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    $\begingroup$ Regarding the reals over the rationals, mightn't the fact that any real (or countable collection of reals) is contained in countable real-closed subfield of $\mathbb{R}$ be regarded as fulfilling the reflection principle? That is, $\mathbb{R}$ is reflecting down to its countable elementary substructures. $\endgroup$ Commented Oct 25, 2012 at 22:41
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    $\begingroup$ The fact that there ought to be a separable example is too trivial to count as a success of Reflection, since if there is any example at all, you can take a separable subspace of it and then you've got a separable example. $\endgroup$ Commented Oct 26, 2012 at 8:30
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    $\begingroup$ The general point I'm making here is that in this context it's the mathematics that tells us to what extent Maximize is an appropriate principle, rather than the principle that is guiding our mathematical expectations. $\endgroup$ Commented Oct 26, 2012 at 8:42
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    $\begingroup$ I agree that this instance of reflection is too trivial to be called a leap, but it is nevertheless an example of reflection. (Reflection is rooted in the Löwenheim-Skolem Theorem, the Compactness Theorem, and relatives. Direct applications of these are too trivial to ever form conjectures, but the reflection idea is what drives me to try these theorems in relevant contexts.) $\endgroup$ Commented Oct 26, 2012 at 12:02
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    $\begingroup$ This discussion, which I find very interesting by the way, leaves me with the feeling that I don't understand very well what the rules of thumb are really saying and what their purpose is. I agree about Banach space theory: in some ways it is a very structureless subject, because any old bunch of functionals can be used to define a norm (as long as you've got enough of them that you don't have just a seminorm), but from time to time it springs surprises -- the almost negligible constraints nevertheless interestingly restrict what you can do. $\endgroup$ Commented Oct 26, 2012 at 12:59