It seems that we can’t have a decentralized system based on trust, and that’s why trustlessness is important in Bitcoin.

To use Bitcoin in a trustless manner, you have to run a fully-validating Bitcoin node. Only then will you be able to verify that the blocks you receive from others are following the consensus rules; for example, that the coin issuance schedule is kept and that no double-spends occur on the blockchain. If you don’t run a full node, you outsource verification of Bitcoin blocks to someone else and trust them to tell you the truth, which means you’re not using Bitcoin trustlessly.

Diagram David Harding has authored an article on the bitcoin.org website explaining how running a full node - or using Bitcoin trustlessly - actually helps you.

The bitcoin currency only works when people accept bitcoins in exchange for other valuable things. That means it’s the people accepting bitcoins who give it value and who get to decide how Bitcoin should work.

When you accept bitcoins, you have the power to enforce Bitcoin’s rules, such as preventing confiscation of any person’s bitcoins without access to that person’s private keys.

Unfortunately, many users outsource their enforcement power. This leaves Bitcoin’s decentralization in a weakened state where a handful of miners can collude with a handful of banks and free services to change Bitcoin’s rules for all those non-verifying users who outsourced their power.

Unlike other wallets, Bitcoin Core does enforce the rules—so if the miners and banks change the rules for their non-verifying users, those users will be unable to pay full validation Bitcoin Core users like you.

— David Harding Full Validation on bitcoin.org (2015) He says that running a full node will help you verify every aspect of the blockchain without trusting anyone else, so as to ensure that the coins you receive from others are genuine. This is great, but there’s one important thing that a full node can’t help you with: it can’t prevent double- spending through chain rewrites:

Note that although all programs—including Bitcoin Core—are vulnerable to chain rewrites, Bitcoin provides a defense mechanism: the more confirmations your transactions have, the safer you are. There is no known decentralized defense better than that.

— David Harding Full Validation on bitcoin.org (2015) No matter how advanced your software is, you still have to trust that the blocks containing your coins won’t be rewritten. However, as pointed out by Harding, you can await a number of confirmations, after which you consider the probability of a chain rewrite small enough to be acceptable.

The incentives for using Bitcoin in a trustless way align with the system’s need for full node decentralization. The more people who use their own full nodes, the more full node decentralization, and thus the stronger Bitcoin stands against malicious changes to the protocol. But unfortunately, as explained in the full node decentralization section, users often opt for trusted services as consequence of the inevitable trade-off between trustlessness and convenience.