From my understanding, in calculus, the concept of infinity mostly commonly relates to potential infinity; the idea that a sequence or series of numbers can be extended indefinitely, but there is no actual limit or endpoint to that sequence. The concept of limits allows us to understand how a function behaves as its input approaches a certain value, even if we can’t get to that value exactly.
For example, the Bolzano-Weierstrass theorem states that every bounded sequence of real numbers has at least one converging subsequence, which means that for a sequence of numbers, if there is a limit, that limit must be a real number. This theorem relies on the idea of potential infinity, as it states that the sequence of numbers can be extended indefinitely, but there is a limit that can be approached but never reached.
Zeno’s paradox is contradictory to the motion and time. It assumes that the distance can be divided into an infinite number of parts (it is potential infinite but not actual infinite), but the infinite series of halfway points does not reflect the physical reality of motion. In reality, motion is continuous, and it does not involve an infinite number of halfway points. One way to resolve this paradox is to recognize that his idea is based on an assumption that the sum of an infinite series of numbers can never be finite. This is not true, as some infinite series can be added up to a finite value (according to Bolzano-Weierstrass theorem).
Thoughts about dividing something by 0: Division is the inverse operation of multiplication, when we divide by 0, we are trying to find a number that, when multiplied by 0, will give us the original value, but since any number multiplied by 0 is always 0, there is no real number that can be used for that, and the result is an infinitely large number represented by infinity.
The second reading for this week is also quite interesting. I have never thought about the concept of 0 because I took it for granted. I would like to quote from the reading “Zero is not a ‘natural’ candidate for acceptance as a number unlike other numerals. It requires a great leap of thought from the concrete to the abstract.
