Why do balloons filled with helium fall up?

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Why do balloons filled with helium fall up and not down like other objects?

They weigh less than the volume of air they displace.

Boats float because they weigh less than the volume of water they displace.

They don't fall up, they float up because their density is less than that of the air.

Falling involves gravity.

I once read about why helium makes your voice squeaky when you inhale it. It's because the lower density causes your vocal chords to vibrate much faster.

Sucking a little helium probably won't hurt you, but if you did it to much you can pass out from the lack of oxygen, and there's a rare chance helium bubbles can enter your bloodstream, which can be fatal.

Even to understand a simple topic like this, taking the time to study science with neotenic¹ curiosity, without hurrying to declare you’ve learned enough, showing how “practical” you are by hastily deciding this or that is worthless because it’s not used in the “real world” (except by the nerds who make the tools needed so others can pretend that area of human knowledge doesn’t exist), or embracing pseudoscience to spite the system for shoving science down your throat, pays off.

The ancients believed things fell or rose because they wanted to go to the place they’d been created for: heavy things belonged at the center of the Earth, and lighter ones, in spherical layers of increasing radius. Modern theories, with concepts like gravitational potential, aren’t all that different from this, particularly when explained in lay terms. The crucial difference is that they’re precise and make testable predictions, but you really need to get into the math to see it. Math is the natural language of the Universe. Those predictions have been tested so thoroughly at this point that we can confidently use those theories to build all sorts of structures and mechanisms which have to deal with gravity on our planet, and to send spacecraft to other bodies in our Solar System, and they work. They wouldn’t work if the theories they’re based on were just the subjective ramblings of white-coated people with too much time on their hands.

So something like a helium balloon in the air, or an air bubble in a bottle of water you’ve just flipped over, is forced up when the denser medium falls to take its place. The math shows you this is equivalent to putting the balloon and an equal volume of air on a seesaw. The system evolves towards the configuration with the most mass in a lowest position. That’s how potential energy is minimized, and you can extract the potential energy lost to reach that configuration to perform some useful work; for example, the balloon can lift a small load that, on its own, would fall through the air, being denser than it.
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¹ neoteny. Retention into adulthood of traits formerly present only in juvenile individuals of the species. Our ability to learn as adults is neotenic. It does decline with age, but it used to decline much earlier and faster.

The other posters are correct, however an easier way to conceptualize it might be for you to think of the denser gas...in this case, air, around the balloon falling down and displacing the balloon filled with the less dense dense gas. The less dense gas cannot hold the weight of the denser gas. It is the same with liquids and liquid gas mixtures. In water, air bubbles float up because the water "caves in" around it, displacing it upwards, or "pushing it up".

What is "up" & "down"?
Results of gravity?
Something we still know very little about. They (CERN) spent billions on a huge machine in the ground outside Geneva to figure out (in extreme detail) what you are asking. Trying to figure out what "mass" and "gravity" are.
It's been running for 10 years, something about a "higgs" particle. That's all they have.

Anyway "fall up" I believe is grammatically incorrect. "fall" automatically implies a "down" vector.
The word you are looking for is "rise".

The same reason that a piece of wood floats on water. Its lighter than the surrounding fluid.

Helium lifts sixty pounds per thousand cubic feet, but hydrogen lifts 80 pounds per thousand cubic feet.

So for a while they experimented with hydrogen balloons. Trouble is that hydrogen explodes and burns, but helium does not.