The Barkhausen criterion for system with positive feedback (being \$B(s)\$ the feedback network transfer function and \$A(s)\$ the gain network transfer function) afirms that for a system to keep oscillating without input signal and without loss, we need the poles of:
\$H(s) = \dfrac{A(s)}{1 - B(s)A(s)}\$
in the imaginary axes at the complex plane.That means there should be a solution to:
\$B(j \omega_o) \cdot A(j \omega_o) = 1\$
and ωo would be the oscillating frequency.
On the other hand, if i have a system with negative feedback (being B(s) the feedback network transfer function and A(s) the gain network transfer function), the overall transfer function will be:
\$H(s) = \dfrac{A(s)}{1+B(s)A(s)}\$
So, in order to have poles in the imaginary axis (no loss), I need a frequency ωd that solves:
\$B(\omega_d).A(\omega_d) = -1\$
That loop transfer function being -1, should mean that the sinusoidal input with frequency ωd wil be shifted by 180º and will keep the same amplitude after passing through the loop \$A(s).B(s)\$, then will be shifted by 180° again because of the negative feedback and will return to its original form to keep oscillating indefinitely.
It will also mean that the transfer function \$H(s)\$ will have infinity gain at jωd which is a requirement to keep a system living without any input signal. Isn't it enough to implement a oscillating system that will keep oscillating any sinusoidal with frequency ωd that enters the loop?
Why do we need positive feedback to implement an oscillating system ?