Blockchain protocols differ in fundamental ways, including the mechanics of selecting users to produce blocks (e.g., proof-of-work vs. proof-of-stake) and the method to establish consensus (e.g., longest chain rules vs. BFT-inspired protocols). These fundamental differences have hindered "apples-to-apples" comparisons between different categories of blockchain protocols and, in turn, the development of theory to formally discuss their relative merits. This paper presents a parsimonious abstraction sufficient for capturing and comparing properties of many well-known permissionless blockchain protocols, simultaneously capturing essential properties of both proof-of-work and proof-of-stake protocols, and of both longest-chain-type and BFT-type protocols. Our framework blackboxes the precise mechanics of the user selection process, allowing us to isolate the properties of the selection process which are significant for protocol design. We illustrate our framework's utility with two results. First, we prove an analog of the CAP theorem from distributed computing for our framework in a partially synchronous setting. This theorem shows that a fundamental dichotomy holds between protocols (such as Bitcoin) that are adaptive, in the sense that they can function given unpredictable levels of participation, and protocols (such as Algorand) that have certain finality properties. Second, we formalize the idea that proof-of-work (PoW) protocols and non-PoW protocols can be distinguished by the forms of permission that users are given to carry out updates to the state.