Is it possible to walk in a friction-less world? Is it possible to walk in a frictionless world. I think you won't be able to walk because walking depends on friction acting on your feet. I think about it like an ice rink because when you try to walk on an ice rink it is incredibly hard as there is less friction acting on your feet. Can it be shown mathematically using principles of mechanics that it is not possible to walk in a frictionless world, or is it only by experiment?
Edit: I do not mean outer space by frictionless world, because you are not being pushed on a surface by gravity. I am looking for a mechanism that would allow a biological specimen to walk on a surface without friction, assuming that all other laws and effects are in place.
Edit: What I mean by walking is that you must move your legs in order to walk. I read the idea about using cleat like shoes to walk. The cleats would make holes into the ground allowing something to push off of horizontally allowing you to walk, but how would that even work? If the world is frictionless making holes in the ground would just turn the material you are standing on into rubble. The rubble would then act like the original material that you wanted to put holes into in the first place. How does material stay together in a frictionless world? 
 A: You cannot walk at all if there is no horizontal component of the force of interaction between you and the ground; by this definition "no friction" is tantamount to "no ability to move horizontally".
But to add to Hapa's ANswer: you can move by throwing stuff. How do you do this? Here's one way if you don't have a handy sack of hammers ready in your pocket just in case the evil ones in charge suddenly switch friction off for a lark:


*

*Looking straight forwards, take a deep breath. As you draw the air in, it is pulled upwards into your nostrils. The third-law-begotten reaction force pulls you downwards, but that's OK because your feet are thrusting against the Earth's surface, so you essentially don't move downwards at all.

*Turn your head to face opposite the direction you want to go in. If needed, use a cyclic shape deformation of your body to achieve this, as in the way an astronaut can rotate in space or the way a torque-free freefalling cat can flip over to land on its feet (you don't need a tail);

*Now breathe swiftly out  through your mouth, pursing your lips for maximum exhaust velocity of the air jet. This will thrust you in the direction you need to go.
Repeat the above cycle to get to where you want to go to, noting that you will also need to make these actions to decelerate when you get there.
A: The purpose of the question, as I read it, is related to an ideal condition (perfectly smooth surface) and a man with regular feet/shoes. In this case, normal daily walking would not be possible, as the horizontal force that moves the person ahead is coming from the friction between the feet and the ground.
F = k*N; 
F : Force (horizontal)
k: friction coefficient (zero in this question)
N: Normal force (vertical in this case, and is equal to the mass * Gravity of the person in question, if he's not jumping and if he's standing still perfectly, on his two feet. This force is irrelevant to the question and answer.)
if k = 0, then F= 0.
However, if it's a question to ask for a solution to walk in a frictionless (super-teflon!) surface, there are options.


*

*as defined before by many, Momentum. Throwing a ball would start moving the person, until he hits something or until the aerodynamic drag (it exists, right?) slows him down to a stop, or until the person throws something else in the other direction. (He must be a perfect thrower to start and stop using only 2 throws, if the second throw is misaligned vectorially, he will keep moving).

*Nailed shoes that crack into the ground. There's no friction, but as soon as the nails are buried in the ground, they allow horizontal forces. The ground must not be a too rigid one (it mustn't be like marble), to walk comfortably.

A: YES walking is possible in a frictionless world with appropriate limb movements. The idea is to twirl the arms and legs to cause propulsion by the Magnus effect. The technique has been developed in a British government ministry https://en.wikipedia.org/wiki/Ministry_of_Silly_Walks
Named for its investigator Gustav Magnus, the Magnus effect causes an aerodynamic force perpendicular to the axis of a rotating cylinder in air. It explains why a flying ball with topspin swerves downward more than would be produced by gravity alone, why backspin has the opposite effect and side-spin causes a swerve to either side. This article shows both aircraft and ships that use the Magnus effect for propulsion.
https://en.wikipedia.org/wiki/Magnus_effect
A: If there is no friction, you can still move by conservation of momentum. Take some stuff with you that you don't need. Throw it away in the opposite of the direction you want to go!
A: Yes. You need ridges (about 2mm tall):
_____|~~~~~|______|~~~~~|____
You can walk on that if your tread is the right shape friction or not. Boot tread will interlock with the ridges providing non-friction-based traction and so the ability to walk.
A: 
Can it be shown mathematically using principles of mechanics that it is not possible to walk in a friction-less world, or is it only by experiment?

