There’s a certain amount we can learn about geometry using only 4 postulates. One thing we can’t learn though is: given a line and a point not on the line, how many lines through the point are parallel to the original line? There are three possible answers (0, 1 and infinity) and all three are logically consistent with the original 4 postulates but they are, of course, incompatible with each other.
For centuries, starting with Euclid, geometers have assumed that one of the possible answers (1) is “right” and the others “wrong” but since the nineteenth century they have known that it was not possible to derive the “right” answer from the other four postulates. If it couldn’t be proven, the only solution was to add the “right” answer as a fifth postulate or basic assumption.
Non-Euclidian geometry began when geometers, in a somewhat paradigmatic move, gave up worrying about which possible parallel postulate was “right” and allowed geometry to branch into three parts. Each part was logically self-consistent so while no two could be “right” at the same time, there was no reason to pick one and call it “right” and the other two “wrong.”
This happened because there was a question in geometry that could not be answered by logical deduction starting with the original four postulates. Could the same situation occur in number theory? Is there some proposition about numbers that is undecidable from the basic axioms? Well dog my cats, there is! And it was discovered by Kurt Godel who proved that there exists some proposition about numbers (call it G) that can neither be proved nor disproved using the basic axioms of arithmetic. In other words, both G and its negation (~G) are logically consistent with the other axioms.
Proceeding as geometers have done we could add G to our list of axioms (call the list of axioms N.) For that matter we could also add ~G. Now, like in geometry, there are two formulations: a “standard” one (N+G) and a “nonstandard” one (N+~G). The problem is that, unlike geometry, we aren’t done because Godel’s method for finding undecidable propositions still works for N+G and for N+~G. I.e. we know that both of these systems still contain an undecidable propostion. In “standard” number theory (N+G) there is a proposition (G’) that can not be decided by deducing from the list of axioms N+G so we can add either G’ or ~G’ as axioms creating another pair of number theories and each of those branches into another pair and so on and so on and so on… We end up with a binary tree of number theories that extends out infinitely.
To get back to your questions, the postulates that define number theory are listed in the book as the definition of TNT. Sadly, I seem to have temporarily misplaced my copy but they include the peano postulates, the definitions of addition and multiplication and the logical rules of inference. None of these basic axioms are redefined to make non-standard number theories. Instead, as undecidable propositions are found, they (or their negations) are added as new axioms. And yes, I’m afraid you might have to re-read the book because Hofstadter’s explaination was a thousand times clearer than mine could hope to be 