in that branch, and cannot detect any influence of the other branches.
Your internal/ external phrasing is different than I'm used to hearing, but the math you've written down looks exactly like the way you would formulate a Many-Worlds type description of the measurement process. If that's not what you're doing, the distinction isn't clear to me.
A: @Sina is making a perfectly good point with this question.
The statement is this:

The measurement problem is not really a problem, because the result of a measurement is not only dependent on the system being "measured", but also on the observer doing the "measuring".

To actually realize this dichotomy between an "internal" and "external" observer, the size of the observer's Hilbert space, given by its dimension $\dim(H_O)$, must be comparable to $\dim(H_S)$ - the dimension of the Hilbert space corresponding to the system under observation. Instead, what we generally encounter is $\dim(H_O) \gg \dim(H_S)$ as is the case for, say, an apparatus with a vacuum chamber and other paraphernalia which is being used to study an atomic scale sample.
In this case the apparatus is not described by the three states $\{|\text{ready}\rangle, |\text{up}\rangle, |\text{down}\rangle\}$, but by the large (infinite?) family of states $\{|\text{ready};\alpha\rangle, |\text{up};\alpha\rangle, |\text{down};\alpha\rangle\}$ where $\alpha$ parametrizes the "helper" degrees of freedom of the apparatus which are not directly involved in generating the final output, but are nevertheless present in any interaction. Examples of these d.o.f are the states of the electrons in the wiring which transmits data between the apparatus and the system.
So if we consider the apparatus to be "internal" then the Hilbert space of the total "system+apparatus" is:
$$ H_{S+O} = H_S \otimes H_O $$
which has as its basis vectors 
$$ \{ \left|\uparrow\right\rangle|\text{ready};\alpha\rangle,
      \left|\uparrow\right\rangle|\text{up};\alpha\rangle,
      \left|\uparrow\right\rangle|\text{down}; \alpha\rangle;
      \left|\downarrow\right\rangle|\text{ready};\alpha\rangle,
      \left|\downarrow\right\rangle|\text{up};\alpha\rangle,
      \left|\downarrow\right\rangle|\text{down};\alpha\rangle  \} $$
It is the states of the form $\left|\uparrow\right\rangle|\text{down};\alpha\rangle$ and $\left|\downarrow\right\rangle|\text{up};\alpha\rangle$ which are "counterfactual" (and you neglected to mention in the question).
At this point I wave my hands and say when an external super-observer looks at the states of the system described by $H_{S+O}$, the conterfactual states of the type mentioned above will interfere destructively due to the presence of the numerous $\alpha$ d.o.f; leaving only the "consistent" states of the type $\left|\uparrow\right\rangle|\text{up};\alpha\rangle$  and $ \left|\downarrow\right\rangle|\text{down};\alpha\rangle$ as the ones with non-negligible amplitudes. So in such cases, their is no contradiction between what the super-observer sees and whatever output the apparatus yields.
It is when the observer and observed systems become comparable in size that we run into all kinds of problems. As far as I know no apparatus has yet been constructed which is described by the same number of d.o.f as the system it is supposed to measure. But we are rapidly approaching that limit with nanotechnology and then this measurement dichotomy will have to dealt with head on.

