Defining foreign key relationships for Rails' models

Question

I have a Comment class with a :foreign_key of post_id in the Post class.

class Comment < ActiveRecord::Base
  belongs_to :post, :class_name => "Post", :foreign_key => "post_id", :counter_cache => true
  belongs_to :author, :class_name => "User", :foreign_key => "author_id"
end

But my CreateComments migration does not define a database-level foreign key:

class CreateComments < ActiveRecord::Migration
  def self.up
    create_table :comments do |t|
      t.column "post_id",       :integer,   :default => 0, :null => false
      t.column "author",        :string,    :default => "",   :limit => 25, :null => false
      t.column "author_email",  :string,    :default => "",   :limit => 50, :null => false
      t.column "content",       :text,      :null => false
      t.column "status",        :string,    :default => "",   :limit => 25, :null => false
      t.timestamps
    end
  end

  def self.down
    drop_table :comments
  end
end

Instead post_id is a simple Integer column.

So, it seems that this foreign key relationship exists only in the mind of Rails, not at the database level.

Is this correct?

Also, is it necessary for the corresponding Post model to also declare its reciprocal foreign key relationship with Comments using the :foreign_key attribute or could that be omitted?

class Post < ActiveRecord::Base
  set_table_name("blog_posts")
  belongs_to :author, :class_name => "User", :foreign_key => 'author_id'
  has_many :comments, :class_name => "Comment",
    :foreign_key => 'post_id', :order => "created_at desc", :dependent => :destroy
  has_many :categorizations
  has_many :categories, :through => :categorizations
  named_scope :recent, :order => "created_at desc", :limit => 5

end