Basic axiom confusion

Question

In mathematical logic,we do know that some statement is unprovable.Such as 0=1 in Peano Arithmetic system.

However,there is still one thing I want to know,let's concentrate our attention on Peano Arithmetic.

Can we prove the consistency of those basic axioms of Peano Arithmetic system(such as the commutative and associative of addition and multiplication)?Intuitively,I mean can we prove that those axioms are not violate to each other? Please include a reason briefly,I am just a beginner on logic.

Thanks!

Answer 1

We can prove the relative consistency of the PA axioms. Assuming a model of set theory, we can construct inside that model a model of PpA. That show that if the axioms of set theory are consistent, then so are the PA axioms.

Answer 2

I am not certain if you're asking how to prove PA is consistent, or how to prove that the basic axioms of PA are consistent.

There are at least three different proof methods to show that PA is consistent:

• The most straightforward is to work in set theory, such as ZFC, and prove that PA has a model. Because this set theory also proves the soundness of first-order logic, this means the set theory proves that PA cannot derive $0=1$.
• There are also syntactic proofs of the consistency of PA. These proofs directly show that there is no derivation of $0=1$ from the axioms of PA, without generating a model. The first syntactic consistency proof was due to Gentzen. It uses transfinite induction up to an ordinal known as $\epsilon_0$. Gentzen's consistency proof set the stage for what is now called "ordinal analysis" in proof theory.
• The second syntactic consistency proof is due to Gödel. It uses very different methods, including the so-called Dialectica interpretation. It is again syntactic: it does not produce a model of PA.

In every case, to prove the consistency of PA we have to use some axioms that are not included in PA. This is due to the second incompleteness theorem, which shows that PA cannot prove its own consistency. The second and third methods assume much less beyond PA, in a particular sense, than the first.

PA itself proves that the set of basic axioms of PA is consistent. In fact, PA has a stronger property: for any finite set $F$ of axioms of PA, PA proves that $F$ is consistent. So we do not need to move to particularly strong systems to prove that the basic axioms of PA are consistent.