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Cardinality (presentation for Math 101 Fall 2008)

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Cardinality Introduction to Analysis December 1, 2008 Samantha Wong
Cardinality ,[object Object],[object Object],[object Object],[object Object]
Some Definitions ,[object Object],[object Object]
Notation ,[object Object],[object Object],[object Object],[object Object]
Ordinal Numbers ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Ordinal Numbers Second Third . . . Two Three . . . First One Ordinal Cardinal
Ordinal Numbers ,[object Object],[object Object]
Some Definitions ,[object Object],[object Object]
Some Definitions (cont’d) ,[object Object],[object Object],[object Object]
A Bit of Cardinal Arithmetic ,[object Object],[object Object],[object Object],[object Object]
Cardinal Numbers ,[object Object],[object Object],[object Object]
The Cardinality of Natural Numbers ,[object Object],[object Object]
Example One ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example Two ,[object Object],[object Object],[object Object],[object Object],[object Object]
Example Three ,[object Object],[object Object],[object Object],[object Object],[object Object]
Definition ,[object Object],[object Object],[object Object],[object Object],[object Object]
Theorem ,[object Object]
Theorem ,[object Object]
The Cardinality of Real Numbers ,[object Object],[object Object],[object Object],[object Object]
The Real Numbers are Uncountable (Proof) ,[object Object],[object Object],[object Object],[object Object]
The Real Numbers are Uncountable (Proof) ,[object Object],[object Object],[object Object],[object Object]
The Real Numbers are Uncountable (Proof) ,[object Object],[object Object]
The Real Numbers are Uncountable (Proof) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
The Real Numbers are Uncountable (Proof) ,[object Object],[object Object],[object Object]
The Real Numbers are Uncountable (Proof) ,[object Object],[object Object],[object Object],[object Object],[object Object]
The Real Numbers are Uncountable (Proof) ,[object Object],[object Object],[object Object],[object Object]
Recall… ,[object Object],[object Object],[object Object],[object Object]
Hmm… ,[object Object]
The Continuum Hypothesis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
The Continuum Hypothesis (cont’d) ,[object Object],[object Object],[object Object]
The Generalized Continuum Hypothesis ,[object Object],[object Object]
The Continuum Hypothesis ,[object Object],l Georg Cantor suggested that no such set exists . Kurt Godel showed that this couldn’t be disproved . Paul Cohen showed that this couldn’t be proved either. 1900 1940 1963
End

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Cardinality