The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise
David Hilbert famously remarked, 'No one will drive us from the paradise that Cantor has created'. This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory; logical objects and logical types; independence results and the universe of sets; and the constructs and reality of mathematical structure. Philosophers and mathematicians will find an abundance of intriguing topics in this text, which is appropriate for undergraduate - and graduate-level courses.
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actual inﬁnite algebraic applications Aristotelian Aristotle arithmetic axiom of choice axiom of reducibility axiomatization belong Cantor classical ﬁnitist constructible sets contains continuum hypothesis deﬁnable deﬁned deﬁnition denumerable descriptive set theory division domain elements existence expression extension extensional ﬁgure ﬁnd ﬁnite number ﬁnite set ﬁrst Frege geometrical intuition Godel inaccessible cardinal indeﬁnitely inﬁnite cardinal inﬁnite numbers inﬁnite ordinal numbers inﬁnite sequence inﬁnite sets inner model introduced justiﬁed kind language of ZF limit logic mathematical mathematician means model of ZF natural numbers negation nominalist notion number class objects one-one correspondence ordinal number paradoxes possible potentially inﬁnite power set power set axiom principle problem proof propositional functions proved quantiﬁcation question rational numbers real numbers reﬂect relation Russell’s sense sequences of rationals set axiom set of points set theory space speciﬁc structure subset sufﬁcient theoretical things tion transﬁnite numbers variable ZF axioms