Is it possible to reveal all the mysteries in Mathematics? How many mathematics theorems are there? It seems that the research and discoveries in the field of mathematics are infinite: new theories are emerging and, at the same time, some old famous hypothesis is still challenging the intelligence of the human kind.
However, originally published in 1931, Gödel’s incompleteness theorem set a logic limit on the scope of mathematics. 80 years have passed; the two Gödel’s incompleteness theorems still stand among the most important theorems in today’s mathematical logic. The incompleteness theorem states that
Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete.
To put in simple words, a mathematics system allows an existence of some statements that can “neither be proved nor disproved,” or in other words, unprovable.
To illustrate the philosophy behind Gödel’s incompleteness theorem, we need to shed light on how the mathematical logic works. Similar to any other fields in science, the operation of mathematics is based on some premises, which are some fundamental axioms in mathematics. For instance, A+B=B+A is one of the axioms in arithmetic calculations.
From these axioms, we can prove more statements and add these statements to our axiom sets, and based on the new axioms, we will be able to generate more axioms, making the whole process an infinite loop. We can model it as a logic tree (as shown below); as its branches expand, the axioms get more and more complex.
Since the age of Greek, mathematicians believed that every true statement can be proved, or in other words, the basic axioms can expand forever and cover all other axioms in the mathematical system. The only difference is that some statements may take significantly longer to prove than others. However, in Gödel’s paper, it is demonstrated that this axiom tree can’t reach to all valid mathematics statements, which means some true statements can’t be proved. To illustrate, let’s examine the following statement:
“This statement can’t be proved by existing axioms.”
Assume this statement is false, then the correct statement would be
“This statement can be proved by existing axioms.”
However, as the statement can be proved, then it must be true. Here we get a contradiction, where it’s both false and true, and since we believe mathematics is consistent (it doesn’t allow the existence of contradictions), we must rebut the original assumption. Thus the statement can’t be false; it must be true. Therefore, we proved that this statement is true and can’t be proved by existing axioms. Since the statement is true, mathematicians would add it to the existing set of axioms to generate more axioms.
Gödel’s idea had a great impact on mathematics. It challenges people’s traditional view on theorems and hypothesis: it may be possible for some statements, such as Goldbach conjecture, Riemann hypothesis, etc., to be true but can never be proved. Today, further research and studies on Gödel’s incompleteness theorem are still carried on by modern mathematicians. For a more detailed introduction and application of Gödel’s incompleteness theorem, read Godel’s Proof by Ernest Nagel and James R. Newman.
Panu Raatikainen (2015) “Neo-logicism and its logic”, History and Philosophy of Logic.