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Line 67: ==References== * {{Citation\|last=Hamkins\|first= Joel David\|url=https://mathoverflow.net/q/44129 \|title=Is the analysis as taught in universities in fact the analysis of definable numbers?\|journal=MathOverflow\|date=October 2010\|accessdate=2016-03-05\|authorlink=Joel David Hamkins}}. * {{Citation\|~~last~~last1=Hamkins\|~~first~~first1=Joel David\|last2=Linetsky\|first2=David\|last3=Reitz\|first3=Jonas\|title=Pointwise Definable Models of Set Theory\|arxiv=1105.4597\|journal=Journal of Symbolic Logic\|volume=78\|issue=1\|pages=139–156\|year=2013\|doi=10.2178/jsl.7801090\|s2cid=43689192}}. * {{Citation \| last1=Kunen \| first1=Kenneth \| author1-link=Kenneth Kunen \| title=[[Set Theory: An Introduction to Independence Proofs]] \| publisher=North-Holland \| ___location=Amsterdam \| isbn=978-0-444-85401-8 \| year=1980}}. *{{Citation \| last= Turing \| first= A.M. \| publication-date = 1937 \| year = 1936 \| title = On Computable Numbers, with an Application to the Entscheidungsproblem \| periodical = Proceedings of the London Mathematical Society \| series = 2 \| volume = 42 \| issue= 1 \| pages = 230–65 \| url = http://www.abelard.org/turpap2/tp2-ie.asp \| doi= 10.1112/plms/s2-42.1.230 }} (and {{Citation \| last = Turing \| first = A.M. \| publication-date = 1937 \| title = On Computable Numbers, with an Application to the Entscheidungsproblem: A correction \| periodical = Proceedings of the London Mathematical Society \| series = 2 \| volume = 43 \| issue = 6 \| pages = 544–6 \| doi = 10.1112/plms/s2-43.6.544 \| year = 1938 }}). Computable numbers (and Turing's a-machines) were introduced in this paper; the definition of computable numbers uses infinite decimal sequences.

Definable real number: Difference between revisions