OK, well this turns out to be easy to explain: > Put it this way. With number theory 1/t is undefined at t = 0. But at the exact left side and right side of t

OK, well this turns out to be easy to explain: > Put it this way. With number theory 1/t is undefined at t = 0. But at the exact left side and right side of t = 0 everything is defined again. There is no mathematical property preventing this from happening with numbers. Same with the ODEs. Undefined at the point of collision with no explicit mathematical rule saying it's not defined everywhere else. So really it's [0, t_0) and (t_0, infinite]. Suppose the first collision takes place at t_0, and the initial conditions are x_0. I agree you can give solutions f_1 on [0, t_0) and f_2 on (t_0, infinity) such that the limit of f_1 as t goes to t_0 from the left is equal to the limit of f_2 as t goes to t_0 from the right (at least in the case of double collisions – I'd have to think a bit about the general case). I do not agree this constitutes a solution of the system of differential equations with initial condition x_0 on [0,infinity). To have a such a solution means, by definition, that there is some function f(t) defined everywhere on [0,infinty) satisfying the differential equation for all times in that interval. As already discussed, that is impossible because at t_0 the coefficients of the ODE are undefined. You might respond that you're not claiming you have a solution on [0,infinity). (It's a little ambiguous to me.) But "a solution of the three body problem with initial conditions x_0" means exactly a function f(t) defined on some interval [0,A) with f(0)=x_0 satisfying the ODE on that interval. And the only way I can understand your remarks is to take A to be infinity. (Note this isn't even a determinism vs. indeterminism issue. It's about what is, definitionally, a solution to an ODE.) (Note also the difference with Norton's dome: in the dome problem, everything –– coefficients of the equation and the solution –– is defined for all positive times.)