Transfinite Induction
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Transfinite induction is an extension of mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. Let P( ) be a property defined for all ordinals . Suppose that whenever P( ) is true for all , then P( ) is also true. Then transfinite induction tells us that P is true for all ordinals. That is, if P( ) is true whenever P( ) is true for all , then P( ) is true for all . Or, more practically: in order to prove a property P for all ordin...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Transfinite induction is an extension of mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. Let P( ) be a property defined for all ordinals . Suppose that whenever P( ) is true for all , then P( ) is also true. Then transfinite induction tells us that P is true for all ordinals. That is, if P( ) is true whenever P( ) is true for all , then P( ) is true for all . Or, more practically: in order to prove a property P for all ordinals , one can assume that it is already known for all smaller .
