1 by dr. saqib hussain introduction to measure theory mth 426
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ByDr. Saqib Hussain
Introduction to Measure Theory
MTH 426
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Ordinal Numbers
Lecture # 12
MTH 426
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Previous Lecture’s Review
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• Similar Sets
• Ordinal numbers
Lecture’s Outline
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Theorem:
Proof:
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Comparison of well ordered sets:
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Theorem:
Proof:
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Theorem:
Proof:
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Theorem:
Proof:
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Theorem:
Proof:
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Theorem:
Proof:
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Theorem:
Proof:
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Ordinal Numbers:
Cardinal number of a well ordered set is called its ordinal number.
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Inequalities in ordinal numbers:
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Theorem:
Proof:
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Theorem:
Proof:
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Ordinal addition
Ordinal multiplication
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Remark:
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Remark:
Ordinal multiplication is non commutative
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Remark:
Ordinal multiplication is associative
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Choice Function:
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Cartesian Product:
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Axiom of choice:
Cartesian product of non empty family of non empty sets is non empty
OrThere exists a choice function for any non empty family of non empty sets.
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Axiom of choice:
Cartesian product of non empty family of non empty sets is non empty
OrThere exists a choice function for any non empty family of non empty sets.
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Zermelo’s Postulate:
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Theorem:
Proof:
Show that axiom of choice is equivalent to Zermelo’s postulate.
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References:
1. Set Theory and Related Topics by Seymour Lipschutz. 2. Elements of Set Theory by Herbert B. Enderton