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Addendum to "countable algebra and set existence axioms"

1985
*
Annals of Pure and Applied Logic
*

Now let I be the ideal generated by all x;(';;t

doi:10.1016/0168-0072(85)90020-x
fatcat:eanq5jpwlba77getxryiz4mame
*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. ...##
###
Clarification of some mathematical misunderstandings about Savage's foundations of statistics, 1954

1993
*
Mathematical Social Sciences
*

It is shown that in his model the probability measure cannot be

doi:10.1016/0165-4896(93)90053-l
fatcat:kxvmcb4d3zalfnf2vwy4nfrh6m
*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. ...##
###
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

2010
*
Journal of Mathematical Economics
*

First, I replace his continuity

doi:10.1016/j.jmateco.2010.07.002
fatcat:bpy4wqpb2fbkbp7axbxaf7eety
*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 ...##
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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]

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,

doi:10.1016/b978-044452208-5/50014-4
fatcat:kyjpcahc3rg7vmknu7uj4vrpdq
*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? ...##
###
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. ...##
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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

1983
*
Annals of Pure and Applied Logic
*

Introduction The purpose of this paper is to answer some special cases of the following question: Which

doi:10.1016/0168-0072(83)90012-x
fatcat:hxtu3cfjabb3bm5zu7rm2tiesa
*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 ...##
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Constructive toposes with countable sums as models of constructive set theory

2012
*
Annals of Pure and Applied Logic
*

The model is constructed as a category of classes, using ideas derived from Joyal

doi:10.1016/j.apal.2012.01.013
fatcat:swz7cbzfbfgdlc3iows3va37nq
*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. ...##
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CARATHÉODORY AND THE AXIOMATIZATION AND ALGEBRAIZATION OF MEASURE THEORY IN THE FIRST HALF OF THE XX CENTURY

2020
*
Revista brasileira de história da matemática (RBHM)
*

However, the fact that the union of a

doi:10.47976/rbhm2015v15n3061-81
doaj:8b8b4561e4da46eb8226079117866809
fatcat:cutqwupv65ajpn7n4jhszn6e3e
*countable*collection of somas*exists*needs an*axiom*: [*Axiom*4.] ... continuum)*and*a*countable**set*always has measure zero. ...##
###
Partially additive categories and flow-diagram semantics

1980
*
Journal of Algebra
*

Elgot imposed two

doi:10.1016/0021-8693(80)90212-4
fatcat:gkowhudg2vff7jd3dydayc2wve
*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. ...##
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Algebraic Independence Relations in Randomizations
[article]

2017
*
arXiv
*
pre-print

For arbitrary T, pointwise

arXiv:1703.10913v1
fatcat:endlvm36lne5jpnknfd36ny7k4
*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. ...##
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Continuum of discrete trajectories in eternal inflation

2015
*
Physical Review D
*

We show that the measure problem is caused by the

doi:10.1103/physrevd.91.023511
fatcat:6hz4ghv4dbfozhgbmii2n3qz4e
*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. ...##
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Stone Duality for Markov Processes

2013
*
2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
*

We prove a Stone-type duality theorem between

doi:10.1109/lics.2013.38
dblp:conf/lics/KozenLMP13
fatcat:dix26nhntvfy7p56uv65gpa5ci
*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. ...##
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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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