I am looking for the syntax to add a column to a MySQL database with a default value of 0
Try this:
ALTER TABLE table1 ADD COLUMN foo INT DEFAULT 0;
From the documentation that you linked to:
ALTER [ONLINE | OFFLINE] [IGNORE] TABLE tbl_name
alter_specification [, alter_specification] ...
alter_specification:
...
ADD [COLUMN] (col_name column_definition,...)
...
To find the syntax for column_definition
search a bit further down the page:
column_definition clauses use the same syntax for ADD and CHANGE as for CREATE TABLE. See Section 12.1.17, “CREATE TABLE Syntax”.
And from the linked page:
column_definition:
data_type [NOT NULL | NULL] [DEFAULT default_value]
[AUTO_INCREMENT] [UNIQUE [KEY] | [PRIMARY] KEY]
[COMMENT 'string']
[COLUMN_FORMAT {FIXED|DYNAMIC|DEFAULT}]
[STORAGE {DISK|MEMORY|DEFAULT}]
[reference_definition]
Notice the word DEFAULT there.