Not sure where you got these numbers from, because they are wrong (unless you have a very old Honda).
Honda's recommendation for most of their recent (2003+) inline-4s is a 10,000 mile oil change interval. For severe service (lots of stop and go city driving), the recommended oil change interval is 5000 miles. Honda's factory oils are were formerly conventional oils (rebranded Mobil 5000) and are now synthetic blends from Conoco Philips, so their recommended intervals do not require the use of fully synthetic oils.
For filters, Honda recommends a filter change every 20,000 miles. Apparently their factory testing showed that the OEM filter could actually last 30,000 miles in between changes, and so they recommended 20k intervals to be conservative.
Here's the recommended service interval for an 03 Civic, for example:
http://owners.honda.com/service-maintenance/minder?year=2003&model=Civic-Sedan#mid^ES2573MW
*There are a few exceptions (like the S2000 which has a 7,500 mile OCI), but in general Honda recommends 10k intervals for standard driving and 5k intervals for severe driving. Again, these intervals do not require fully synthetic oil.