Results for 'infinite permutation groups'

986 found
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1. Homogeneity of infinite permutation groups.Saharon Shelah & Simon Thomas - 1989 - Archive for Mathematical Logic 28 (2):143-147.

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2. Supplements of bounded permutation groups.Stephen Bigelow - 1998 - Journal of Symbolic Logic 63 (1):89-102.
Let λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation group B λ (Ω), or simply B λ , is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation group G acting on Ω is a supplement of B λ if B λ G is the full symmetric group on Ω. In [7], Macpherson and Neumann claimed (...)

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3. Binary Relations and Permutation Groups.Hajnal Andréka & Ivo Düntsch - 1995 - Mathematical Logic Quarterly 41 (2):197-216.
We discuss some new properties of the natural Galois connection among set relation algebras, permutation groups, and first order logic. In particular, we exhibit infinitely many permutational relation algebras without a Galois closed representation, and we also show that every relation algebra on a set with at most six elements is Galois closed and essentially unique. Thus, we obtain the surprising result that on such sets, logic with three variables is as powerful in expression as full first order (...)

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4. On -Homogeneous, but Not -Transitive Permutation Groups.Saharon Shelah & Lajos Soukup - 2023 - Journal of Symbolic Logic 88 (1):363-380.
A permutation group G on a set A is ${\kappa }$ -homogeneous iff for all $X,Y\in \bigl [ {A} \bigr ]^ {\kappa }$ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. G is ${\kappa }$ -transitive iff for any injective function f with $\operatorname {dom}(f)\cup \operatorname {ran}(f)\in \bigl [ {A} \bigr ]^ {\le {\kappa }}$ and $|A\setminus \operatorname {dom}(f)|=|A\setminus \operatorname {ran}(f)|=|A|$ there is a $g\in G$ with $f\subset g$.Giving a partial answer to a question (...)

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5. The cofinality spectrum of the infinite symmetric group.Saharon Shelah & Simon Thomas - 1997 - Journal of Symbolic Logic 62 (3):902-916.
Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem is the main result of this paper. Theorem. Suppose that $V \models GCH$ . Let C be (...)

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6. Automorphism Groups of Arithmetically Saturated Models.Ermek S. Nurkhaidarov - 2006 - Journal of Symbolic Logic 71 (1):203 - 216.
In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that ifMis a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2be countable arithmetically (...)

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7. Failure of n -uniqueness: a family of examples.Elisabetta Pastori & Pablo Spiga - 2011 - Mathematical Logic Quarterly 57 (2):133-148.
In this paper, the connections between model theory and the theory of infinite permutation groups are used to study the n-existence and the n-uniqueness for n-amalgamation problems of stable theories. We show that, for any n ⩾ 2, there exists a stable theory having -existence and k-uniqueness, for every k ⩽ n, but has neither -existence nor -uniqueness. In particular, this generalizes the example, for n = 2, due to Hrushovski given in 3. © 2011 WILEY-VCH Verlag (...)

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8. Extended anonymity and Paretian relations on infinite utility streams.Tsuyoshi Adachi, Susumu Cato & Kohei Kamaga - 2014 - Mathematical Social Sciences 2014 (72):24-32.
We examine the range of anonymity that is compatible with a Paretian social welfare relation (SWR) on infinite utility streams. Three alternative coherence properties of an SWR are considered, namely, acyclicity, quasi-transitivity, and Suzumura consistency. For each case, we show that a necessary and sufficient condition for a set of permutations to be the set of permissible permutations of some Paretian SWR is given by the cyclicity of permutations and a weakening of group structure. Further, for each case of (...)

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9. Additive representation of separable preferences over infinite products.Marcus Pivato - 2014 - Theory and Decision 77 (1):31-83.
Let X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }$$\end{document} be a set of outcomes, and let I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{I }$$\end{document} be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} on XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }^\mathcal{I }$$\end{document} admits an additive representation. That is: there exists a linearly ordered abelian (...)

