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In this position paper, we initiate a systematic treatment of reaching consensus in a permissionless network. We prove several simple but hopefully insightful lower bounds that demonstrate exactly why reaching consensus in a permission-less setting is fundamentally more difficult than the classical, permissioned setting. We then present a simplified proof of Nakamoto's blockchain which we recommend for pedagogical purposes. Finally, we survey recent results including how to avoid well-known painpoints in permissionless consensus, and how to apply core ideas behind blockchains to solve consensus in the classical, permissioned setting and meanwhile achieve new properties that are not attained by classical approaches.