Philosophy Stack Exchange is a question and answer site for those interested in the study of the fundamental nature of knowledge, reality, and existence. It only takes a minute to sign up.
Sign up to join this community
What is the minimum number of axioms you need, apart from definitions and usage of the notation, such that you have a system that does not contradict itself?
I would just think that the answer is simply 1, but then, what does Godel's incompleteness theorem say about it? Would there exist a statement that cannot be proved?
If we have an axiom system with a finite number of axioms, we can always reduce them to only one, replacing the set of original axioms with their conjunction.
Thus, every non-trivial axiom system that is finitely axiomatized can be formulated in an equivalent form with a single axiom.
Gödel's Incompleteness Theorems apply to systems that (in addition to otehr conditions) have a set of axioms that is finite or at least decidable; Robinson arithmetic, for example, is finitely axiomatized and it is enough for G's Theorem.
GIT only applies to axiom systems that can express the natural numbers. Given any such one-axiom system, it would be incomplete.
An example of a one-axiom system that's incomplete would be the conjunction of the Godel-Bernays axioms. http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
In other words, NGB can be finitely axiomatized so that you could then take the logical conjunction of each of the axioms to make a single statement that encapsulates all of the axioms. It would then be incomplete.
A really simple axiomatic system to consider:
Alphabet: |
Axiom: ||
Rule of inference: We can append | to the end of every statement.
From the only axiom, we infer |||.
From |||, we infer ||||.
From ||||, we infer |||||.
And so on.
Obviously no contradictions are possible, but we cannot derive |.
A little more complicated system with something like true and false statements:
Alphabet: ~ |
Axiom 1: |
Axiom 2: ~||
Rule of inference: You can append || to the end of each statement.
Note that we cannot derive both statement x (not beginning with '~') and statement ~x. So, no "contradictions." You cannot derive ~|. You also cannot derive ||.
So, I'm guessing, you will need at least one axiom and one rule of inference in any axiomatic system. If you want to model "true" and "false" statements, you will probably need at least two axioms.
