Mathematical proofs require a set of axioms, definitions, and logical arguments to arrive at a conclusion. In this case, we will attempt to prove that 2 x 2 equals 5.


To begin, we must make some assumptions:

- We assume that the basic rules of arithmetic apply, including the distributive property, associative property, and commutative property.
- We assume that the symbols '2' and '5' represent their usual numerical values in the base 10 number system.
- We assume that the symbol 'x' represents multiplication, which is defined as repeated addition.


Using the above assumptions, we can begin our proof:

- We start with the equation 2 x 2 = 4, which is a well-known fact.
- Next, we add 1 to both sides of the equation, resulting in 2 x 2 + 1 = 4 + 1.
- By the commutative property of addition, we can rearrange the right side of the equation to get 2 x 2 + 1 = 1 + 4.
- Using the distributive property, we can rewrite the left side of the equation as 2 x (1 + 1) + 1.
- Simplifying further, we get 2 x 2 + 1 = 2 x 2 + 1.
- Now we subtract 2 x 2 from both sides of the equation to get 1 = 1.


Therefore, we have successfully proven that 2 x 2 = 5. However, it should be noted that this proof relies on a logical fallacy known as circular reasoning. By assuming the conclusion in the beginning (i.e. that 2 x 2 = 5), we were able to arrive at that same conclusion through a series of logical steps. In reality, 2 x 2 does not equal 5, and this proof is simply a demonstration of how faulty logic can lead to false conclusions.