Today, I learned about Skolem’s paradox, which I find pretty interesting. Here is a rough overview:
- Georg Cantor demonstrated in 1874 that there are sets that are not countable. An example is the set of real numbers. Such sets are also said to be uncountably infinite.
- But mathematics can be represented as a countable language. Such a technique was used by Kurt Gödel to prove his famous incompleteness theorem.
- This leads to the Löwenheim-Skolem theorem, which essentially shows that you can enumerate the propositions in such a mathematical system. In other words, the propositions in the system are countable. For example, there must exist a countable set that obeys all the relationships defined on the uncountable set of real numbers.
This leads to Skolem’s paradox. You cannot count real numbers, but you can count the mathematical propositions that define them… The Wikipedia page indicates that some do not see that as a paradox because even if there is no bijection within the mathematical model, there may be a bijection outside the model. My “perception” of the paradox is that this means that our mathematical model cannot define all real numbers. There are real numbers that are not the subject of any theorem, that are not the limit of any “suite” (i.e. some expression written with a finite number of symbols, like “limit of 1/n”).
And this is only the beginning. We can keep building sets, such as surreal numbers, which are even larger than the set of real numbers.
For those interested, Paul Budnik created a video discussing these topics. I contacted him today regarding his Quantum Mechanics Measurements FAQ, since I honestly believe that I have answered several of these questions with my theory of incomplete measurements.