Number of Distinct Axiomatic Systems

Question

This may well be a vaguely formulated question. Please bear with me and help me modify it to make it meaningful and rigorous, or show that it is hopelessly meaningless.

I understand an axiomatic system is a set of consistent statements. Suppose the number of such statements in each system is finite. Is the number of distinct systems is (countably) infinite?

My conjecture is it is true. I suppose to show it, I assume the contrary, and construct a new system that is distinct from all the prior systems. But I can not see how to do this.

Answer 1

Well, there's a first-order theory where every model has exactly one element, and a first-order theory where every model has exactly two elements, and so on. There are infinitely many of these first-order theories, none of them equivalent to each other.

These first-order theories can be constructed explicitly by an algorithm. Here's an example of a first-order theory where every model has exactly four elements:

• $a \ne b$.
• $a \ne c$.
• $a \ne d$.
• $b \ne c$.
• $b \ne d$.
• $c \ne d$.
• For all $e$, $e = a$ or $e = b$ or $e = c$ or $e = d$.

This pattern can be continued to get first-order theories where every model has exactly $n$ elements, for any chosen constant $n$. Simply have axioms asserting that no two of $n$ constants are equal to each other, and one final axiom asserting that everything is equal to one of these $n$ constants.

Conversely, any axiomatic system with only finitely many axioms can be written as a string of symbols from a finite alphabet, meaning it can be encoded as an integer. So there must be only countably many such axiomatic systems, up to choice of symbols.