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Book Review: Foundations of infinitesimal stochastic analysis

D. N. Hoover
<span title="1989-10-01">1989</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/iwegpttow5b4vhkhs3scejn5fi" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
If one did want to construct the set of reals, it would be defined as a set which contained standard parts for all finite rationals.  ...  The theory left over would be the model theory of structures occurring in analysis.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0273-0979-1989-15832-1">doi:10.1090/s0273-0979-1989-15832-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3xc6f7tuqzgitnkcf3yi33jszy">fatcat:3xc6f7tuqzgitnkcf3yi33jszy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190430222403/http://www.ams.org/journals/bull/1989-21-02/S0273-0979-1989-15832-1/S0273-0979-1989-15832-1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/81/94/81941577be44dd3ff98fafb11d170e0b05cb3c5f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0273-0979-1989-15832-1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Page 3230 of Mathematical Reviews Vol. 58, Issue 5 [page]

<span title="">1979</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
In nonstandard analysis one works with a standard universe which consists of all the sets one can obtain from a basic set X by iterating the power set operation an arbitrary finite number of times.  ...  One also works with a non- standard universe of sets together with a mapping which assigns to each standard set A its nonstandard counterpart * A.  ... 
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Page 464 of Mathematical Reviews Vol. , Issue 84b [page]

<span title="">1984</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
In the second part the existence of nonstandard models (containing infinitesimal or infinite numbers) for the theories of natural numbers, rational numbers and real numbers is proved by the upward Léwenheim-Skolem  ...  The results in (1) are proved using methods of saturated models. The first part of (3) is proved by giving an interpretation of Kelley- Morse set theory in the system under consideration. C. A.  ... 
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Page 4811 of Mathematical Reviews Vol. , Issue 83m [page]

<span title="">1983</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
We let X denote an internal set in the enlargement, @ an internal algebra of subsets of X, and v:@-—*[0, 00] an internal additive set function. °y denotes the standard part of v.  ...  Wozniak, Czeslaw 83m:03078d On the nonstandard model of the theory of elasticity. (Russian summary) Bull. Acad. Polon. Sci. Sér. Sci. Tech. 29 (1981/82), no. 11-12, 225-230.  ... 
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Representing measures in potential theory and an ideal boundary

Peter A. Loeb
<span title="">2010</span> <i title="Duke University Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nrfr3w3zrze3xivnrnrjtkx3ey" style="color: black;">Illinois Journal of Mathematics</a> </i> &nbsp;
A primary motivating application of the author's work on measure theory was a nonstandard construction of standard representing measures for positive harmonic functions.  ...  That work yielded new standard weak convergence methods for constructing such measures on spaces of extreme harmonic functions in very general settings.  ...  If A is a set in the standard model, then * A, called the nonstandard extension of A, is the set in the nonstandard model with the same name and formal properties as A.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/ijm/1348505536">doi:10.1215/ijm/1348505536</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fxkhacjqcjd23hejqbxmeg7p3e">fatcat:fxkhacjqcjd23hejqbxmeg7p3e</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20150830080739/http://www.math.uiuc.edu:80/~loeb/BurkholderArticle.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a2/87/a287ebae43ccac66f47571e22fe370af9188f3dc.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/ijm/1348505536"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Book Review: Nonstandard methods in stochastic analysis and mathematical physics

A. E. Hurd
<span title="1987-10-01">1987</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/iwegpttow5b4vhkhs3scejn5fi" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
The second part of the book begins with a long chapter on applications of nonstandard analysis to stochastic processes which explores at length the ideas mentioned in the previous paragraph.  ...  A large part of the book under review is devoted to applications of nonstandard analysis to various aspects of probability theory, and the foundations are presented in Chapter three.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0273-0979-1987-15607-2">doi:10.1090/s0273-0979-1987-15607-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/f77lhpng4jd5thzptdttwbx3da">fatcat:f77lhpng4jd5thzptdttwbx3da</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180725193248/http://www.ams.org/journals/bull/1987-17-02/S0273-0979-1987-15607-2/S0273-0979-1987-15607-2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/0b/fb/0bfb358d131c7d27b6dba60604198e58fd0eafd4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0273-0979-1987-15607-2"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Page 35 of Mathematical Reviews Vol. , Issue 2000a [page]

<span title="">2000</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
35 03H Nonstandard models and model theory. The author studies the logic L, which is first- order logic extended by quantification over functions.  ...  Use of internal set theory (a version of nonstandard analysis), developed by Edward Nelson, shows that Thomson’s lamp is chimerical: its copy within set theory yields a contradiction.  ... 
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Computable quotient presentations of models of arithmetic and set theory [article]

