If infinite cardinals are not completed totalities but potential infinities, the second premise—that any non-empty set of ordinals has a least element—cannot be straightforwardly applied.

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Completed totalities(metaphysics and infinity): Collections or wholes that are finished and fully determined, with a definite end point and total size.
Infinite cardinals(mathematical logic and set theory): Numbers that describe the size of infinite sets—basically, different ways of measuring 'how infinite' something is.
Least element(set theory and mathematics): The smallest or first item in a collection when things are arranged in order.
Potential infinities(philosophy of mathematics): Things that are infinite in the sense that they can keep going forever, but aren't 'finished'—like counting numbers that never stop increasing.
Premise: A premise is a statement or fact that you assume to be true as a starting point for reasoning or making an argument. Think of it as the foundation or building block you use to reach a conclusion—for example, "All dogs are animals" and "My pet is a dog" are premises that lead to the conclusion "My pet is an animal." Premises are essentially the evidence or claims you offer before drawing a final conclusion.

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