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10. Isomorphism types of maximal cofinitary groups.Bart Kastermans - 2009 - Bulletin of Symbolic Logic 15 (3):300-319.
A cofinitary group is a subgroup of Sym(ℕ) where all nonidentity elements have finitely many fixed points. A maximal cofinitary group is a cofinitary group, maximal with respect to inclusion. We show that a maximal cofinitary group cannot have infinitely many orbits. We also show, using Martin's Axiom, that no further restrictions on the number of orbits can be obtained. We show that Martin's Axiom implies there exist locally finite maximal cofinitary groups. Finally we show that there exists a (...)

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11. A many permutation group result for unstable theories.Mark D. Schlatter - 1998 - Journal of Symbolic Logic 63 (2):694-708.
We extend Shelah's first many model result to show that an unstable theory has 2 κ many non-permutation group isomorphic models of size κ, where κ is an uncountable regular cardinal.

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12. A many permutation group result for unstable theories.Mark D. Schlatter - 1998 - Journal of Symbolic Logic 63 (2):694-708.
We extend Shelah's first many model result to show that an unstable theory has 2κmany non-permutation group isomorphic models of size κ, where κ is an uncountable regular cardinal.

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13. Cardinal invariants related to permutation groups.Bart Kastermans & Yi Zhang - 2006 - Annals of Pure and Applied Logic 143 (1-3):139-146.
We consider the possible cardinalities of the following three cardinal invariants which are related to the permutation group on the set of natural numbers: the least cardinal number of maximal cofinitary permutation groups; the least cardinal number of maximal almost disjoint permutation families; the cofinality of the permutation group on the set of natural numbers.We show that it is consistent with that ; in fact we show that in the Miller model.

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14. Some results on permutation group isomorphism and categoricity.Anand Pillay & Mark D. Schlatter - 2002 - Journal of Symbolic Logic 67 (3):910-914.
We extend Morley's Theorem to show that if a theory is κ-p-categorical for some uncountable cardinal κ, it is uncountably categorical. We then discuss ω-p-categoricity and provide examples to show that similar extensions for the Baldwin-Lachlan and Lachlan Theorems are not possible.

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15. Combinatorial and recursive aspects of the automorphism group of the countable atomless Boolean algebra.E. W. Madison & B. Zimmermann-Huisgen - 1986 - Journal of Symbolic Logic 51 (2):292-301.
Given an admissible indexing φ of the countable atomless Boolean algebra B, an automorphism F of B is said to be recursively presented (relative to φ) if there exists a recursive function $p \in \operatorname{Sym}(\omega)$ such that F ⚬ φ = φ ⚬ p. Our key result on recursiveness: Both the subset of $\operatorname{Aut}(\mathscr{B})$ consisting of all those automorphisms which are recursively presented relative to some indexing, and its complement, the set of all "totally nonrecursive" automorphisms, are uncountable. This arises (...)

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16. The cofinality of the infinite symmetric group and groupwise density.Jörg Brendle & Maria Losada - 2003 - Journal of Symbolic Logic 68 (4):1354-1361.
We show that g ≤ c(Sym(ω)) where g is the groupwise density number and c(Sym(ω)) is the cofinality of the infinite symmetric group. This solves (the second half of) a problem addressed by Thomas.

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17. Maximal subgroups of infinite symmetric groups.James E. Baumgartner, Saharon Shelah & Simon Thomas - 1992 - Notre Dame Journal of Formal Logic 34 (1):1-11.

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18. Normal subgroups of infinite symmetric groups, with an application to stratified set theory.Nathan Bowler & Thomas Forster - 2009 - Journal of Symbolic Logic 74 (1):17-26.

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19. An infinite superstable group has infinitely many conjugacy classes.I. Aguzarov, R. E. Farey & J. B. Goode - 1991 - Journal of Symbolic Logic 56 (2):618-623.

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20. On linearly ordered sets and permutation groups of countable degree.Hans Läuchli & Peter M. Neumann - 1988 - Archive for Mathematical Logic 27 (2):189-192.

