The concept of infinity has been evaluated many times throughout history. Only recently, in the nineteenth century, has major progress evolved in the field. The chapter “Beyond Infinity” answers the questions, “what is mathematics and why should I study it?” by reviewing several mathematician’s theories of infinity. First, the author mentioned Galileo who theorized that a line which measured 3 inches long contained the same amount of points as a line twice it’s length. The author also referred to Bernhard Bolzano, a mathematician who later on attempted to define infinity as well, but failed to do so. Archamedis was also referred to, for he developed a system for infinity called ‘myriad’. Using this, he was able to estimate the number of grains of sand there are on a beach. The chapter was focussed on matmatician named Cantor who became well known towards the end of the nineteenth century. Cantor believed that one set of numbers is equivalent to another set if they can be paired together. This was referred to as his “stepping-stone” process. He also used this to define numbers beyond infinity. Cantor assigned the first letter of the Hebrew alphabet, along with a subscript, to represent the number of elements in an “ordinary infinite set”. The first letter (Aleph) with zero for a subscript represented real numbers. Aleph with a numeral one stood for real and irrational numbers. Cantor recognized infinity as a verb, rather than a noun. This was uncommon, for it contradicted previous Platonic theories. Therefor, many mathematicians’ dismissed his theories at first. Also, he was criticized for not having an “absolute infinity”. Some mathematicians eventually started to accept Cantor’s theories. He then went on to prove that there are an infinite equal amount of fractions and whole numbers, and that the set of irrational numbers is larger than wh...
