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If you give a little thought to it, it is apparent that an object that has zero length in the direction that it travels can only move in that direction in discrete jumps. That is to say, it cannot smoothly pass through every possible point between two positions but must do so in discrete steps.
Invoking the idea of making infinitely small movements in infinitely small periods of time doesn't solve this because no matter how small the step is it will still be greater than 0. I guess this can be shown mathematically by the statement: 1/infinity > 0
Actually, thinking about it even further, numbers themselves are also subject to this zero thickness boundary issue. While we can say that the number of values between say 0 and 1 is infinite, each value has zero 'thickness' and therefore every possible value requires a discrete separation from any other number.
Now obviously, any movement of the axis in that direction requires a displacement greater than zero, otherwise it'll still be in the same place and won't have moved at all, but once it has moved then the distance that it has travelled, however small that might be, will still be greater than the 'thickness' of the axis with the result that there will be a 'gap' between the two positions - you can't have two individual parallel axis actually touching each other without being them being in exactly the same place - they need to be separated and even if the separation is infinitely small, it must still be greater than zero.
I must say that I don't see this as a maths problem but as a logic problem. Maths is great but it only works within rules that make it self consistant. Dealing with infinity is an area where, imho, it doesn't work very well, producing answers that solve but which may mean nothing, for example ∞ + 1 = ∞, 1 * ∞ = ∞, ∞ * ∞ = ∞ etc. At the same time, maths might allow you to work with sizes smaller than the planck length but would the answers actually mean anything?
The problem is that you're assuming that your object is infinitely thin, but that your steps in space can't be. You can't have it both ways.