Divisibility in βN and N

The paper first covers several properties of the extension of the divisibility relation to a set N of nonstandard integers, including an analogue of the basic theorem of arithmetic. After that, a connection is established with the divisibility in the Stone-Čech compactification βN, proving that the...

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Bibliographic Details
Published inReports on mathematical logic Vol. 54; no. 54; pp. 65 - 82
Main Author Sobot, Boris
Format Journal Article
LanguageEnglish
Published Kraków Wydawnictwo Uniwersytetu Jagiellońskiego 2019
Jagiellonian University Press
Jagiellonian University-Jagiellonian University Press
Subjects
Analytic Philosophy
Enlargement
Logic
Mathematical problems
Methodology and research technology
Theorems
StoneČech compactification
ultrafilter
nonstandard integer
divisibility
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Summary:The paper first covers several properties of the extension of the divisibility relation to a set N of nonstandard integers, including an analogue of the basic theorem of arithmetic. After that, a connection is established with the divisibility in the Stone-Čech compactification βN, proving that the divisibility of ultrafilters introduced by the author is equivalent to divisibility of some elements belonging to their respective monads in an enlargement. Some earlier results on ultrafilters on lower levels on the divisibility hierarchy are illuminated by nonstandard methods. Using limits by ultrafilters we obtain results on ultrafilters above these finite levels, showing that for them a distribution by levels is not possible.
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ISSN:0137-2904
2084-2589
DOI:10.4467/20842589RM.19.003.10651