{"title":"Rainer Stuhlmann-Laeisz. Gottlob Freges Grundgesetze der Arithmetik: Ein Kommentar des Vorworts, des Nachworts und der einleitenden Paragraphen. [Gottlob Frege's Basic Laws of Arithmetic: A Commentary on the Foreword, the Afterword and the Introductory Paragraphs]","authors":"Matthias Wille","doi":"10.1093/philmat/nkab011","DOIUrl":"10.1093/philmat/nkab011","url":null,"abstract":"","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/philmat/nkab011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42276960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson's transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields (surreals, Levi-Civita field, Laurent series) may have advantages over hyperreals in probabilistic modeling. We show that probabilities developed over such fields are less expressive than hyperreal probabilities.
{"title":"Internality, transfer, and infinitesimal modeling of infinite processes","authors":"Emanuele Bottazzi;Mikhail G Katz","doi":"10.1093/philmat/nkaa033","DOIUrl":"10.1093/philmat/nkaa033","url":null,"abstract":"A probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson's transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields (surreals, Levi-Civita field, Laurent series) may have advantages over hyperreals in probabilistic modeling. We show that probabilities developed over such fields are less expressive than hyperreal probabilities.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/philmat/nkaa033","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46761265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We argue that, under the usual assumptions for sufficiently strong arithmetical theories that are subject to Gödel's First Incompleteness Theorem, one cannot, without impropriety, talk about the Gödel sentence of the theory. The reason is that, without violating the requirements of Gödel's theorem, there could be a true sentence and a false one each of which is provably equivalent to its own unprovability in the theory if the theory is unsound.
{"title":"There May Be Many Arithmetical Gödel Sentences","authors":"Kaave Lajevardi;Saeed Salehi","doi":"10.1093/philmat/nkaa041","DOIUrl":"10.1093/philmat/nkaa041","url":null,"abstract":"We argue that, under the usual assumptions for sufficiently strong arithmetical theories that are subject to Gödel's First Incompleteness Theorem, one cannot, without impropriety, talk about the Gödel sentence of the theory. The reason is that, without violating the requirements of Gödel's theorem, there could be a true sentence and a false one each of which is provably equivalent to its own unprovability in the theory if the theory is unsound.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/philmat/nkaa041","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47318243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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