There is a story of a rabbit and the turtle. In this story the rabbit gave the turtle a head start of around a hundred meters. After the turtle passed the hundred the rabbit started to sprint to the finish line. He covered half the distance between the turtle and himself, which means that there is now a distance gap of 50 meters between the turtle and the rabbit. He continues to sprint and covers again half the distance. This mean that there is a distance of 25 meters between them. He continues to move forward but for some reason he can not reach the turtle because there seems to be infinitely many space between them.
Of coarse we know that the story is not true, if we sprint then we can reach the turtle and overcome so called endless distance. But the real question is why?
This sounds counter intuitive but it explains why we could cross seemingly endless spaces. Imagine infinity not as one set, but infinity as separate sets with a quantitative "size" property. Imagine infinities as fruit. We get grapes apples and watermelons. All of the fruit have sets of infinite numbers, but yet they are different sizes, the watermelon is a bigger infinity than the grape.
Here is a set that we can compare to a apple. let the set of sums be: 1+ (1/2) + (1/4) + (1/8)....
This set adds half of the previous term. This pattern continues for ever and ever. But we know that that answer is 2.
We can compare the grape to 1 + (1/3) + (1/9) + (1/12)+....
we know if this continues for ever then the answer will be 3/2 or 1.5.
It this smaller than the first set that continues for ever.
Lets think about clapping hands each time you walk a certain distance (You know to put the thought in another light).
lets say I walk 1 meter and clap my hands. Then I walk half the distance and then clap my hands. If I where to continue this pattern infinitely times then I would have move forward for what would have seemed forever and clapped my hands for ever and yet I only moved 2 meters.
In calculus we use integration to ind the area underneath function. The basic idea of integration is finding small rectangles that fit in the area under the curve, to approximate the the surface area. e do something like this:
This image is my own creation
But if we want to find the total , perfect area then we need to create infinitely amounts of rectangles that fit a finite surface area. We can also conclude that bigger surface areas is a type of bigger infinity.
In conclusion. Infinities is more complex than just something that continues on and on forever! This concept is still too big for our minds to truly comprehend, but luckily we can make it a bit easier for our brains to understand through logical reasoning.
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