LessonsThe Originator of the World: Necessary Existence, Infinite Regress and the Correspondence ProofQuiz

فإن كان بإزاء كل واحد من الأولى واحد من الثانية، كان الناقص كالزائد وهو محال
If every member of the first has a counterpart in the second, the shorter would equal the longer, which the classical proof calls impossible.

The historical inference assumes that a genuinely larger ordered collection cannot be equal in number to the collection formed by removing a member. On that assumption complete matching would erase the difference between the whole and the shortened whole.

Important qualification: Modern set theory allows a bijection between an infinite set and a proper subset. This step is therefore presented as a premise of the classical argument, not as a generally accepted result of modern mathematics.

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First alternative: equality of shorter and longer

The Originator of the World: Necessary Existence, Infinite Regress and the Correspondence Proof

فإن كان بإزاء كل واحد من الأولى واحد من الثانية، كان الناقص كالزائد وهو محال
If every member of the first has a counterpart in the second, the shorter would equal the longer, which the classical proof calls impossible.

The historical inference assumes that a genuinely larger ordered collection cannot be equal in number to the collection formed by removing a member. On that assumption complete matching would erase the difference between the whole and the shortened whole.

Important qualification: Modern set theory allows a bijection between an infinite set and a proper subset. This step is therefore presented as a premise of the classical argument, not as a generally accepted result of modern mathematics.