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21. Groupwise density and the cofinality of the infinite symmetric group.Simon Thomas - 1998 - Archive for Mathematical Logic 37 (7):483-493.
We study the relationship between the cofinality $c(Sym(\omega))$ of the infinite symmetric group and the cardinal invariants $\frak{u}$ and $\frak{g}$ . In particular, we prove the following two results. Theorem 0.1 It is consistent with ZFC that there exists a simple $P_{\omega_{1}}$ -point and that $c(Sym(\omega)) = \omega_{2} = 2^{\omega}$ . Theorem 0.2 If there exist both a simple $P_{\omega_{1}}$ -point and a $P_{\omega_{2}}$ -point, then $c(Sym(\omega)) = \omega_{1}$.

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22. Unbounded families and the cofinality of the infinite symmetric group.James D. Sharp & Simon Thomas - 1995 - Archive for Mathematical Logic 34 (1):33-45.
In this paper, we study the relationship between the cofinalityc(Sym(ω)) of the infinite symmetric group and the minimal cardinality $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{b}$$ of an unbounded familyF of ω ω.

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23. Russell's alternative to the axiom of choice.Norbert Brunner & Paul Howard - 1992 - Mathematical Logic Quarterly 38 (1):529-534.
We prove the independence of some weakenings of the axiom of choice related to the question if the unions of wellorderable families of wellordered sets are wellorderable.

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24. A model-theoretic proof for P ≠ NP over all infinite Abelian groups.Mihai Prunescu - 2002 - Journal of Symbolic Logic 67 (1):235 - 238.
We give a model-theoretic proof of the fact that for all infinite Abelian groups P ≠ NP in the sense of binary nondeterminism. This result has been announced 1994 by Christine Gabner.

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25. A Model-theoretic Proof For {$\roman P\neq {\rm Np}$} Over All Infinite Abelian Groups.Mihai Prunescu - 2002 - Journal of Symbolic Logic 67 (1):235-238.

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26. Uniformization Problems and the Cofinality of the Infinite Symmetric Group.James D. Sharp & Simon Thomas - 1994 - Notre Dame Journal of Formal Logic 35 (3):328-345.
Assuming Martin's Axiom, we compute the value of the cofinality of the symmetric group on the natural numbers. We also show that Martin's Axiom does not decide the value of the covering number of a related Mycielski ideal.

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27. Separations and transfers in the polynomial hierarchy of infinite abelian groups.M. Bourgade - 2001 - Mathematical Logic Quarterly 47 (4):493-502.

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29. Subgroups of small index in infinite symmetric groups. II.Saharon Shelah & Simon Thomas - 1989 - Journal of Symbolic Logic 54 (1):95-99.

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30. Invariance and Definability, with and without Equality.Denis Bonnay & Fredrik Engström - 2018 - Notre Dame Journal of Formal Logic 59 (1):109-133.
The dual character of invariance under transformations and definability by some operations has been used in classical works by, for example, Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this article, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in L∞∞ so as to cover the cases that are of interest in the logicality debates, getting McGee’s theorem about (...)

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31. Infinite Systems in SM Explanations: Thermodynamic Limit, Renormalization (semi-) Groups, and Irreversibility.Chuang Liu - 2001 - Philosophy of Science 68 (S3):S325-S344.
This paper examines the justifications for using infinite systems to ‘recover’ thermodynamic properties, such as phase transitions, critical phenomena, and irreversibility, from the micro-structure of matter in bulk. Section 2 is a summary of such rigorous methods as in taking the thermodynamic limit to recover PT and in using renormalization group approach to explain the universality of critical exponents. Section 3 examines various possible justifications for taking TL on physically finite systems. Section 4 discusses the legitimacy of applying TL (...)

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32. Actions by the classical Banach spaces.G. Hjorth - 2000 - Journal of Symbolic Logic 65 (1):392-420.
The study of continuous group actions is ubiquitous in mathematics, and perhaps the most general kinds of actions for which we can hope to prove theorems in just ZFC are those where a Polish group acts on a Polish space.For this general class we can find works such as [29] that build on ideas from ergodic theory and examine actions of locally compact groups in both the measure theoretic and topological contexts. On the other hand a text in model (...)

