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Newton's first law establishes the concept of inertia and inertial frames; the second law (F = ma) quantifies how forces cause acceleration in inertial frames; the third law (action-reaction) ensures momentum conservation in isolated systems. Together, these three statements form the complete axiomatic foundation of classical mechanics.
You already understand F = ma from your prerequisite study of Newton's second law, and you know what an inertial reference frame is. The formal treatment here shows why all three laws are needed together, and what each one contributes that the others cannot supply.
Newton's first law is not merely the special case of the second law with F = 0. It is the definition of what counts as an inertial frame. From your prerequisite: an inertial frame is one that is not accelerating. But how do you identify one? The first law answers this operationally: a frame is inertial if and only if a body subject to no net force moves at constant velocity in that frame. This is the test. Without the first law, you have no way to know whether a given frame is valid for applying F = ma — the second law would be circular (force causes acceleration, but how do you measure force without already knowing your frame is inertial?). The first law breaks the circularity.
Newton's second law in its most general form is F = dp/dt, where p = mv is linear momentum. For constant mass, dp/dt = m(dv/dt) = ma. The momentum form is more fundamental: it applies to variable-mass systems (rockets expelling propellant) and generalizes naturally to special relativity. The proportionality constant m between force and acceleration is the inertial mass — it measures resistance to change in motion, not weight. This is experimentally the same as gravitational mass (to 14 decimal places), a deep coincidence that Einstein elevated to a postulate in general relativity.
Newton's third law states that forces always come in equal and opposite pairs: if A exerts force F on B, then B exerts force −F on A simultaneously, of the same type. The key word is *simultaneously* — this law has no counterpart in relativity, where the concept breaks down for fields that propagate at finite speed. But in classical mechanics, the third law guarantees that internal forces within an isolated system cancel in pairs: when you sum all forces over every particle in an isolated system, the internal forces sum to zero, leaving only external forces. This is precisely why the total momentum of an isolated system is conserved (dp_total/dt = ΣF_external = 0) — momentum conservation is a theorem derived from the third law, not an independent postulate. The three laws form a logically closed system: the first identifies valid frames, the second governs dynamics within them, and the third ensures global consistency through action-reaction pairs.