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Every group has a terminating transfinite automorphism tower
The automorphism tower of a group is obtained by computing its automorphism
group, the automorphism group of THAT group, and so on, iterating
transfinitely. Each group maps canonically into the next using inner
automorphisms, and so at limit stages one can take a direct limit and continue
the iteration. The tower is said to terminate if a fixed point is reached, that
is, if a group is reached which is isomorphic to its automorphism group by the
natural map. This occurs if a complete group is reached, one which is
centerless and has only inner automorphisms. Wielandt [1939] proved the
classical result that the automorphism tower of any centerless finite group
terminates in finitely many steps. Rae and Roseblade [1970] proved that the
automorphism tower of any centerless Cernikov group terminates in finitely many
steps. Hulse [1970] proved that the the automorphism tower of any centerless
polycyclic group terminates in countably many steps. Simon Thomas [1985] proved
that the automorphism tower of any centerless group eventually terminates. In
this paper, I remove the centerless assumption, and prove that every group has
a terminating transfinite automorphism tower.Comment: 4 pages, to appear in the Proceedings of the American Mathematical
Society, see also
http://scholar.library.csi.cuny.edu/users/hamkins/papers.html#MyAutoTower
The modal logic of arithmetic potentialism and the universal algorithm
I investigate the modal commitments of various conceptions of the philosophy
of arithmetic potentialism. Specifically, I consider the natural potentialist
systems arising from the models of arithmetic under their natural extension
concepts, such as end-extensions, arbitrary extensions, conservative extensions
and more. In these potentialist systems, I show, the propositional modal
assertions that are valid with respect to all arithmetic assertions with
parameters are exactly the assertions of S4. With respect to sentences,
however, the validities of a model lie between S4 and S5, and these bounds are
sharp in that there are models realizing both endpoints. For a model of
arithmetic to validate S5 is precisely to fulfill the arithmetic maximality
principle, which asserts that every possibly necessary statement is already
true, and these models are equivalently characterized as those satisfying a
maximal theory. The main S4 analysis makes fundamental use of the
universal algorithm, of which this article provides a simplified,
self-contained account. The paper concludes with a discussion of how the
philosophical differences of several fundamentally different potentialist
attitudes---linear inevitability, convergent potentialism and radical branching
possibility---are expressed by their corresponding potentialist modal
validities.Comment: 38 pages. Inquiries and commentary can be made at
http://jdh.hamkins.org/arithmetic-potentialism-and-the-universal-algorithm.
Version v3 has further minor revisions, including additional reference
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