It all depends on your definition of walk.
If you mean, slide along a horizontal surface thanks to friction, then by definition no, it is not possible. Your critical angle is zero, so the horizontal component of your motion by whatever force you can bring to bear against the pavement will be zero.
You can play with the definition of friction (electrostatic/electromagnetic coupling between the walking appendages and the pavement), so that you can walk thanks to a different type of friction.
If you are interested on horizontal travel then you can use traction, sort of large scale friction: by either exploiting irregularities on the pavement or creating them by pinching or stabbing. Some sea-bottom dwellers 'walk' by stabbing the sand and using that for traction. I would not call that walk however; it's closer to a sort of horizontal climbing. Also, this is a bit like cheating; what we call friction in everyday life also involves traction, so when we say "No friction" we should imply also no traction, and "no judicious punching holes in the pavement".
Another possibility is to employ reaction; by either swimming through the air (you would need some sort of 'wing'), or directly using jet propulsion (conservation of momentum). For that you would need large lungs and powerful thoracic muscles. There's a scene in Arthur C. Clarke's The Sands of Mars in which, if I remember correctly, a character is stranded inside an habitat "bubble" so that he hasn't traction (the bubble is smooth on the inside) and has insufficient friction (due to being in microgravity). He resorts to throwing away his clothes for momentum, and supplementing that with vigorous puffing and spitting (at that point his Captain enters the habitat bubble).
It is possible that whatever force neutralizes friction will also change the viscosity of air and make flight inefficient. This leaves jet (or "squid") propulsion.
A: Although friction is not one of the four basic forces of nature, it exists because those basic forces exist.  Friction is the resistance to motion of two objects held against each other.
Friction that allows us to walk depends on gravity to convert our mass to weight which holds our feet against the surface where static friction enables the soles of our shoes to push off against the surface of the Earth.
Even when ice skating, kinetic friction keeps the blades sliding against the ice in a forward direction, and static friction allows the skates to push off to propel the skater forward.
Some friction is caused by molecular attraction, such as when Vibram soles are able to maintain grip on smooth rock.  This is the result of electromagnetic forces in the rubber molecules of Vibram attracting rock molecules.
Even a rocket engine, which otherwise could be used to propel you in a frictionless environment, depends on the difference in force between friction caused by great pressure of propellant exhaust on a small area (exit pressure) and the lesser friction of propellant exhaust flowing through a larger area (free stream pressure).  The equation for rocket thrust shows this:
$$\text{rocket thrust} = \text{mass flow rate} \times \text{exit velocity} + (\text{exit pressure} - \text{free stream pressure}) \times \text{exit area}$$
The coefficient of friction is a measure of how strongly two surfaces will stick together.  It's the ratio between the force necessary to induce sliding, and the pressure holding the two surfaces together.  It can be used to calculate the amount of friction:
$$F = u \times N$$
$F$ is the frictional force
$u$ is the coefficient of friction
$N$ is the normal force (perpendicular to both surfaces, which presses them together)
If surfaces could slide with no force at all, the numerator of their coefficient of friction would be zero, the coefficient itself would be zero, and the frictional force between those two surfaces would be zero.  Here is a list of coefficients of friction of various materials.  The maximum coefficient of friction is one, and the minimum is zero.
Friction exists everywhere in the Universe.  To have no friction, there would have to be no gravity, no electromagnetic force, no gluons to hold atomic nuclei together, and no bosons to allow the buildup of heavy nuclei in stars.  All would be chaos.  Newton's third law would be inoperative.  The universe would be a soup of uniform density with no structure.  Entropy would tend to be maximal.  You would not be able to walk.
A: Some people have suggested throwing something out--that's not walking.
However, a world being frictionless does not mean that it is perfectly flat.  You can walk by exploiting this fact--if a foot is in a depression you can develop a horizontal force no greater than the sine of the angle of the walls of the depression times your weight.
Note, furthermore, that if you are standing you will inherently slide into such depressions.
While this will be far more like a toddler's first tentative steps than what we call walking it's at least locomotion based on movement of your legs.
A: If there is no friction, then you can not rely on it providing traction. However, other forces must still exist, or you'd fall through the floor! So, there is simple artificial solution to being able to walk in frictionless world.
All that is needed is sufficiently contoured surface, something like this kind of knobs covering the surface:
   ####     ####     ####
    ##       ##       ##
###########################

Then all you need is shoes with this kind of soles:
############################
         #         #
        ###       ###

