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LOCATION:60 Evans Hall
SEQUENCE:0
DTEND:20090828T180000
DTSTART:20090828T161000
UID:philosophy.berkeley.edu:events:554
DTSTAMP:20091124T190325
DESCRIPTION:Cantor’s theory of cardinal numbers offers a way to generalize 
 arithmetic from finite sets to infinite sets using the notion\nof one-to-on
 e correspondence between two sets. As is well known\, all countably infinit
 e sets have the same ‘size’ in this account\, namely that of the cardinalit
 y of the natural numbers. However\, throughout the history of reflections o
 n infinity another powerful intuition has played a major role: if a collect
 ion A is properly included in a collection B then the ‘size’ of A should be
  less than the ‘size’ of B (part-whole principle). This second intuition wa
 s not developed mathematically in a satisfactory way until quite recently. 
 In this talk I begin by reviewing the contributions of some thinkers who ar
 gued in favor of the assignment of different sizes to infinite collections 
 of natural numbers. Then\, I present some recent mathematical developments 
 that generalize the part-whole principle to infinite sets in a coherent fas
 hion. Finally\, I show how these new developments are important for a\nprop
 er evaluation of a number of positions in philosophy of mathematics which a
 rgue either for the inevitability of the Cantorian notion of infinite numbe
 r (Gödel) or for the rational nature of the Cantorian generalization as opp
 osed to that\, offered by Bolzano\, based on the part-whole principle (Kitc
 her).
SUMMARY:Logic Colloquium\nPaolo Mancosu\nMeasuring the size of infinite col
 lections of natural numbers: Was Cantor's theory of infinite number inevita
 ble?
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