# Gödel's incompleteness wrt weakend versions of ZFC

1. May 26, 2013

### phoenixthoth

Suppose for the sake of argument that we look at ZFC with the axiom of infinity removed.

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#First_incompleteness_theorem

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#Second_incompleteness_theorem

We would then be in a position where the hypotheses of Gödel's theorems are not satisfied, correct? Basically, I want to remove, for the sake of argument, a minimal amount of axioms of ZFC so that ZFC minus some axiom(s) leads to a theory that does not include arithmetical truths and is not capable of expressing elementary arithmetic.

Is it possible that ZFC- (my shorthand for ZFC with some axiom(s) removed) is consistent and complete?

2. May 26, 2013

### MathematicalPhysicist

Long time no see, I am just being nostalgic...

3. May 30, 2013

### Preno

No, obviously any subtheory of an incomplete theory is also incomplete.

Also, ZFC without the axiom of infinity is basically Peano Arithmetic (or rather, replacing the Axiom of infinity by its negation yields a theory which is bi-interpretable with PA).