Let me start with stating my definitions in measuring the length of objects:
The way to measure an object properly is to measure it when it's static. So take the ultra resolution (which can describe the exact form) picture of the object. Now, we want to notate the length with some objective value. Fortunately, there is another object snapped which is shorter than the sample object. Now we can compare this short object with the sample and say n pieces of short objects can maximally fill the sample, and we can say that this is the length of the sample. In general,
Notice that u is a unit and the length coefficient can only be a natural number. For further generalization, there is a shorter unit so we can get more accurate length.
This is the concept what I'm thinking about length in real life (this demonstrates the difference as compared to mathematically defined concept). I strongly support this way and from now on I'll interpret the following assumptions with this view.
Assumption 1. The unit length of the space is 0.
Objection of 1. From the definition 1, let n≠k and L1=n[u(i=∞)]≠L2=k[u(i=∞)]. Since [u(i=∞)]=0, L1=L2 which leads to contradiction. For intuitive approach, the assumption implies that the unit does not local the space, nothing can be measured or comparable and is the contradiction that the objects have their independent localized space(length) and the distance from others exist.
Assumption 2. The unit length of the space is non-zero.
Interpreting of 2. From the definition 1, we want to find the smallest(or shortest) unit. If [u(i→∞)] where number of n is getting bigger which intuitively implies that the smallest unit is getting smaller with respect to time proceeds(take the picture when t=0 and t=1 or arbitrary non zero point), and this situation can be equivalent with mathematically defined continuous curve (or space). And [u(i=k)] where k is some number which implies that the space is statically quantized. And the last interpretation, when [u(i=f(t))] which means that the smallest unit changes with respect to time and also quantized.
I have thought hard about these final circumstances but I can't go any further. The three circumstances can stand independently in my opinion and this can't happen in reality. Thus my knowledge is not good enough to determine this problem. Any clue for determining the particular circumstance?