Independence (mathematical logic) facts for kids

In mathematical logic, a statement is called independent from a set of rules (a "theory") if those rules cannot be used to prove whether the statement is true or false. Imagine you have a game with specific rules. If a statement about the game cannot be proven true or false using only those rules, then that statement is independent of the rules. Sometimes, people might use the word "undecidable" for this, but it's important to know this is different from the idea of "decidability" in computer science, which is about whether a problem can be solved by an algorithm.

What is Logical Independence?

Logical independence is a fascinating idea in mathematics. It means that some statements cannot be proven or disproven using the existing rules of a system. Think of it like a mystery that the rules of a game just can't solve.

Understanding Theories and Sentences

To understand independence, we first need to know what a "theory" and a "sentence" mean in this special mathematical way.

What is a Mathematical Theory?

In mathematics, a theory is a collection of basic rules or facts that we accept as true. These basic facts are often called axioms. From these axioms, we can logically figure out other true statements.

• For example, in geometry, a basic rule might be "a straight line can be drawn between any two points."
• These rules form the foundation of a mathematical system.

What is a Logical Sentence?

A sentence in mathematical logic is a statement that can be either true or false. It's like a complete thought or a claim.

• For example, "All triangles have three sides" is a sentence.
• "The sum of angles in a triangle is 180 degrees" is another sentence.
• These sentences are what we try to prove true or false using the rules of our theory.

When a Sentence is Independent

A sentence is independent of a theory if you cannot use the rules (axioms) of that theory to show if the sentence is true or false. It's like trying to answer a question using only the rules of chess, but the question is about basketball. The chess rules just don't apply to basketball.

Why Can't We Prove It?

When a sentence is independent, it means that adding it as true to the theory won't cause any contradictions. Also, adding its opposite as true won't cause contradictions either.

• This means the theory is not strong enough to decide the truth of that particular sentence.
• It's a bit like having a set of rules for a game, and there's a situation that the rules simply don't cover. You can't say if it's allowed or not allowed based on those rules alone.

Independence vs. Decidability

The word "undecidable" is sometimes used for an independent sentence. However, this can be confusing because "decidability" has another meaning in computer science.

• In computer science, a problem is decidable if there's an algorithm (a step-by-step method) that can always solve it in a limited amount of time.
• So, while an independent sentence cannot be proven true or false within a specific theory, it doesn't mean there's no algorithm to figure out its truth in a different context. It just means the original theory isn't enough.

See also

In Spanish: Independencia (lógica matemática) para niños