Why is adding attributes to an already instantiated object allowed?

A leading principle is that there is no such thing as a declaration. That is, you never declare "this class has a method foo" or "instances of this class have an attribute bar", let alone making a statement about the types of objects to be stored there. You simply define a method, attribute, class, etc. and it's added. As JBernardo points out, any __init__ method does the very same thing. It wouldn't make a lot of sense to arbitrarily restrict creation of new attributes to methods with the name __init__. And it's sometimes useful to store a function as __init__ which don't actually have that name (e.g. decorators), and such a restriction would break that.

Now, this isn't universally true. Builtin types omit this capability as an optimization. Via __slots__, you can also prevent this on user-defined classes. But this is merely a space optimization (no need for a dictionary for every object), not a correctness thing.

If you want a safety net, well, too bad. Python does not offer one, and you cannot reasonably add one, and most importantly, it would be shunned by Python programmers who embrace the language (read: almost all of those you want to work with). Testing and discipline, still go a long way to ensuring correctness. Don't use the liberty to make up attributes outside of __init__ if it can be avoided, and do automated testing. I very rarely have an AttributeError or a logical error due to trickery like this, and of those that happen, almost all are caught by tests.

Just to clarify some misunderstandings in the discussions here. This code:

class Foo(object):
    def __init__(self, bar):
        self.bar = bar

foo = Foo(5)

And this code:

class Foo(object):
    pass

foo = Foo()
foo.bar = 5

is exactly equivalent. There really is no difference. It does exactly the same thing. This difference is that in the first case it's encapsulated and it's clear that the bar attribute is a normal part of Foo-type objects. In the second case it is not clear that this is so.

In the first case you can not create a Foo object that doesn't have the bar attribute (well, you probably can, but not easily), in the second case the Foo objects will not have a bar attribute unless you set it.

So although the code is programatically equivalent, it's used in different cases.

Python lets you store attributes of any name on virtually any instance (or class, for that matter). It's possible to block this either by writing the class in C, like the built-in types, or by using __slots__ which allows only certain names.

The reason it works is that most instances store their attributes in a dictionary. Yes, a regular Python dictionary like you'd define with {}. The dictionary is stored in an instance attribute called __dict__. In fact, some people say "classes are just syntactic sugar for dictionaries." That is, you can do everything you can do with a class with a dictionary; classes just make it easier.

You're used to static languages where you must define all attributes at compile time. In Python, class definitions are executed, not compiled; classes are objects just like any other; and adding attributes is as easy as adding an item to a dictionary. This is why Python is considered a dynamic language.

No, python is flexible like that, it does not enforce what attributes you can store on user-defined classes.

There is a trick however, using the __slots__ attribute on a class definition will prevent you from creating additional attributes not defined in the __slots__ sequence:

>>> class Foo(object):
...     __slots__ = ()
... 
>>> f = Foo()
>>> f.bar = 'spam'
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: 'Foo' object has no attribute 'bar'
>>> class Foo(object):
...     __slots__ = ('bar',)
... 
>>> f = Foo()
>>> f.bar
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: bar
>>> f.bar = 'spam'

It creates a radius data member of my_circle.

If you had asked it for my_circle.radius it would have thrown an exception:

>>> print my_circle.radius # AttributeError

Interestingly, this does not change the class; just that one instance. So:

>>> my_circle = Circle()
>>> my_circle.radius = 5
>>> my_other_circle = Circle()
>>> print my_other_circle.radius # AttributeError