Extension of the indentation fracture toughness estimation method to very small length scales often requires the use of an indenting punch much more acute than the oft-used Vickers probe. Experimental results for very acute, sharp probes have motivated a new approach to the indentation fracture mechanics of radial crack development. An extension of the standard two-component (residual elastic-plastic + elastic contact) stress-field model of radial fracture is proposed, based on the concept that a sufficiently acute indenter can also act as a 'wedge,' prying open the surface-located radial cracks. In this, the first of a two-part series, a three-component wedging indentation model is constructed, and some general characteristics of the model are explored. In particular, the implications of the three-component stress field of the model for the description of radial crack development during load-unload indentation cycles of acute probes are considered. Explicit predictions of crack development are compared with the qualitative features of experimental observations, providing a basis for the quantitative comparisons in Part II.