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Addendum to "countable algebra and set existence axioms"

Harvey M. Friedman, Stephen G. Simpson, Rick L. Smith
1985 Annals of Pure and Applied Logic  
Now let I be the ideal generated by all x;(';;t and $!tnf -1 for n EN. We claim that I exists.  ...  Let f, g :N +-N be given with Vi, j ((f(i) # go')) and f, g both one-to-one. Let R, = Q[x,, : n EN] be the polynomial ring over the rationals with countably many indeterminates.  ... 
doi:10.1016/0168-0072(85)90020-x fatcat:eanq5jpwlba77getxryiz4mame

Clarification of some mathematical misunderstandings about Savage's foundations of statistics, 1954

Peter Wakker
1993 Mathematical Social Sciences  
It is shown that in his model the probability measure cannot be countably additive, that the set of events must be a u-algebra and not just an algebra, that Savage did not characterize all atomless finitely  ...  additive probability measures, and that the state space in his model, while infinite, does not have to be uncountable.  ...  Acknowledgement The research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.  ... 
doi:10.1016/0165-4896(93)90053-l fatcat:kxvmcb4d3zalfnf2vwy4nfrh6m

Page 3779 of Mathematical Reviews Vol. , Issue 85i [page]

1985 Mathematical Reviews  
O3E Set theory 85i:03158 Friedman, Harvey M. (1-OHS); 85i:03157 Simpson, Stephen G. (1-PAS); Smith, Rick L. (1-FL) Countable algebra and set existence axioms. Ann. Pure Appl.  ...  The purpose of this paper is to answer some special cases of the following question: Which set existence axioms are needed in order to prove the theorems of countable algebra?  ... 

Simple axioms for countably additive subjective probability

Igor Kopylov
2010 Journal of Mathematical Economics  
First, I replace his continuity axioms P6 and P7 with a simple modification of Arrow's (1970) Monotone Continuity.  ...  Aside from subjective expected utility, I characterize exponential time discounting in a setting with continuous time and an arbitrary consumption set.  ...  a finite decimal expansion; • S = ∞ i=1 ˝i is a product of finite or countable sets ˝i, and A is any algebra that contains sets {s ∈ S : s i = ω i } for all i and all ω i ∈ ˝i; • S is endowed with a Hausdorff  ... 
doi:10.1016/j.jmateco.2010.07.002 fatcat:bpy4wqpb2fbkbp7axbxaf7eety

Page 208 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 56, Issue 2 [page]

1950 American Mathematical Society. Bulletin of the American Mathematical Society  
then a+csb-+c—whence n-a=n-b implies a=); if a; <5; for all 7, 7 of a finite or countable set, then c exists such that a; Sc); for all i, 7.  ...  Since the proof of the postulates for addition of cardinal numbers does not involve the Axiom of Choice for uncountable sets, deduc- tions from the postulates are also independent of this Axiom, pro- vided  ... 

The fourth head of βℕ [chapter]

Ilijas Farah
2007 Open Problems in Topology II  
We start with a concrete problem about continuous maps on thě Cech-Stone remainder of the natural numbers, N * = βN \ N, and gradually zoom out to the larger picture.  ...  Clopen algebras of other Parovičenko spaces are also countably saturated, and this allows one to apply back-and-forth methods to relate these and similar algebras.  ...  Is it possible to construct a nontrivial map betweenČech-Stone remainders of locally compact countable spaces without using additional axioms of set theory?  ... 
doi:10.1016/b978-044452208-5/50014-4 fatcat:kyjpcahc3rg7vmknu7uj4vrpdq

Page 485 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 80, Issue 2 [page]

1955 American Mathematical Society. Transactions of the American Mathematical Society  
Assume that there exists a g-isomorphism 7 carrying the o-algebra {A} of Baire sets in To: onto a g-algebra {7A} that consists of all Baire subsets of the Baire set T7941 =BCY.  ...  It is easy to see that B consists of at least N; points and that every set 7A is countable or has countable complement in B.  ... 

