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If the set of numbers.remainingThat is your big mistake! Start to count, continue, continue, continue, .. . What you can determine that you can count. The set of not counted numbers remains infinite. But you cannot determine a first element. All your following waffle is worthless, because it violates this fundamental truth. Simply try it instead of "proving" counterfactual nonense.
does not hold a first element,
then the set of numbers.remaining
is the empty set.
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