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Motivation: What does not have to be proved in mathematics?

(related to Part: Methods of Mathematical Proving)

Mathematical proofs are the strength of mathematics. In the following statements, however, do not have to be proven:

Mathematical Axioms

Axioms are small, well-understood set of sentences, which are used to axiomatize a given mathematical branch, i.e. to show that its claims can be derived from these axioms. Typically, there are two types of axioms:

1. Logical axioms Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g. the a straight line is determined by two distinct points),

2. Non-logical axioms Non-logical axioms are actually defining properties for the domain of a specific mathematical theory (e.g. the commutativity law).

Modern mathematics admits multiple, equally "true" systems of logic, based on distinct systems of axioms. The system of axioms used in BookOfProofs (work in progress) can be found in the Building Block Index.

Vacuously True Statements

A proposition: \(S\) is "vacuously true", if it resembles statements, which can be reduced to the following basic forms:

1. \(\forall x: P(x) \Rightarrow Q(x)\), where it is the case that \(\forall x: \neg P(x)\).
   As examples, consider the following statements (all being vacuously true):

   • "If the Sun is not a star, then it doesn't shine."
   • "If the Earth is a disk, then our ship going to the west is lost!"
   • "If \(2\times 2 \neq 4\) then \(2\times 2 +1\neq 5\)."

2. \(\forall x \in \emptyset: Q(x)\), where the set \(\emptyset\) is empty.
   As examples, consider the following statements (all being vacuously true):

   • "All cell phones in an empty box are turned off."
   • "All cell phones in an empty box are turned on."
   • "Since all elements \(\alpha\in\emptyset\) are transitive sets, so \(\emptyset\) is an ordinal number."
   • "There is only one way to permute \(0\) objects. So, \(0!=1\)."

3. \(\forall \xi: Q(\xi)\), where the symbol \(\xi\) is restricted to a type that has no representatives.
   As examples, consider the following statements (all being vacuously true):

   • "Our services cover all adult persons under the age of 18."
   • "Let \(X\) be the set of integers \(x\) such that \(x^2 < 0\). All such \(x\) are even, i.e. divisible by \(2\)."

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Proofs: 1 2 3 4

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References

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1. Multiple Authors: Wikipedia