Can Incompleteness Complete Mathematics I: The theory and how Gödel proved it

 

Erdem Akder

This is the first part of the article about Gödel’s theory of incompleteness. This part will delve into the theory and the methodology that Gödel followed to demonstrate that mathematics is incomplete. The second part will concentrate on the implication of the theory and its relationship with epistemology and physics.

 

 

 

 

 

 

 

 

 

 

 

 

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The theory of incompleteness is a significant breakthrough for mathematics and the philosophy of mathematics. Kurt Gödel developed the theory in 1931. He came up with this theory during an era of conflict for mathematicians.

At the end of 1800, Georg Cantors' set theory caused a dichotomy among mathematicians. Some of them said “Mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true.”(4) Mathematicians who followed this ideology called themselves Intuitionists, and they took a side against Cantors’ set theory. However, some mathematicians pursued the idea that “all mathematics can be reduced to rules for manipulating formulas without any reference to the meanings of the formulas.”(5) This philosophy has been pursued by the Formalists who
supported Cantor and praised his set theory.

David Hilbert was the leader of Formalists and deeply believed that they could put mathematics on an axiomatic basis. He asked twenty-three famous mathematical problems in 1900; the second of those problems helped him to construct three new questions. He asked, is mathematics complete, consistent, decidable? This article will be mainly about the first problem
and its answer.

Kurt Gödel, at age 25, found an answer to the first question with his paper "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I" in 1931. He proved that mathematics is not complete. Before exploring how he proved it, we must understand what it means to be incomplete. Most basically, Gödel proved that some mathematical statements are not provable.

He proved incompleteness by inventing a new number system. He found Gödel numbers, in which each mathematical statement refers to a specific number, a Gödel number. In his book “Gödel’s Proof” Ernest Negel says elementary signs can be divided into two: constant signs and variables. There are exactly 12 constant signs which are mapped to integers 1-12. Those constant signs are for example signs like “+” referring to plus or “0” referring to zero. Here is the complete chart of the mapping of constant signs.

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For variables, he established a similar system of mapping. For integer variables, he mapped each of them to a prime number greater than 12, and for each sentential variable, he mapped them to the square of a prime number greater than 12. He does the same with a cube of primes greater than 12 for predicate variables. Now, let’s examine a statement to understand the system. The statement there is an x such that x is the immediate successor of y can be written as (ヨx) (x=sy). Then, he assigns a single number to this statement by raising a prime number to the power of the number that each sign maps in order. For this statement, our number is 2^8 × 3^4 × 5^13 × 7^9 × 11^8 × 13^13 × 17^5 × 19^7 × 23^17 × 27^9. Let’s call this number a; another statement (ヨx) (x=ss0) can also be written and we can call it b. Now, the combination of these two statements gives the gödel number 2a3b. This is how Gödel’s system works. However, there is one statement which drags it into the incompleteness theorem. The statement says “The formula that has Gödel number G is not demonstrable.” The gödel number this statement refers to is “G,” which is the most crucial part of the theory. We can understand this statement by comparing it to a similar one. Imagine a card that says, “The statement on the other side of this card is wrong,” and on the other side of the card, it says, “The statement on the other side of this card is true.” When the case is considered, for a while, readers can understand it is an obvious paradox. So, when we look at the Gödel number G, we see a statement that contradicts the number it has mapped if we consider it as a false statement, but according to Hilbert and Formalists, there cannot be inconsistencies in mathematics. So, we must accept that the statement is true. Now, there is a statement that is not demonstrable, and since the system maps these statements to a certain number, we now have an unprovable equation in mathematics. Which means, mathematics is incomplete.

In summary, Gödel developed a system capable of translating verbal and logical statements into mathematical equations, revealing the existence of mathematical statements that are undecidable within the system.

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Works Cited

1)https://www.youtube.com/watch?v=HeQX2HjkcNo&t=1244s&ab_channel=Veritasium
2)https://www.youtube.com/watch?v=O4ndIDcDSGc&t=669s&ab_channel=Numberphile
3)https://www.quantamagazine.org/how-godels-proof-works-20200714/
4)https://plato.stanford.edu/entries/intuitionism/
5)https://www.britannica.com/topic/formalism-philosophy-of-mathematics
6)Nagel, E., Newman, J. R., & Hofstadter, D. R. (2001, October 1). Gödel’s Proof. NYU Press.

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