# System with infinite number of axioms

Assume we have a set of axioms $A_0$. There exists a statement that can be formulated with these axioms that cannot be proven to be true with this system. Assume we give such a statement axiomatic status and add it to $A_0$ and construct a new set of axioms $A_1$. Continue this process indefinitely. Assume we end up with a set of axioms $A_{\infty}$, that contains infinite number of axioms. Is this set effectively generated? Is this system complete? Is it consistent?

I don't know anything about these things, apart from what can be read in Wikipedia about Gödel's incompleteness theorems. Just a thought I had.

-
If you are generating the $A_i$ by a process, then you can't get completeness - there are statements that are unprovable that can't be resolved by that process. If the $A_i$ are not enumerated by a process, then they are not an effective axiom system. –  Thomas Andrews Feb 20 '13 at 15:06

Second, if your original theory $A_0$ is effectively generated, then your $A_\infty$ will be too -- there's a mechanical, deterministic process that will eventually print every axiom in $A_\infty$.
$A_\infty$ will be consistent too, if $A_0$ is -- by the "compactness" property of formal logic which says that every inconsistent system of axioms has a finite subsystem that's also inconsistent. (If there's a proof of a contradiction in the system, then because proofs are finite by definition, the proof depends on only finitely many of the axioms). Every finite subset of $A_\infty$ will be a subset of one of the $A_n$'s, and all of these are consistent by construction.
Since $A_\infty$ is effectively generated and (because it extends $A_0$) sufficiently rich, we can repeat the Gödel process on it, and get at Gödel sentence for $A_\infty$. Add that to $A_\infty$ to get $A_{\infty+1}$, and proceed ad nauseam. (In this context it is traditional to write $\omega$ instead of $\infty$, and there's a theory of the necessary numbers "beyond infinity" under the name "ordinal numbers").