A spectrum is a sequence $X_0,X_1,...$ of spaces together with structure morphisms $\Sigma X_n\to X_{n+1}$. To get the usual model for the stable homotopy category based on the category of spectra, one "inverts" the suspension functor (or the shift functor) which is not an isomorphism in the category of spectra. It seems to me that this is a kind of allowing the objects to be indexed over the integers. So why does one not define a spectrum to be indexed over the integers or at least bounded below?

A spectrum is a sequence $X_0,X_1,...$ of spaces together with structure morphisms $\Sigma X_n\to X_{n+1}$. To get the usual model for the stable homotopy category based on the category of spectra, one "inverts" the suspension functor (or the shift functors) which is not an isomorphism in the category of spectra. It seems to me that this is a kind of allowing the objects to be indexed over the integers. So why does one not define a spectrum to be indexed over the integers or at least bounded below?