In the United States, given that the ZIP code defines both the city and the state, why do we have to include all three on envelopes? Or do we?

To make things easier for the Post Office.

"mail travels more efficiently, and therefore more quickly, when senders use the code"

https://en.wikipedia.org/wiki/ZIP_code

For any letter addressed within the United States, the USPS requires two pieces of information on the envelope.

Address of the recipient: Placed on the front (non opening) side of the envelope in the center. Generally, the name of the addressee should be included above the address itself. A ZIP+4 code will facilitate delivery.


https://en.wikipedia.org/wiki/US_Mail#Addressing_envelopes

I'm pretty sure we don't. Try sending something to someone with just their name, street address, and zip code.

Mr. John Doe
123 Main St.
___________98765

But be sure to write the zip code well to the right; I've put in blank lines to space it since this format will put it to the left if I don't.

"A ZIP code's address and the city name written on the same line do not necessarily mean that that address is within that city....Thus, from the perspective of the U.S. Postal Service, the city of Centennial does not exist—it is part of Aurora, Englewood or Littleton. In the ZIP-code directory, Centennial addresses are listed under those three cities."
https://en.wikipedia.org/wiki/ZIP_code#ZIP_codes_only_loosely_tied_to_cities

We had a similar situation when we lived in California, and mail with our city's name or the P.O. name arrived just the same. Right now in Alaska, our zip code belongs to a city 30 miles from us.

The reason why the USPS requires all the information on the envelope that seems a bit redundant is because people often accidentaly mix up the ZIP code numbers, by having the city and state listed the USPS can then cross check.

Also, in many rural areas, more than one town often share the same ZIP code.

http://maps.huge.info/zip.htm

I think the only reason for putting the city and state on is to verify the zip code, which can be written wrong or illegibly.

My friends and family criticize my sloppy writing, but the postmaster back in California assured me it was 100% legible, unlike much she saw. Illegibility of addresses, according to her, is a major problem for postal workers. And she greatly encouraged the use of zip + four, which in my case told the letter carrier not only which state, city, and neighborhood I lived in, but even in which apartment building.

http://www.ehow.com/how_6399440_address-envelopes-usps.html