it is unclear to me how the Pythagorean theorem assumes the existence of a continuum. in order to "rectify" the diagonal length of a right triangle with its sides, one only needs to solve a quadratic equation:
A2 + B2 = C2
and for such one only needs algebraic numbers, which do not form a continuum (in fact they are countable).
it is unclear whether a continuum is a good abstract model of "the universe we live in". i think it is, but studying the very fine structure of the actual world has some practical obstacles: the limits of our technology (it takes a lot of energy to break apart an atom, for example). from far enough away, discrete and continuous systems can look very much alike....and there is no way of knowing at the moment on which level of resolution from the "bottom layer" of the universe we reside.
the "real numbers" are, in a sense, poorly named...one tends to think they have some sort of distinguished existential property, which is no more true of them then any logical structure. while i am not a "fictionalist" per se (one who believes that all mathematics is merely a game humans play for diversion), such a philosophical stance does have the attraction of avoiding some of the usual ontological dilemmas that are part and parcel of mathemaitcs (what IS a number, how do we know we are RIGHT about how they behave?).
the Archimedian property of a number system is "stronger" (more restictive) than the Pythagorean theorem: in the hypperreals, for example, the Pythagoren theorem is "still true" (the usual algebraic rules still hold), but the Archimedean property fails (there are hyperfinite numbers larger than any integer). they are not "equivalent".
the continuum hypothesis, does not, by the way, actually assert the existence of a continuum...rather it says something about the size of a particular continuum (R, the real numbers, or equivalently the interval (0,1)): that it is "the next largest infinity" after the "first infinity" of the set of natural numbers. in other words, it asserts something about "the boundary of uncountablility" which has proved extremely difficult to prove. what is known (at least as i understand it) is that the axioms of set theory alone (that is Zermelo-Fraenkel plus the Axiom of Choice) are not sufficiently strong to force the continuum hypothesis on us. set theory says relatively little about "which sets exist", we have three main ways of "making bigger sets from smaller sets":
1) form the power set
2) take the union of a collection of already established sets
3) form the cartesian product of a family of sets
of these, only (1) affects "the level of infiniteness": (2) corresponds to "addition of cardinals" (that is |AUB| is bounded by |A|+|B|, and we have equality if A and B are disjoint), (3) corresponds to "multiplications of cardinals" |AxB| = |A|*|B|, while (1) corresponds to "exponentiation" (|P(A)| = 2|A|).
talking about functions lifts us from (3) to (1). collections of functions are usually *much* larger than the sets they "act upon". for example, the real numbers may be thought of as a line (determined by a starting point 0, and locating a "unit length" 1 somewhere). but the set of polynomial functions on R (which is a very restricted class of functions) takes up "infinite dimensions" (we can choose any real number for every possible (non-negative) power of x).
i don't think the "iffs" are quite as cut-and-dried as you present them. existence of SOME irrational numbers does not entail existence of ALL irrational numbers: for example the existence of √2 does not help us understand how we might obtain pi. futhermore, one doesn't need "algebraicness" to get "large sets", the set of all sequences of natural numbers is already quite large, without any algebraic structure superimposed on it.