The consistency guarantee requires that RW histories are compatible with some sequentially consistent history on a non-concurrent RW register. Defining a total order on operations is sufficient, I believe, but not necessary (does it matter what order two consecutive reads happened in?).
Paxos is, fundamentally, a quorum-based system that deals with reordering of messages. It sacrifices liveness for correctness - if the proposer does not hear back from a majority of nodes (in the case of, e.g. a partition), the protocol will not complete (availability is sacrificed).
My point is not that there is a 'vital packet' in every protocol, the omission of which will cause either a lack of availability or consistency (although I can certainly design protocols that way!) - it's that for every protocol there is a network partition which causes it to be either unavailable or inconsistent. That network partition might be dropping ten messages, or just one. Retransmitting would make sense, but in real life message failures are often highly temporally correlated :(
The proof of this, by the way, is in a very famous paper by Fischer, Lynch and Patterson called "The Impossibility of Distributed Consensus With One Faulty Process". One take away is that one slow-running process can take down any protocol. It may take a few missed messages, but only a single node...
Paxos sacrifices latency
“The network will be allowed to lose arbitrarily many messages sent from one node to another”
The theory behind all this really does hold this point up. I have another blog post with much more detail on the theory here: http://the-paper-trail.org/blog/?p=49, but I warn you it may be heavy going.
If a node you can contact is required to guarantee consistency, there will be some times that it will have to refuse your request because other nodes are not contactable.
The author's point was that in any distributed system there is a non-zero probability of a network failure. While both clients and server nodes can retry connections, there is a non-zero probability that the problem will persist longer than your "availability agreement" allows. In that case, you have a choice - return potentially inconstent data or refuse the request.
What you seem to be arguing is that the probabilities of failure - in particular of repeated failure - while non-zero, are effectively zero. The author would disagree (as they point out, the probabilities combine exponentially as the number of nodes increase.) I think he's right and that you are wrong.
So it is tolerant of partition and it sacrifices availability in favour of consistency. So it is CP, not CA.
But what does it mean to tolerate a partition? As if the system has a choice?
Any CA system is claiming to be consistent and available as long as the network doesn't partition. That's the strongest statement you can make under the CAP theorem, and Paxos certainly falls in that camp.
My problem with the original article was that it claimed that any individual network or node failure was a partition affecting the consistency or availability of the system. Paxos is a clear counterexample to that, as it tolerates a lot more than that without sacrificing consistency or availability.
Once the network actually partitions (or half the nodes become unreachable), then you are correct. The CAP theorem comes into play again and we must sacrifice either C or A, and Paxos chooses A.
Or does any system subject to hardware or power failure fail to count as "available"?
A distributed system is considered available if "every request received by a non-failing node [results] in a response." It does not mean you cannot retransmit or retry.
Similarly, the consistency guarantee only requires that there exist a total order on operations. Failures are ok, as we're allowed to retransmit, retry, and otherwise tolerate faults. There is no inconsistency, nor is anything eventual.
My point is that a "temporary" partition is just a fault, and as long as the fault is shorter than the allowed response time of the system, it doesn't make a difference.