We discuss some recent work by Tim Maudlin concerning Black Hole Information Loss. We argue, contra Maudlin, that there is a paradox, in the straightforward sense that there are propositions that appear true, but which are incompatible with one another. We discuss the significance of the paradox and Maudlin's response to it.The black hole information loss paradox (Hawking, 1976) is a widely discussed (putative) puzzle that arises when one attempts to make sense of physicists' expectation that global quantum dynamics will be unitary in light of the postulated phenomenon of black hole evaporation (Hawking, 1974(Hawking, , 1975 Unruh, 1976), which is apparently a consequence of Hawking radiation. 1 In a provocative recent manuscript, Tim Maudlin (2017) forcefully argues that "There is no 'information loss' paradox." The "solution" to the "paradox", he claims, requires no new physics and indeed, was already available, though not appreciated, in 1975. The appearance of paradox arises only because of persistent errors, both mathematical and conceptual in character, by prominent members of the physics community.There is much to admire in Maudlin's treatment of the subject. We agree, for instance, with his discussion of the (non-)role of "information" in the putative paradox. What is really at issue is predictability, or even better, retrodictability, and not information (at least in the sense of information theory): as we will describe in more detail below, the puzzle concerns whether any specification of data on a particular surface is sufficient to retrodict the physical process by which that data came about. We also think that Maudlin draws helpful attention to the core issues, which sometimes get obscured in physicists' discussions of solutions to the paradox. For instance, as Maudlin correctly observes, one can only expect unitary evolution between states defined on a special class of surfaces, known as Cauchy surfaces; any conceptually adequate discussion of black hole information loss will need to consider the conditions under which suitable Cauchy surfaces are available in the first place.Still, we do not agree with his central claim. As we will presently argue, we take it that there is a paradox, in the precise sense that there are well-motivated and widely accepted Email addresses: jmanchak@uci.edu (JB Manchak), weatherj@uci.edu (James Owen Weatherall) 1 See Unruh and Wald (2017) for a recent (opinionated) review of the issues and the literature over the last four decades.

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Leibnizian metaphysics underpins the universally held view that spacetime must be inextendible – that it must be “as large as it can be” in a sense. But here we demonstrate a surprising fact within the context of general relativity: the property of inextendibility turns out to be unstable when attention is restricted to certain collections of “physically reasonable”spacetimes.

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Wilhelm (2021) has recently defended a criterion for comparing structure of mathematical objects, which he calls Subgroup. He argues that Subgroup is better than SYM * , another widely adopted criterion. We argue that this is mistaken; Subgroup is strictly worse than SYM * . We then formulate a new criterion that improves on both SYM * and Subgroup, answering Wilhelm's criticisms of SYM * along the way. We conclude by arguing that no criterion that looks only to the automorphisms of mathematical objects to compare their structure can be fully satisfactory.