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33. Explaining Universality: Infinite Limit Systems in the Renormalization Group Method.Jingyi Wu - 2021 - Synthese (5-6):14897-14930.
I analyze the role of infinite idealizations used in the renormalization group (RG hereafter) method in explaining universality across microscopically different physical systems in critical phenomena. I argue that despite the reference to infinite limit systems such as systems with infinite correlation lengths during the RG process, the key to explaining universality in critical phenomena need not involve infinite limit systems. I develop my argument by introducing what I regard as the explanatorily relevant property in RG (...)

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34. Infinite systems in SM explanations: Thermodynamic limit, renormalization (semi-) groups, and irreversibility.Chuang Liu - 2001 - Proceedings of the Philosophy of Science Association 2001 (3):S325-.
This paper examines the justifications for using infinite systems to 'recover' thermodynamic properties, such as phase transitions (PT), critical phenomena (CP), and irreversibility, from the micro-structure of matter in bulk. Section 2 is a summary of such rigorous methods as in taking the thermodynamic limit (TL) to recover PT and in using renormalization (semi-) group approach (RG) to explain the universality of critical exponents. Section 3 examines various possible justifications for taking TL on physically finite systems. Section 4 discusses (...)

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35. Finite covers with finite kernels.David M. Evans - 1997 - Annals of Pure and Applied Logic 88 (2-3):109-147.
We are concerned with the following problem. Suppose Γ and Σ are closed permutation groups on infinite sets C and W and ρ: Γ → Σ is a non-split, continuous epimorphism with finite kernel. Describe the possibilities for ρ. Here, we consider the case where ρ arises from a finite cover π: C → W. We give reasonably general conditions on the permutation structure W;Σ which allow us to prove that these covers arise in two possible (...)

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36. Classification of -Categorical Monadically Stable Structures.Bertalan Bodor - 2024 - Journal of Symbolic Logic 89 (2):460-495.
A first-order structure $\mathfrak {A}$ is called monadically stable iff every expansion of $\mathfrak {A}$ by unary predicates is stable. In this paper we give a classification of the class $\mathcal {M}$ of $\omega$ -categorical monadically stable structure in terms of their automorphism groups. We prove in turn that $\mathcal {M}$ is the smallest class of structures which contains the one-element pure set, is closed under isomorphisms, and is closed under taking finite disjoint unions, infinite copies, and (...)

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37. The permutations with N_ non-fixed points and the sequences with length _N of a set.Jukkrid Nuntasri & Pimpen Vejjajiva - forthcoming - Journal of Symbolic Logic:1-10.
We write$\mathcal {S}_n(A)$for the set of permutations of a setAwithnnon-fixed points and$\mathrm {{seq}}^{1-1}_n(A)$for the set of one-to-one sequences of elements ofAwith lengthnwherenis a natural number greater than$1$. With the Axiom of Choice,$|\mathcal {S}_n(A)|$and$|\mathrm {{seq}}^{1-1}_n(A)|$are equal for all infinite setsA. Among our results, we show, in ZF, that$|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$for any infinite setAif${\mathrm {AC}}_{\leq n}$is assumed and this assumption cannot be removed. In the other direction, we show that$|\mathrm {{seq}}^{1-1}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$for any infinite setAand the subscript$n+1$cannot be (...)

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38. Construction of models from groups of permutations.Miroslav Benda - 1975 - Journal of Symbolic Logic 40 (3):383-388.

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39. Elementary equivalence of infinite-dimensional classical groups.Vladimir Tolstykh - 2000 - Annals of Pure and Applied Logic 105 (1-3):103-156.
Let D be a division ring such that the number of conjugacy classes of the multiplicative group D ∗ is equal to the power of D ∗ . Suppose that H is the group GL or PGL, where V is a vector space of infinite dimension ϰ over D . We prove, in particular, that, uniformly in κ and D , the first-order theory of H is mutually syntactically interpretable with the theory of the two-sorted structure 〈κ,D〉 in the (...)

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40. Exclusion Principles as Restricted Permutation Symmetries.S. Tarzi - 2003 - Foundations of Physics 33 (6):955-979.
We give a derivation of exclusion principles for the elementary particles of the standard model, using simple mathematical principles arising from a set theory of identical particles. We apply the theory of permutation group actions, stating some theorems which are proven elsewhere, and interpreting the results as a heuristic derivation of Pauli's Exclusion Principle (PEP) which dictates the formation of elements in the periodic table and the stability of matter, and also a derivation of quark confinement. We arrive at (...)

