How does Bitcoin enable trust?

[Ppimp]

What are we trusting in exactly when we say that Bitcoin now enables decentralized trust on the network?
Are we trusting that the encryption algorithm for ECC public key encryption isn't broken and so when someone signs a transaction they MUST have the private key?
Are we trusting that the transactions on the network are valid? And if it is that we are trusting the transactions are valid, are what we really saying is that we TRUST that 51% of the hashing power is controlled by absolutely unbiased actor who lets every transaction through (first come first serve, transaction fee descending).
In either scenario I don't see where it 'blows my mind' for enabling trust on the network? I feel like i'm missing something, what is the special type of trust that Bitcoin block chain technology enables that can be applied as a platform to other logical problems? What is it about the solving of the Byzantine General's problem that makes the power of Bitcoin unique?

[H.W.Z]

OP, you don't get how the trust works in bitcoin. Bitcoin implements a trust model of trust by computation. Trust in the network is ensured by requiring participants to demonstrate proof-of-work, by solving a computationally difficult problem. The cumulative computing power of thousands of participants, accumulated over time in a chain of increasing-difficulty proofs, ensures that no miners or even collection of miners can cheat, as they lack the computation to override the trust. As proof-of-work accumulates on the chain of highest difficulty (the blockchain), it becomes harder and harder to dispute. In bitcoin, a new proof-of-work is added every 10 minutes, with each subsequent proof making it exponentially more difficult to invalidate the previous results.no one miner is trusted, and no one needs to be trusted. There is no central authority or trusted third party in a distributed consensus network.

[amaclin]

ensures that no miners or even collection of miners can cheat

This works only while difficulty is increasing.
Otherwise cheating is possible and cheap