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    $\begingroup$ It's worthwhile to mention that DeWitt's criticism to the result of Choquet-Bruhat and Geroch no longer applies. Jan Sbierski recently proved the existence of a maximal Cauchy development without using Zorn's lemma: On the Existence of a Maximal Cauchy Development for the Einstein Equations - a Dezornification, in Annales Henri Poincaré 17 (2016) 301-329. $\endgroup$ Commented Jan 20, 2017 at 14:21
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    $\begingroup$ It's interesting to note, though, that by loosening the notion of "space", as @AndrejBauer puts it in his answer, one can indeed make the spectral theorem for commutative C*-algebras (a.k.a. Gelfand duality) constructive. More precisely, one replaces "compact Hausdorff topological space" by "compact, completely regular locale in a Grothendieck topos" in the statement of Gelfand duality - this was shown by B. Banaschewski and C.J. Mulvey in A Globalisation of the Gelfand Duality Theorem, Ann. Pure Appl. Logic 137 (2006) 62-103. $\endgroup$ Commented Jan 20, 2017 at 14:54
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    $\begingroup$ @AbdelmalekAbdesselam Agreed, also because Gelfand duality is actually equivalent to the Boolean prime ideal theorem, which is a bit weaker than AC. Nonetheless, the axiom of (countable) dependent choices (normally acronymed DC) is still non-constructive. A large part of analysis can indeed be done just with DC (Baire's theorem, for instance, is equivalent to it), but DeWitt's criticism still stands - he begs for an actual construction of the choice function, even if it is a countably dependent one. Thanks for the physics.SE link, I'll have a look. $\endgroup$ Commented Jan 21, 2017 at 1:02
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    $\begingroup$ This also begs the question of whether separable Hilbert spaces suffice for applications, specially regarding (mathematical) physics: physics.stackexchange.com/questions/90004/separability-axiom-really-necessary/ $\endgroup$ Commented Jan 21, 2017 at 1:02
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    $\begingroup$ One can get away with a strict form of finitism as one's mathematical foundation for physics, as shown by Ye (dx.doi.org/10.1007/978-94-007-1347-5, free draft copy here: phil.pku.edu.cn/cllc/people/fengye/…). From a review [1]: "most of the mathematics necessary for modern theoretical physics can be developed within a strict finitist framework. In Chapter 8, Ye outlines semi-Riemannian geometry sufficient for proving a version of Hawking’s singularity theorem." ([1] journals.uvic.ca/index.php/pir/article/view/13181/4184) $\endgroup$ Commented Jan 21, 2017 at 1:06