I hope this answer makes sense. However, such questions always lie in treacherous territory. So if I've made some tautological error which invalidates everything I have said, please point it out !
A: The answer to Sina's question is that there is a problem, but one does need
to be precise about what you mean by 'the same physics'.  In the conceptual
framework of QM you are always allowed to push the boundary outwards...to 
include more and more of the universe.  This change in boundary never affects
the 'experimental results' or 'measurements' in the following sense: we must
assume there is a second measurement apparatus which measures the dial or 
read-out of the first one.
micro-System ----> | 1st boundary ---> 1st apparatus -----> | 2nd boundary > final apparatus.
The probabilities which will be measured by the second, final apparatus do not depend 
on whether we use the first boundary or use the second boundary.
In this sense, it is not possible to experimentally detect any difference in the system
no matter where we draw the boundary.  
But, is this what Sina means about 'the physics of the first apparatus' (If I may be allowed to paraphrase his question).  Wigner posed this same question very clearly more than once, 
and I recommend reading his papers on this.  If we choose the first boundary, then 'the physics' of the first apparatus is a stochastic process, a non-unitary evolution.  If we choose 
the second boundary, the 'physics' of the first apparatus entangled with the micro system
is now deterministic and unitary.  For Wigner and many others, this is a physical difference and is a serious problem.  For Bayesians and many others, it doesn't count as truly 'physical' because it does not lead to any differences in predictions of experimental results.
So the answer to Sina's question, which is the same as Wigner's question, is, it depends on what you mean by 'physical' and what you mean by 'is a problem'.
A: If you want to study how to move the boundaries and correctly get the right answers, you can study the SSC (Sufficiency of Subspace Correlations) theorem.  There is an entire interpretation (called the Ithaca Interpretation) built around it, and it sounds like exactly what you are describing.  You can look up Mermin, or "correlations without correlata" if you want more references.
However, you are doing it wrong.  For instance, probability isn't observed with a single system, you have to have a bunch of identically prepared systems.  And so if you expect to discuss probability theoretically or experimentally in either case bring up a whole bunch of subsystems, and then your system should have the whole bunch of subsystems.  And then it won't be a probability but a relative frequency.  And you won't evolve to one state or the other, you will evolve to a state where the whole system has a relative frequency of those outcomes, and it will have that frequency with effectively 100% probability.  
That's a game changer.  If you throw away what is really happening then drawing the line looks like it changes things, and that's just a sign that you have not learned how to draw the line correctly.
A: There is an infinite nested system of boxes like matryoshka dolls with infinitely many levels. Label each level by i. Is each box an open system, or a closed system? It's actually possible to have both. At the ith level, there is a time period $T_i$ — starting from the big bang which was the beginning of everything at all levels — when either exactly, or to a very good approximation, the ith box is decoupled from the rest of the i+1th box outside the ith box taken as its environment. The box is effectively a closed system, and we have a factorized unentangled state between the environment and the system. After $T_i$, significant coupling occurs and the system and environment gets entangled, and the system evolves to an open system. It now appears the i+1th box is a closed system, but after some time $T_{i+1} > T_i$, it couples to the rest of the next higher box, etc. .
Copenhagen or many worlds? Both! Suppose you identify yourself inside the i+1th box but outside the ith box at a given moment. Then, before $T_i$, you impose the many worlds interpretation for the ith box taken as a closed system. Then, coupling happens. Information about the ith box which reaches you is considered "collapsed" as in Copenhagen. You become an idealist and think only those things you know about the ith box exists and owe their existence to you, that in fact, the ith box is in you. Then, you start thinking about the i+1th box as a whole which you identify as your universe. After a while, you start to apply systems thinking to yourself, and start identifying less with yourself, and more with the entire box you are in. There comes a conversion moment when you suddenly "jump" up to the next higher level and identify yourself outside the i+1th box but inside the i+2th box, although it would take some time for you to discover the boundaries of the i+2th box. Then, you realize you were really outside the i+1th box all along, but were only falsely identified with it because you were accessing records and memories from that box, but confusing that for the real thing. This is the moment of transcendence! The new box transcends, yet includes all the earlier boxes. Information about the i+1th box which never reaches the i+2th box reverts back to the many worlds interpretation, including those which were falsely identified as "collapsed" at an earlier level.
All the levels above you are temporarily interpreted as classical.
This is the new and improved version of the von Neumann chain and the Heisenberg cut.
Is Being timeless, or Becoming in time? Both! When you identified yourself with the i+1th level, as more and more information reaches you, you think time passes in a process of Becoming with more and more collapses as your knowledge keeps growing. When you transcend the system, you realize these were all timeless memories after all. Becoming has become Being.
God is the nested chain of all the higher levels than the one you currently identify with. God isn't a specific entity at any given level, but transcends yet includes all levels. Any collapse of the wavefunction can only come from God by passing through all the levels above. The nested system of boxes goes on without end, which is why the kabbalists call God Ein Sof.   You always falsely identify yourself with a given level, but you are really God all along.
After a while, you start noticing an algorithmic pattern for the transcendent jumps up each level, and try to contain God within this algorithmic pattern. But you realize this is only a transfinite induction ordinal jump after all in a larger transfinite box. God can't be contained because God is always transcendent.
See Ken Wilber for more details.