Countable algebra and set existence axioms11Research partially supported by NSF grants MCS-79-23743, MCS-78-02558, and MCS 8107867. Simpson's research was also supported by an Alfred P. Sloan Research Fellowship

Harvey M. Friedman, Stephen G. Simpson, Rick L. Smith
1983 Annals of Pure and Applied Logic  
Introduction The purpose of this paper is to answer some special cases of the following question: Which set existence axioms are needed in order to prove the theorems of countable algebra?  ...  The algebraic theorems which we consider are of a traditional sort: existence and uniqueness of countable algebraic closures; existence and uniqueness of countable real closures; prime and maximal ideals  ... 
doi:10.1016/0168-0072(83)90012-x fatcat:hxtu3cfjabb3bm5zu7rm2tiesa

Constructive toposes with countable sums as models of constructive set theory

Alex Simpson, Thomas Streicher
2012 Annals of Pure and Applied Logic  
The model is constructed as a category of classes, using ideas derived from Joyal and Moerdijk's programme of algebraic set theory.  ...  Our proof of point 4 involves a detour through models of algebraic set theory in the sense of Joyal and Moerdijk [18], extending the work of [4] [5] [6] .  ...  This paper was written while the authors were on sabbatical at the Universities of Ljubljana and Sussex respectively. They thank Andrej Bauer and Bernhard Reus for hosting these visits.  ... 
doi:10.1016/j.apal.2012.01.013 fatcat:swz7cbzfbfgdlc3iows3va37nq

CARATHÉODORY AND THE AXIOMATIZATION AND ALGEBRAIZATION OF MEASURE THEORY IN THE FIRST HALF OF THE XX CENTURY

Carmen Martínez Adame
2020 Revista brasileira de história da matemática (RBHM)  
However, the fact that the union of a countable collection of somas exists needs an axiom: [Axiom 4.]  ...  continuum) and a countable set always has measure zero.  ... 
doi:10.47976/rbhm2015v15n3061-81 doaj:8b8b4561e4da46eb8226079117866809 fatcat:cutqwupv65ajpn7n4jhszn6e3e

Partially additive categories and flow-diagram semantics

Michael A Arbib, Ernest G Manes
1980 Journal of Algebra  
Elgot imposed two axioms on an algebraic theory to define an iterative algebraic theory.  ...  This is the context of Theorem 4.2 with X an algebraic theory. Exactly which set of axioms should be imposedis not yet clear.  ... 
doi:10.1016/0021-8693(80)90212-4 fatcat:gkowhudg2vff7jd3dydayc2wve

Algebraic Independence Relations in Randomizations [article]

Uri Andrews, Isaac Goldbring, H. Jerome Keisler
2017 arXiv   pre-print
For arbitrary T, pointwise algebraic independence in T^R satisfies extension for countable sets, has finite character, has local character with the smallest possible bound, and satisfies base monotonicity  ...  if and only if algebraic independence in T does.  ...  The relation | aω ⌣ over N satisfies extension and full existence for all countable sets A, B, B, C.. Proof. We first prove full existence for countable sets. Let A, B, C be countable subsets of K.  ... 
arXiv:1703.10913v1 fatcat:endlvm36lne5jpnknfd36ny7k4

Continuum of discrete trajectories in eternal inflation

Vitaly Vanchurin
2015 Physical Review D  
We show that the measure problem is caused by the countable additivity axiom applied to the maximal σ-algebra of countably infinite sample spaces.  ...  We discuss eternal inflation in context of classical probability spaces defined by a triplet: sample space, σ-algebra and probability measure.  ...  The author is grateful to Alan Guth, Mahdiyar Noorbala, Ken Olum and Alex Vilenkin for very useful discussions and comments on the manuscript. The work was supported in part by Templeton Foundation.  ... 
doi:10.1103/physrevd.91.023511 fatcat:6hz4ghv4dbfozhgbmii2n3qz4e

Stone Duality for Markov Processes

Dexter Kozen, Kim G. Larsen, Radu Mardare, Prakash Panangaden
2013 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science  
We prove a Stone-type duality theorem between countable Aumann algebras and countably-generated continuous-space Markov processes.  ...  Our results subsume existing results on completeness of probabilistic modal logics for Markov processes.  ...  Acknowledgments We would like to thank Jean Goubault-Larrecq, Ernst-Erich Doberkat, Robert Goldblatt and Larry Moss for helpful discussions.  ... 
doi:10.1109/lics.2013.38 dblp:conf/lics/KozenLMP13 fatcat:dix26nhntvfy7p56uv65gpa5ci

Page 866 of Mathematical Reviews Vol. 18, Issue 11 [page]

1957 Mathematical Reviews  
Finally some known results on the existence of countably additive measures on A/J and in particular on the existence of countably additive two-valued measures on A/J, if A is the Boolean algebra of all  ...  He proves that the subsystems of propositional algebra L(C, E) consisting of the consequences of the axioms (1) and (2), L(C, EZ, K) consisting of the consequences of the axioms (1), (2) and (3), and L  ... 
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