To Infinity and Beyond
Take a look at the picture below:
A train is traveling and has to turn on one of two roads. Both have people laying down, just spaced out differently. The two different rail tracks go on forever. With way would you pull the level and have the train travel? You would probably choose the track with people spaced out further. Even though both tracks continue forever, and theoretically, infinite people would die, the spaced-out one seems to have a lesser infinity.
Imagine the counting numbers: 1, 2, 3, 4, 5, etc.
There are infinite counting numbers as you can always add one to the previous value. Now imagine the numbers between 0 and 1. Again, there are infinite values. In this case, the larger the set, the closer to 0 the numbers start at. So how do we measure the size of sets?
Georg Cantor established his method of comparing sets while also developing set theory, a mathematical branch. Cantor used a one-to-one rule when looking at infinite sets. Let’s say you have the set [1, 2, 3] and the set [x, y, z]. In this case, each element in each set has a corresponding match in the other set: 1->x, 2->y, and 3->z. Using Cantor’s rule, these sets would be the same size. Going back to the train problem, in one small section of the track, there would be more people than in another. This means that each element does not match equally to the other set; therefore, the track with more people would be a larger infinity. But what about different size sets? Cantor proved that the set from [0,1] is larger than the set of all Natural (or counting) numbers. Cantor also stated that there are infinite possible infinities. Specifically, Cantor was referring to Mathematical infinities: lines and number sets.
With their being mathematical infinities, that suggests that there are also other classifications. Along with mathematical, there are physical and metaphysical infinities. These don’t have to do with number sets but can also be quantified to some degree. Physical deals with infinities in time and dimensions. Metaphysical takes a more theological approach dealing with thought and gods. Cantor separated these three types of infinity making quantifying and studying them easier. Cantor also stated that these mathematical infinities actually exist, not theoretical.
With these different measures of infinity, set theorists and quantify sets and compare them to each other. So next time you think of this training problem, remember, that even though both tracks are infinite, one does in fact have fewer casualties than another.
Sources:
https://en.wikipedia.org/wiki/Georg_Cantor
https://www.britannica.com/science/infinity-mathematics
https://www.businessinsider.com