Is Peano arithmetic consistent in predicate logic? If so, what are the implications? Peano arithmetic can't prove its own consistency, as proven by Gödel's incompleteness theorem. Many seem to think that this means that it is possible that Peano axioms are inconsistent.
However, if the Peano Axioms are formulated in predicate logic, it seems as though they are satisfiable. From that it follows that they are consistent. Thus it's impossible to ever prove anything contradictory from the axioms in predicate logic. (Is this correct?)
So since that is so, should we not accept that the axioms in fact are consistent? It seems to me that no matter what is proven/provable from the axioms, the result/thorem must still be consistent. How should one think about this distinction?
 A: In order to say that a collection of axioms is satisfiable, you must provide a model satisfying them. It certainly appears that standard arithmetic satisfies the Peano axioms, but can you prove it? Can you even prove that it makes sense to talk about "standard arithmetic"? It's not even obvious that infinite sets exist, without an axiom saying they do! Can you show that there isn't a largest integer, without assuming something equivalent to the Peano axioms?
The thing is, if you can provide a finite model satisfying a set of axioms, you can check the axioms directly and thus prove that they're satisfied. For example, to show that the set of sentences $\{a < b, b < c\}$ is consistent, all we have to do is provide the example of three objects $a$, $b$, and $c$, and a relation stating $a < b$ and $b < c$. But Peano Arithmetic has no finite models, so we can't provide anything so directly inspected. In principle, it could be that we've completely misunderstood how the natural numbers work, and that Peano's axioms don't actually hold of them.
Now, most mathematicians take it as assumed that $PA$ is indeed consistent, because it's inconvenient to assume otherwise, but it is definitely possible that a contradiction is provable from $PA$.
As to the distinction "in predicate logic": the Incompleteness Theorem is about predicate logic. Moving to predicate logic doesn't free you from Incompleteness, it puts you right in the crosshairs.