Now when you walk, the knobs at the bottom of the shoes will hook onto the knobs on the floor, and then you will be able to push against them without your feet slipping, same way as the floor stops you from falling because of the gravity.
The key is having the hooks, having angle of 90 degrees or more so that the sliding will stop at the hook shape. If angle is less, if then the surfaces will slide the wrong way and slip off without getting hooked.
The most practical way to achieve this would probably be to simply have velcro shoes and floors, though I suspect the material might need to be altered (stronger? different shapes?) to allow proper hooking without friction helping things.
So to answer the title question: Yes, it is possible to walk, if surfaces are suitable. It might be possible to walk on some natural surfaces which would provide hooking action, and it would be easy to walk on artificial surfaces built for the purpose.
A: I assume by zero friction you mean no roughness or deformity in the ground. Perfectly smooth. Even so there is a way to walk. As you plunge your foot into the ground you compress it a little based on the atomic theory of matter. This impression allows your foot to be slightly lower than adjacent atoms and can therefore push away from them. On ice this results in melting the top layers of atoms resulting in the slippery experience we have. In sand you would easily be able to walk as you dig in with each step. 
A: I agree with you. The existence of microwelds on the surface and the object is essentially what causes our feet/shoes/socks/whatever to push back and forward upon the surface to propel us forward. Without something to back up on (or more accurately, to hook on to), we can't expect to propel forward. Newton's Third Law says every force has an equal and opposite reaction. The microwelds on our feet essentially "latch" on and push into the microwelds of the surface, and the surface pushes back with equal force to propel our bodies forward. However, without the existence of these microwelds, there would be no friction, and there wouldn't be any thing to "latch on" and "push back" onto. It would just be two smooth surfaces rubbing against each other. Completely smooth.
I can't explain in completely mathematical terms, but here is how I think it would go.
The coefficient of friction would be $0$, therefore the entire time, no friction would exist. Keep in mind that air resistance is also a type of friction, so we can also say that is non-existent. If we lift up our feet and put it forward, essentially, we are putting weight forward in respect to our center of mass, therefore propelling the body forward with a weight/propulsion force. But since there is no force to keep the rest of our body back, or our feet planted to stabilize, the weight would make us faceplant right onto the cruel, heartless, floor.
A: In addition to the answers given already which cover the momentum and traction sides of the problem, I'd also like to toss in magnetism. You could just use magnetic fields to either push or pull you in the direction you want to go. Bonus points for using simple electromagnets that could be switched on and off so you can accelerate/decelerate at will.
A: Newton precisely expressed it in the law of inertia. A body (in this case a normal person with shoos trying to walk on frictionless surface) maintains its state of motion unless unbalanced external force acts up on it. This means, the person on the frictionless surface maintains his state of rest, if he was at rest, or moves with the velocity that he was moving with. He can't change his velocity. He also can not change his state of rest unless he do something to change his state by changing the force acting upon him. 
The question here is walking on frictionless world not moving. Walking is different from travelling in space using action-reaction phenomena, radiation (light traveling in space), molecular vibration (sound and heat). Walking is moving only with the aid of friction. You drag your one leg to the ground a little bit backward and lifting you other leg forward. That means the other leg is supported by friction force with the surface (see Fig.).

According to Newton's second law:
             $$   F_net = ma
$$
              $$  F-f=ma
$$
But if there is no friction to support the backward leg then there will be no walking. Because nothing supports the person at the back. There could be motion, however, he maintains his original velocity. This time it is not walking rather sliding. So, motion may not be zero but walking is. The motion is when f=0,
             $$   F=ma
$$
If the person has a means, to have exerted up on an external force F, then he can move in space but that is not a walk. That may be is propulsion like a rocket or radiation like light. 
Rocket propulsion equation is (non-relativistic)
   $$dp/dt = d/dt(mv)
$$
   $$dp/dt = v dm/dt + m.dv/dt
$$
The first term on the right is rate of change of mass. A person can thus propel in space if either he has to throw some mass away in the opposite direction of his motion or he has to gain mas from the opposite direction. Then he can move in space like a rocket.  Radiation is also another way of motion carried by EM (Electromagnetic) wave like light. 
To sum up, if there is no friction, there will be no walking because there is no support on the other leg but there could be motion depending upon the force needed for the motion. 
A: There are at least 3 answers to your question depending on what you mean by "world"?
1 - if you mean "universe," the answer is no, because there would not be "anything" that could walk, or a place to walk on.
2 - if you mean "earth,"  the answer is no, because there would be no earth.
3 - if you mean "surface," the answer is no, because a horizontal force component can not be created by walking. 
