Mathematical True Statements that are Impossible to Prove
EducationMathematical True Statements that are Impossible to Prove
In almost all subjects especially mathematics, there are many things that cannot be proved but can be assumed. The subject of mathematics contains anomalies. It seems nothing is provable but can be proved only by making assumptions which are more or less unprovable themselves.
Like for example in the case of the addition of 1 plus 1 which equals to 2. This particular statement is not provablenor can be disproved. However, it can be made true or proved that it is what it isuntil and unless one assumesit to be true and one will not be able to do the required calculations related to mathematics at all.
Another example is the existence of the root of minus 1 i.e., i. Now we know that +1 has a root that is 1.Theoretically,there should also be a root of i or -1. But so far, this can be just an assumption that cannot be proved and never will.
The generation of Random Number. Several programs and machines’ softwares have claimed that they can generate random numbers. However, these generally have a sequence that keeps on repeating after a point in time, that actually proves not to be truly random. In reality, it is not possible to prove neither disapprove that a particular program or a machine can perform the generation of random numbers indefinitely.
In Mathematics point of view, the greatest Prime number is the infinitya value or term commonly used in mathematical expression or as a number. This is also one of the mathematical statementsthat can never be proved or disproved.
Some researchers and mathematicians highlighted that there are three points in the formalization of mathematics. These are:
- In Mathematics a large number of true statements exist and that they cannot be proved.
- Also, their consistency within the systemduring the formalization cannot be proved.
- Another point to be noted is that there is no particular problem in deciding or checking whether a given statement has a proof or not. Hence, it remains undecidable.
What does it exactly mean when one says formalization of mathematics?
This statement, “the formalization of mathematics”, means a formal language is used for the description of certain statements for framing and writing mathematical statements or in other words a definition that is quite precise making it easily understandable from a mathematical view point that the given statement is said to be true. A formal language is sound andconsistent if the true statement has a proof that is valid.
These three points are actually proven by various people. The first and second points have been proven by Kurt Friedrich Gödel in the year 1931 whereas the third point has been proven by two others i.e., Church and Turing in 1936. The result of these two that is the Church-Turing resultgives a different path in provingKurt Friedrich Gödel’s results, and hence there were presented for proofs instead of the original ones of Gödel’s.
Let us take a look at Kurt Friedrich Gödel’s theorems:
Gödel’s First Theorem:
There exist true propositions that have no proof in a consistent formalization of mathematics that satisfy assumptions (1) and (2).
Proof:
Assume that there is a formal system that meets assumptions (1) and (2) and has a proof for all true statements, leading to a contradiction. We can observe that this implies that the Acceptance problem can be solved, which isn’t true. Let (‹M›, w) be the Acceptance problem’s input. We create the statements S‹M›, w, which is true only if M accepts w, and (not S‹M›, w), which is true only if M rejects w. Then, in lexicographic order, we enumerate all strings P and check if P is a proof that (notS‹M›,w) is true and if P is a proof that (notS‹M›,w) is true for each of them.
Gödel’s Second Theorem:
The consistency of any formalization of mathematics that satisfies the assumptions (1), (2), and (3) cannot be proven.
Proof:
Consider the algorithm shown below:
- Input: A Turning Machine M with a description ‹M›
- Make the sentence S true only if and only if M accepts the input ‹M›
- In Lexicographic order, for every string P,
- Stop and reject if P is a proof that S is true.
- Stop and reject if P is a proof that S is true.
Learn from more examples: Mathematical Induction from Class 11 Maths
A Turing machine, MG, implements this algorithm. Consider MG’s behavior on input ‹MG›: if it accepts, there’s a significance that MG doesn’t accept on input ‹MG›, which is inconsistent. If it refuses, we observe that MG accepts on input ‹MG›, which contradicts consistency once again. But what we just presented is a proof that MG enters an infinite loop, and in particular, a proof that ¬S is true, which can be formalized in our system. However, if such a proof exists, MG must come to a halt, and we have a contradiction.