I've recently taken an interest in arithmetic and how it can be axiomatized. I haven't been reading very deeply about some of the known axioms I've come across, like Peano axioms, which by my interpretation seems to include axioms for only two operations, addition and multiplication, concerning only natural numbers.
Second-order arithmetic:
From what I read real numbers can be formalized in second-order arithmetic, but I have a hard time understanding how this is done since the axioms about operations mentioned here are only for addition and multiplication as well?
From wikipedia:
- The formal theory of second-order arithmetic (in the language of second-order arithmetic) consists of: the basic axioms, the comprehension axiom for every formula φ (arithmetic or otherwise), and the second-order induction axiom. This theory is sometimes called full second-order arithmetic to distinguish it from its subsystems. Because full second-order semantics imply that every possible set exists, the comprehension axioms may be taken to be part of the deductive system when these semantics are employed.
The bolded axioms is what I think explains the subsets of natural numbers (rest of the real numbers?)?
I've studied geometry to some extent and there the axiomatization seems more specific and maybe "easier" to follow.
Self-verifying theories:
On wikipedia again, I found an article about Self-verifying theories which seems to include the four basic operations, investigated first by Dan Willard, that says:
- Addition and multiplication are not function symbols of Willard's language; instead, subtraction and division are, with the addition and multiplication predicates being defined in terms of these.
Are there any specific and well known systems to read about that falls in to this category?
Question(s):
To clarify I would like to know the idea of Self-verifying theories and Second-order arithmetic respectively. Perhaps also how they differ? This possibly requires a long and not so less complicated answer, but any links or redirections are much appreciated.