Michał Tomasz Godziszewski, Joel David Hamkins
<span title="2017-02-27">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient presentations of models of arithmetic and set theory.  ...  And similarly no nonstandard model of finite set theory has a computable quotient presentation.  ...  Consider the standard system of any nonstandard model of arithmetic, which is the collection of traces on the standard N of the sets that are coded inside the model.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1702.08350v1">arXiv:1702.08350v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ypkwaioenzasdjmr75be4l4zfq">fatcat:ypkwaioenzasdjmr75be4l4zfq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200830221415/https://arxiv.org/pdf/1702.08350v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5b/ac/5bac29b57fcd1a753fca8458ea3369062d507eb0.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1702.08350v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Page 3416 of Mathematical Reviews Vol. , Issue 97F [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
property, i.e., any countable model of ZFC can be embedded in a model of BST as the class of all standard sets (this is not true for IST and the minimal transitive model of ZFC).  ...  In Part II, External universes over the universe of bounded set theory, two extensions of BST by external sets are designed—the elementary external set theory (EEST) and the Hrbaéek set theory (HST).  ... 
<span class="external-identifiers"> </span>
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Internal approach to external sets and universes

Vladimir Kanovei, Michael Reeken
<span title="">1995</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nzzezfqw65bhpbeksnoskjzuza" style="color: black;">Studia Logica: An International Journal for Symbolic Logic</a> </i> &nbsp;
To solve this problem, we consider BST, bounded set theory, a modification of I S T which is, briefly, a theory for the family of those I S T sets which are members of standard sets.  ...  We show that B S T is strong enough to incorporate external sets in the internal universe in a way sufficient to develop the most advanced applications of nonstandard methods.  ...  Part 1: this article The system of foundations for nonstandard mathematics we propose is based on bounded set theory BST, a minor modification of Nelson's IST, and, at the same time, the "internal part  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bf01061236">doi:10.1007/bf01061236</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kqe2boy6vfbezbofqxoszwnwd4">fatcat:kqe2boy6vfbezbofqxoszwnwd4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809134322/http://lab6.iitp.ru/en/pub/en_sl_1995_kr.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f7/a7/f7a7d725c1fdcfbf2dbdc6d3a2a7fb22443640cc.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bf01061236"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Page 1585 of Mathematical Reviews Vol. 57, Issue 5 [page]

<span title="">1979</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
“In part II, we continue the study of the use of different models of ZF set theory as carriers for the mathematics of quantum mechanics.  ...  Benioff, Paul A. 57 #12210b Models of Zermelo Frankel set theory as carriers for the mathe- matics of physics. II. J. Mathematical Phys. 17 (1976), no. 5, 629-640.  ... 
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A Natural Model of the Multiverse Axioms

Victoria Gitman, Joel David Hamkins
<span title="">2010</span> <i title="Duke University Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/un57kpou5jawnm7yjqoeumbno4" style="color: black;">Notre Dame Journal of Formal Logic</a> </i> &nbsp;
If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of [Hama].  ...  The research of the second author has been supported in part by grants from the CUNY Research Foundation and the National Science Foundation.  ...  The research of the first author has been supported by grants from the CUNY Research Foundation.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/00294527-2010-030">doi:10.1215/00294527-2010-030</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tlqqmpijibeu5cgwyci2rizwii">fatcat:tlqqmpijibeu5cgwyci2rizwii</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20110719182508/http://websupport1.citytech.cuny.edu/faculty/vgitman/files/papers/multiverse.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e6/95/e69532c47099fad73cc81cbb20900a1bae9785a0.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/00294527-2010-030"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Book Review: An introduction to nonstandard real analysis

Robert M. Anderson
<span title="1987-04-01">1987</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/iwegpttow5b4vhkhs3scejn5fi" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
Suppose OE is an internal *-finite set, i.e. an infinite set constructed in a nonstandard model which satisfies all the formal properties of finite sets.  ...  the parts of the backward shift operator.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0273-0979-1987-15523-6">doi:10.1090/s0273-0979-1987-15523-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5ogp2gz755ewxfrklwgepgkhcq">fatcat:5ogp2gz755ewxfrklwgepgkhcq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180723161035/http://www.ams.org/journals/bull/1987-16-02/S0273-0979-1987-15523-6/S0273-0979-1987-15523-6.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/da/41/da41cebca3af8e7035b5a9d8af2ab3c21f830ecc.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0273-0979-1987-15523-6"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

A natural model of the multiverse axioms [article]

Victoria Gitman, Joel David Hamkins
<span title="2011-04-22">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms introduced by Hamkins.  ...  The research of the second author has been supported in part by grants from the CUNY Research Foundation and the National Science Foundation.  ...  The research of the first author has been supported by grants from the CUNY Research Foundation.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1104.4450v1">arXiv:1104.4450v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5w5dg5hlczdpzoqwk3gizd74ce">fatcat:5w5dg5hlczdpzoqwk3gizd74ce</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171006045054/https://core.ac.uk/download/pdf/2191323.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/88/1c/881c38e0359902db9ad882ce31f409e5273fd927.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1104.4450v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Page 2105 of Mathematical Reviews Vol. 57, Issue 6 [page]

<span title="">1979</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
and the standard part of a nonstandard admissible.  ...  Some model-theoretic properties of Z- + bounded A,-collection are examined. The third section deals with standard parts of nonstandard admissible sets (i.e., nonstandard w-models of KP).  ... 
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