The concept of numbers is way more complicated than it sounds. In this article, we will provide an introduction to how they are mathematically constructed from the most fundamental level in set theory. This should give you an insight into what they truly mean and are. Beware that, as a result, this article is relatively technical.
Natural Numbers
Natural numbers appear to us as the simplest numbers. They seem to be the fundamental blocks of mathematics. But they are not. Nearly all mathematical theories are rather based on set theory. In these theories, all mathematical objects are sets. The way they interact is then defined by the axioms of the theory. In this context, natural numbers exist only if these axioms allow the construction of sets which perfectly match what we would expect from natural numbers.
Well, it shouldn’t be too hard to find such axioms…
Well, in fact, it’s a pretty tough task to find such axioms which don’t imply any contradiction… which is essential! In 1901, Bertrand Russell proved the necessity of doing this carefully, as he showed that the axioms of the time, now known as naive set theory, led to the Russell’s paradox. This led to the foundational crisis of mathematics: During the following decades, mathematicians were lost, as no appropriate set of axioms were found.
What about now? Have they found the appropriate set of axioms?
Well, they found a set of axioms that seems appropriate, called the Zermelo–Fraenkel set theory with the axiom of choice, or ZFC. ZFC comes from Zermelo’s first axioms introduced in 1908 and was then slowly improved until the 1920s.
What do you mean by “seems appropriate”? We don’t even know???
Hehe… Let’s not get too much into this because it would take a while! You can learn more on the axiomatization of mathematics in my article on self-reference. Let’s now just stick with ZFC. Now, the question is: How can we define sets which correspond to natural numbers? And we need a preliminary result…
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου