| A History of Numbers
| Propositional Logic
| Logical Completeness
| The Liar's Paradox
| Basic Methods of Mathematical Proof
| Integers and Natural Numbers
| Irrational Numbers
| Imaginary Numbers
| The Euler Equation
Everything we do depends on the consistency of maths. Without consistency, we couldn't be sure that 1 + 1 always equalled 2. We would worry that the calculations used to create the plane we flew in suddenly no longer applied, or that the number 5 had decided to change its meaning from one day to the next.
Logically speaking, consistency represents the notion that a set of logical statements is free from any inconsistency. (See the entry on Propositional Logic for more information on logical sentences.)
The laws of maths have to be consistent if our world is not to fall apart one sunny afternoon. Luckily, they are.
Using Models to Define Consistency
How do we define consistency? First, we need to examine models.
A 'model' of a set of sentences is an assignment of the values true and false to the individual propositions of the logic, such that each sentence in the set becomes true. Think about that for a minute or two before reading this example.
Let's assume that our set of sentences contains the following:
- 'Bert is a gunslinger'
- 'Bob is a gunslinger or Louise is a pugilist'
- 'Louise is a pugilist and Leslie is a bartender'
We can then assign truth values to the propositions. One assignment may be:
- 'Bert is a gunslinger' is true
- 'Bob is a gunslinger' is false
- 'Louise is a pugilist' is true
- 'Leslie is a bartender' is false
Now with this assignment1, the statement 'Louise is a pugilist and Leslie is a bartender' from the first list above becomes false, and so this assignment is not a model of the sentences. However, if we have the same assignment as in the second list, but change 'Leslie is a bartender' to true, then we do have a model of the sentences.
Why Use Models?
Why would we be interested in models? It's because they give us an easy way to define consistency. If a set of sentences has a model, then it is said to be 'consistent'; otherwise it is 'inconsistent'.
Consider a set of statements that contains the sentences:
- 'Bert is a gunslinger'
- 'Bert is not a gunslinger'
However we assign truth or falsehood to the proposition 'Bert is a gunslinger', one of the two sentences will be false, and hence the set is inconsistent.
A 'tautology' is a particularly interesting type of sentence, in that it is always true independent of the assignment of truth values to propositions. In other words, every assignment is a model of a tautology. An example would be 'Either Bert is a gunslinger or Bert is not a gunslinger'. Any assignments of truth to 'Bert is a gunslinger' will make the sentence true. Note that any set of tautologies will be consistent, as they can all be assigned to being true.
Why Maths is Consistent
Here we prove the statement 'all sentences are semantic consequences of an inconsistent set'. We'll then use this result to show that maths is consistent.
Before we prove the statement, we need to know what a semantic consequence is. A semantic consequence of a set of sentences is a sentence which evaluates to true in every model of the set. In particular of course, each member of the set is a semantic consequence of the set.
In short, a semantic consequence of a set is a sentence which is always true, given that the sentences in the set are true.
However, seeing as an inconsistent set of sentences has no models, then trivially any sentence is true in all models of the inconsistent set (that is, all of none). Therefore all sentences are semantic consequences of an inconsistent set.
Therefore, because we know not all sentences are true in maths ('1+1=3' is an example), maths can't be inconsistent.
Semantic Consistency of Propositional Logic
Consistency can also used in the context of whole logical systems. Completeness is defined as the property that every problem has a solution. In terms of propositional logic, semantic completeness shows that every tautology has a proof. Consistency is the other side of this coin, and shows that everything that can be proved is always true (a tautology).
Unfortunately, these pleasant properties of propositional logic do not extend to more complex logical systems, but nevertheless, weaker versions of completeness and consistency are often found associated with these more complex logics.