I didn't ignore what you wrote, I disagreed with it. It may be clear to you that potential infinity is the way to assess the volume of the towers but I think it

I didn't ignore what you wrote, I disagreed with it. It may be clear to you that potential infinity is the way to assess the volume of the towers but I think it is incorrect. I don't think that volume is the appropriate thing to use for the sort of size comparison you want to make. The argument you made is that we should consider one tower to have twice the volume of the other because we can fit it inside twice. I don't think this is a useful notion, since it is trivial to to fit one tower inside the other tower arbitrarily many times if you slice them up a bit. I think the sense in which one tower is twice as big as the other is captured by another quantity you mentioned already - the cross-sectional area. You don't need this potential vs actual infinity stuff at all. You just talk about whichever of the volume and cross-section is relevant to you. > That's completely wrong. You are operating under the assumption that potential infinity is a number. No type of infinity is a number, but I was indeed working under the assumption that it could be used as a cardinality since you told me you could understand volume with it, and I told you I understand volume as the cardinality of a set of unit-volumes filling the universe. If potential infinity can not be understood as a cardinality it seems entirely impossible to talk about it being the volume of the universe. > Consider the set described by the interval [0, 1]. Is it countable or uncountable? Neither, it is finite. We are talking about volumes here (used as short hand for lengths/areas/whatever is appropriate for the dimension), not cardinalities. Its volume is 1. If the underlying space is continuous then its cardinality is going to be uncountable, if the space is discrete the cardinality will be finite (not countable but finite), but either way the volume (defined by the appropriate measure) will be 1. > Otherwise the tower would have both countable and uncountable volume at the same time Nope, by fairly standard arguments your tower can be partitioned up into an countable number of unit cubes, and it emphatically can't be partitioned up into an uncountable number of unit cubes. You can use essentially the same argument that says one can't partition the real line up into uncountably many unit intervals.