Post codes are maintained by the Royal Mail so you need to ask them, not the council.

Post code lists used to be published and were to be found in local libraries - and very useful they were too. Don't know if libraries still hold them.

You can search for postcodes on the royal mail website. However, they restrict the number of searches you can do in a day so finding all the codes in se23 would take a long time.

The Multimap maps show postcode boundaries. They are sometimes arbitrary, but would show if a street is within SE23.