The integers can be placed in a one-to-one correspondence... — Carmelics

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The integers can be placed in a one-to-one correspondence with the natural numbers.

Modality & Possibility

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1 reason for

2 reasons against

Reasons For

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Reason for

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1.The integers can be reordered as 0, -1, 1, -2, 2, -3, 3, … by alternating between positive and negative integers.
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2.This alternating ordering has the same order type as the natural numbers.
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3.Any ordering with the same order type as the natural numbers enables a one-to-one correspondence with the natural numbers.
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Reasons Against

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Reason against 1 of 2

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1.A one-to-one correspondence requires a definite rule mapping each element, but no finite rule can be actually applied to infinitely many integers.
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2.Potential infinity (as Aristotle distinguished from actual infinity) permits only endless progression, not completed totalities amenable to bijection.
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3.Therefore, the claimed correspondence presupposes a completed infinite totality, which strict finitists and intuitionists like Brouwer deny is mathematically legitimate.
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Reason against 2 of 2

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