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A system is said to be stable when any of its poles are <0.
However I don't get why that is the case. Negative poles mean negative angular frequency, and negative angular frequency is equal to positive angular frequency, so the definition contradicts itself. What am I missing?
A system is said to be stable when any of its poles are <0.
I'm not sure who said that, but they're wrong or they've been misquoted.
A Linear Time Invariant system that can be described by ordinary differential equations is stable if the real part of every blessed one of its poles is less than zero.
This is because for any pole $a$ in a system transfer function, nearly any response of the system to an input will have an element of the response that is proportional to $e^{a t}$. For any $a$ such that $\mathcal R (a) < 0$, $\lim_{t \to \infty} e^{a t} = 0$. But for any $a$ such that $\mathcal R (a) > 0$, $\lim_{t \to \infty} \left | e^{a t} \right | = \infty$.
So even one pole with a positive real part will cause the whole system to be unstable.
Negative poles mean negative angular frequency
That is incorrect. In the Laplace domain, any sinusoidal component to a signal is caused by components with $s = a + j\omega,\ \omega \ne 0$. Your "angular frequency" is the $j\omega$ part, and it is orthogonal to the real number line.
and negative angular frequency is equal to positive angular frequency
That is also incorrect.
You are probably starting with $\cos j \omega = \cos -j \omega$ and jumping to a conclusion, bolstered by the fact that in a system with all real-valued gains, the poles and zeros will all be purely real or will occur in complex-conjugate pairs.
However, while I've never seen it used in a control system, it is common in communications systems to convert a signal centered on a carrier down to baseband as an inphase/quadrature pair -- and an inphase/quadrature pair acts just like a complex number.
In this context (and in some hypothetical control systems context where it makes sense to have complex-valued gains), negative frequencies do have meaning, and a negative frequency does not "equal" a positive frequency -- because $e^{j\omega t} \ne e^{-j \omega t}\ \forall \ t$.
