I've been reading Tao's Analysis 1 recently and I've learnt about the Peano Axioms. I believe I have a good understanding of the axioms and I have been able to complete the problems in the chapter. However, I am aware that Tao is adopting a rather unconventional (or non-standard) approach to the topics in the book. Therefore, I always like looking up additional info online to complement the concepts I learn in his book.
My problem concerns the difference between first and second order logic. In Tao's book, the axiom of induction is described as an 'axiom schema', which is a template for creating an infinite number of axioms for every property P(n). I'm perfectly fine with this concept, but then I looked up the Peano Axioms on Wikipedia, and the article there mentions a distinction between using first and second order logic to frame the axiom of induction.
The article seems to state that the axiom is in second-order logic when it
"quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers)"
, but is in first-order logic when treated as an axiom schema. I understand the phrase 'quantifies over predicates' as being something like $\forall P$ (for all predicates). What I don't understand is 1. how is this different from an axiom schema (having an axiom for every P(n)), and also 2. why does the Wikipedia article say this is equivalent to 'sets of natural numbers'?
(Sorry if this question might seem obvious for experts out there. I'm a physics student with a mathematical bent and I'm still quite new to an axiomatic treatment of maths at such a foundational level. I've only taken an intro math course on logic so I understand things like conjunction, implication, truth-tables, for all, there exists and all that but I don't really understand the difference between first and second order logic. Thank you very much in advance for taking the time to read through my question.)