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