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41. Cardinal invariants of infinite groups.Jörg Brendle - 1990 - Archive for Mathematical Logic 30 (3):155-170.
LetG be a group. CallG akC-group if every element ofG has less thank conjugates. Denote byP(G) the least cardinalk such that any subset ofG of sizek contains two elements which commute.It is shown that the existence of groupsG such thatP(G) is a singular cardinal is consistent withZFC. So is the existence of groupsG which are notkC but haveP(G) (...)

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42. The permutations with n_ non‐fixed points and the subsets with _n elements of a set.Supakun Panasawatwong & Pimpen Vejjajiva - 2023 - Mathematical Logic Quarterly 69 (3):341-346.
We write and for the cardinalities of the set of permutations with n non‐fixed points and the set of subsets with n elements, respectively, of a set which is of cardinality, where n is a natural number greater than 1. With the Axiom of Choice, and are equal for all infinite cardinals. We show, in ZF, that if is assumed, then for any infinite cardinal. Moreover, the assumption cannot be removed for and the superscript cannot be replaced by (...)
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43. Reverse mathematics, young diagrams, and the ascending chain condition.Kostas Hatzikiriakou & Stephen G. Simpson - 2017 - Journal of Symbolic Logic 82 (2):576-589.
LetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$ to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

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44. There are 2ℵ⚬ many almost strongly minimal generalized n-gons that do not interpret and infinite group.Mark J. Debonis & Ali Nesin - 1998 - Journal of Symbolic Logic 63 (2):485 - 508.
Generalizedn-gons are certain geometric structures (incidence geometries) that generalize the concept of projective planes (the nontrivial generalized 3-gons are exactly the projective planes).In a simplified world, every generalizedn-gon of finite Morley rank would be an algebraic one, i.e., one of the three families described in [9] for example. To our horror, John Baldwin [2], using methods discovered by Hrushovski [7], constructed ℵ1-categorical projective planes which are not algebraic. The projective planes that Baldwin constructed fail to be algebraic in a dramatic (...)

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46. Infinite barbarians.Daniel Nolan - 2019 - Ratio 32 (3):173-181.
This paper discusses an infinite regress that looms behind a certain kind of historical explanation. The movement of one barbarian group is often explained by the movement of others, but those movements in turn call for an explanation. While their explanation can again be the movement of yet another group of barbarians, if this sort of explanation does not stop somewhere we are left with an infinite regress of barbarians. While that regress would be vicious, it cannot be (...)

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47. Pareto Principles in Infinite Ethics.Amanda Askell - 2018 - Dissertation, New York University
It is possible that the world contains infinitely many agents that have positive and negative levels of well-being. Theories have been developed to ethically rank such worlds based on the well-being levels of the agents in those worlds or other qualitative properties of the worlds in question, such as the distribution of agents across spacetime. In this thesis I argue that such ethical rankings ought to be consistent with the Pareto principle, which says that if two worlds contain the same (...)

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48. Elimination of unbounded quantifiers for some poly-regular groups of infinite rank.Philip Scowcroft - 2007 - Annals of Pure and Applied Logic 149 (1-3):40-80.
This paper extends theorems of Belegradek about poly-regular groups of finite rank to certain poly-regular groups of infinite rank. A model-theoretic property aiding these investigations is the elimination of unbounded quantifiers, and the paper establishes both a general model-theoretic test for this property and results about bounded quantifiers in the special context of ordered Abelian groups.

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49. Selective and Ramsey Ultrafilters on G-spaces.Oleksandr Petrenko & Igor Protasov - 2017 - Notre Dame Journal of Formal Logic 58 (3):453-459.
Let G be a group, and let X be an infinite transitive G-space. A free ultrafilter U on X is called G-selective if, for any G-invariant partition P of X, either one cell of P is a member of U, or there is a member of U which meets each cell of P in at most one point. We show that in ZFC with no additional set-theoretical assumptions there exists a G-selective ultrafilter on X. We describe all G-spaces X (...)