To see how the sum '2 + 2 = 4' is true, it is necessary to create a formal logical proof. Before I get to that proof, though, it is important that I explain the ground rules the proof is based upon.
First, logicians and mathematicians use the symbol '∀' to indicate universal quantification. So the form '∀x' is read "all x," "any x," or "every x" where the variable x expresses something that is an element of a set.
Next, there is an important rule in logic and mathematics referred to as "universal elimination." Universal elimination, often denoted '∀E', is the logic that if all elements of a set have a particular property, then a single element of that set, an "instance," has that property. So, for example, since all prime numbers have the property that they are only evenly divisible by 1 and themselves, then 17, an instance of the prime numbers, is only evenly divisible by 1 and itself. Universal elimination is a rule that is often used in logical and mathematical proofs to arrive at a conclusion.
To form mathematical proofs involving operations on the nonnegative integers; 0, 1, 2, 3 and so forth; logicians and mathematicians use what is called the "successor function." The successor function is usually denoted with successive letter s's. The "successor" of any nonnegative integer z is the least of the integers greater than z, or z + 1. So the successor of 0 is 1 which is denoted 's0, the successor of 1 is 2 which is denoted denoted 'ss0' and so on.
Finally, there are two important successor-axioms we need to know to prove 2 + 2 = 4:
A1: ∀x(x + 0) = x
A2: ∀x∀y(x + sy) = s(x + y)
So the proof that 2 + 2 = 4 is the following:
It's important to understand that this proof is based on selected ground rules. In particular the successor function assumes that there is no upper limit to the nonnegative integers which is to say that all nonnegative integers have successors.
First, logicians and mathematicians use the symbol '∀' to indicate universal quantification. So the form '∀x' is read "all x," "any x," or "every x" where the variable x expresses something that is an element of a set.
Next, there is an important rule in logic and mathematics referred to as "universal elimination." Universal elimination, often denoted '∀E', is the logic that if all elements of a set have a particular property, then a single element of that set, an "instance," has that property. So, for example, since all prime numbers have the property that they are only evenly divisible by 1 and themselves, then 17, an instance of the prime numbers, is only evenly divisible by 1 and itself. Universal elimination is a rule that is often used in logical and mathematical proofs to arrive at a conclusion.
To form mathematical proofs involving operations on the nonnegative integers; 0, 1, 2, 3 and so forth; logicians and mathematicians use what is called the "successor function." The successor function is usually denoted with successive letter s's. The "successor" of any nonnegative integer z is the least of the integers greater than z, or z + 1. So the successor of 0 is 1 which is denoted 's0, the successor of 1 is 2 which is denoted denoted 'ss0' and so on.
Finally, there are two important successor-axioms we need to know to prove 2 + 2 = 4:
A1: ∀x(x + 0) = x
A2: ∀x∀y(x + sy) = s(x + y)
So the proof that 2 + 2 = 4 is the following:
- ∀x∀y(x + sy) = s(x + y) by A2
- ∀y(ss0 + sy) = s(ss0 + y) by ∀E on 1
- (ss0 + ss0) = s(ss0 + s0) by ∀E on 2
- (ss0 + s0) = s(ss0 + 0) by ∀E on 2
- (ss0 + ss0) = ss(ss0 + 0) by Transitive Equality on 3 and 4
- ∀x(x + 0) = x by A1
- (ss0 + 0) = ss0 by ∀E on 6
- (ss0 + ss0) = ssss0 by Transitive Equality on 5 and 7
It's important to understand that this proof is based on selected ground rules. In particular the successor function assumes that there is no upper limit to the nonnegative integers which is to say that all nonnegative integers have successors.
is true for all integers n≥1.Jul 7, 2021
