Here's how I see the layers, listing the bottom first (sometimes there's a little overlap between adjacent layers):

  1. Axioms of logic (some statement is either true or false, etc), the possible logical operations (AND, OR, NOT etc), when you can do deduction (when it's allowed to go from some premises to some conclusion: when all the premises are true, then the conclusion has to be true. If any premise is false, *it doesn't matter either way*).
  2. Axioms of natural numbers:
    1. There exists a zero, which is a natural number
    2. For any natural number, there's an unique successor
    3. zero is not a successor of any natural number
    4. If two natural numbers n and m have the same successor then they are equal
    5. If some set X has the zero and of each of its elements also the successor then X is the set of natural numbers (or bigger)
    (*Those are all of them*)
  3. (A little) set theory: a set S is countable infinite <=> there exists a bijection of S with the set of natural numbers: Let S={a, b, c, d, ...}, N={0, 1, 2, 3, ...}. Then define a bijection between S and N such that 0 <-> a, 1 <-> b, 2 <-> c, 3 <-> d and so on and thus *count* the elements in S.
  4. Field axioms and Algebra (a + b = b + a, ...)
  5. How natural numbers will not be complete under the field operations (solve a + 1 = 0 for a) -> introduction of whole numbers
  6. How whole numbers will not be complete (when trying to solve a b = 1 for a) -> introduction of rational numbers
  7. How rational numbers will not be complete (when trying to solve a a = 2 for a) -> introduction of real numbers, necessiting introduction of *limits of sequences* in order to be able to represent a real number as *the limit of* a sequence of intervals which are exceedingly getting smaller (I mean the distance between the two ends of the interval - those ends are RATIONAL numbers - gets smaller and smaller) and closer to the "true value". The limit enables you to find the true value without doing the (infinite number of) steps of switching to the next element in said sequence, given that you can get "as close as you like to some limit", just define that trying to get close enough (*what is close enough?*) an infinite number of steps is the same as actually reaching it (and also making sure that this does not violate any of the axioms or conclusion that have been established before). Think of it like the number-guessing game where Alice thinks of a number and Bob has to guess it. So Bob guesses something and Alice tells him whether Bob's number is too small or too big or neither. Then Bob guesses again, either ad infinitum (when Alice's number was an irrational number) or some countable number of steps (when Alice's number was a rational number).
  8. How it makes sense to introduce limits of (countable-infinite elements) series (sums) as a labour-saving device in the same vein as above.
  9. How then it makes sense to find functions where it doesn't matter when you take such a limit: before or after applying the function. Those are called continuous functions and are the most important functions in the history of the world.
  10. Turns out these allow you to use all the normal axioms to define and use continuous functions as if the limits WERE NOT LIMITS but just one single rational number as the Classical Greeks had (!!). This means you can reuse most of the results of the Greeks and just replace "rational number" by "real number" in their proofs. So do so.
  11. It was found useful to be able to be able to go from a global description to a local description of the same facts in physics. As far as we know, all physical effects are simplest to describe as local effects. However, we are interested in the global effects. This was traditionally called the infinitesimal calculus, the physical effects were infinitesimal (infinitely small :-) ). Derivation is going from a global to a local description. Integration is going from a local to a global description (except when we can't do the integration because our tools suck, then we solve differential equations instead - or throw up our hands in defeat - that happens).
  12. How it makes sense to be able to get the limit of a series (a sum) of "very fine" areas, called Riemann integral (the simplest one).
  13. How real numbers will not be complete (when trying to solve i i = -1 for i) -> introduction of complex numbers
  14. How suddenly complex numbers (complex-valued fields) are complete under *all* field operations and functions (*what is close enough?*) (!!!).
  15. Introduction of *common* functions which are the everyday-speak of all STEM "as convenient short-hands" (and of course in the end are defined using combinations of the stuff above and in the end are footed on the axioms): exp, ln, sin, cos, sinh, cosh, √, connection properties between those, ..., planar (radial-angular) representation of complex numbers etcetc. Note that exp is the most useful and cool function of all of those. Almost everything else in this list can some way or another be reduced to combinations of exp. There was a time when almost every calculation was done using exp. Multiplication, Exponentiation (aha :-) ), Division, √, you name it. So exp is your friend.