Xjyuk

Not a world, precisely, but an adventure that I once ran: "The Illusory Castle". It was a puzzle that, once the solution was known, was quite easy. An artifact the party needed had been taken for safe-keeping by a legendary illusionist. When the party tracked down the artifact, they came to a castle with a groundskeeper who confirmed that the artifact was indeed held within the castle. He also told them that the entire place was an illusion, and ONLY the artifact was real. However, banishing the entire illusion would trap the artifact in the illusionist's pocket dimension, so they needed to extract the artifact first.

I had planned on this taking an hour or two to solve, including encounters, but it ended up taking most of a weekend. This was the premise of the puzzle:

1. The castle was set to "accommodate" its guests. Whatever they looked for, it provided. If they looked for a door, there was one, if they looked for a secret passage, they found it. If they asked if there was a guardian in the next room . . . yep . . .is he a badass? . . . yep . . . wait, are there traps we have to avoid while fighting the guardian? . . . yep.


2. If the players looked for the artifact, the castle showed them an illusion of the artifact, not the real thing.


3. Any time they got tpw'd, they woke up back at the groundskeeper's shack with some wounds. Not dead, but worse off than last run.

Solution: There was one constant every time they went through, a sword that always looked the same and, if the players examined their wounds, were the only "real" wounds that the place left on them.

That is the only example I have of something similar to what you ask. A whole world built on a similar concept would be like a dream plane . . . and either a nirvana or nightmare realm . . . or both, depending on how one set up the rules.

This would only work if there was only one observer and it was God.

Everything would break down as soon as another observer was added.

Mathematics (and logic as a branch of mathematics) is an "absolute" science, in the sense that it is not bound to experimental verification. Physicists keep verifying Einstein's relativity theory, no mathematician keeps verifying Pythagoras' theorem.

In physics or any other science one makes an hypothesis, design an experiment to falsify that hypothesis, and based on the outcome accept or reject the hypothesis. This endlessly goes on. I.e. Newtonian gravity works well if it is about throwing stones at your enemy. It fails when it comes to control the timing of GPS satellite signal.

In mathematics one starts from axioms, which are a set of statements which are taken as true (like Euclid's five postulates founding Euclidean geometry), and from them derives/proves a set of theorems. Those theorems are true within the boundaries defined by the axioms, and don't need falsification. I.e. in Euclidean geometry the sum of two sides of a triangle is always bigger than the third one. That's proven and independent from the observer and the precision of the observation. There is no observer in mathematics.

I can think of two sets of examples.

There is one such kind of world and you go there everyday, even if you don't remember. We call such worlds dreams.

In a dream, you may find yourself light as a feather and yet be stuck to the ground as if by great weight, while everyone around you is moving normally.

In a dream, you may accelerate in the vacuum of space without the need to expel reaction mass.

In a dream, your clothes may quantum-tunnel themselves off of you when you are about to speak to a crowd.

If you wish to see some really creative works related to such worlds, I highly recommend a series of comic books called The Sandman, written by Neil Gaiman and published by Vertigo.

The image above is XKCD #52

Also, magic.

True magic is different from science, in the sense that it is not some technology which we cannot understand. It is the modification of reality through the mystic/"occult". In many works of fiction, the engine that channels, powers and drives such magic is the mind.

Consider Marvel's House of M, where the Scarlett Witch one day got tired of whole struggle of mankind vs. mutants. On a whim she said "no more mutants", resulting in ~90% of them losing their powers.

In the world of Marvel comics, there are other beings who can reshape the world with their thoughts: Lord Chaos and Master Order, the Living Tribunal, Shuma Gorath, the Beyonder, the Shaper of Worlds. Dr. Strange is usually either dealing with those directly or being consulted by others about those beings.

We use Lagrangian mechanics to identify a set of shrewdly chosen coordinate axes on which a seemingly intractable problem becomes static.

Assuming there was some Lagrangian where some extremely cleverly chosen axes of mental state were stable, what might that look like? I'm thinking that things like The Heroes Journey and TV Tropes - libraries of a general "shape" nearly all of humankind's stories take - would probably fit.

What would this look like? It would be a world where bad guys always wear black, are often somehow disfigured. Good guys generally wear white. Good people are beautiful and healthy. The solution to almost every problem requires going on a journey and facing some obstacle. Teachers are wise, but most times impossible to understand. And so on.

In literature, Torg tried to do this explicitly with their pulp reality The Nile Empire. Games Workshop' Chaos has touched on these ideas, but almost exclusively looked at the monsterous aspect of it. Not surprisingly, since GW admits taking a lot of ideas from him - Michael Moorcock's Eternal Champion explores living in a world like this, where you might or might not (hard to tell) be the observer in charge. And in modern theater Inception had semi-stable traversable internal worlds of the mind, that followed externally predictable rules as viewed from a carefully chosen perspective.

As stated in other answers, some of the ideas from your question are impossible due to the different points of view, beliefs and thoghts of everyone, for example:

• 
  A world where all the observer wants is true: if I want to be rich and someone feels envy from my wealth, desiring my poverty, the physics of this world would be in conflict.

But... this leds to a new idea for me, where you could complete the "laws" of this universe stablishing some priorities, like some "karma gravity" making the "good" people (in a global moral/ethic way) to get their desires come true prior to the desires of an "evil" man with opposite ideas. "Karma" could be changed by age, weight, etc.

Similar things happen in the following ideas:

• A world where all the observer fears is true




• A world where all the observer believes is true

Every universe where everyone can define/created/generate changes, matter, feelings by their minds will (unconsciously or not, represented by fears, desires, ideas, etc.) will suffer by the opposition and the non-coincidence of the different people thinking.

Finally, we left with "the world where all the observer believes is false".
In my mind, this option you propose can be perfectly plausible, as René Descartes stated:

"I suppose therefore that all things I see are illusions; I believe that nothing has ever existed of everything my lying memory tells me. I think I have no senses. I believe that body, shape, extension, motion, location are functions. What is there then that can be taken as true? Perhaps only this one thing, that nothing at all is certain.", Meditations on First Philosophy.

In our universe already, we can suppose thet our beliefs are false.

In order to comment an example, we never live the "present itself". We are always reacting to what happened a few microseconds (if we are looking at something a few metter far from us), a few minutes (properly, eight minutes if we look for changes on the Sun surface) or hundred of years (300 years, approximately, if we look at Polaris star in the night sky) back in time. We will never know what happening in "real precised" real time.

We can also asume that our senses are not perfect. Is our blood really red? If nobody tells the colour blind people about our different (and more frequent) perception, they would defend that blood is blue. Are we all colour blind compared to another race? Who is the truth owner?

It should not be difficult for the writer/story designer to make that the reality everyone perceives is wrong (partially at least). Every mistakes/misbeliefs could get assembled all as an entire lie for all. What makes me remember a couple of examples...

I hope you can find inspiration to create your world with this.

Good luck! And don't let your sense lies to you... to much.

TL:DR;

One of Einstein's students at Princeton (Kurt Godel- he's kinda famous) showed that every system of logical, well defined rules has truths about it that can't be proven using its own logic. Every system of math and logic needs a different system to fully describe itself according to Godel's Incompleteness Theorem. There's an infinite number of worlds that have made these different systems, and thus different discoveries about these systems.

I'm going to be brutally honest and say your examples make no sense to me, but in terms of mathematics it's been formally proven that you can define an infinite number of unique worlds with different (but totally valid) logic and ways of doing math because none of them will be a complete description of all logic.

If you're a maths junkie, there's actually a great example of this in the real world. A renaissance mathematician (Gerolamo Cardano) was competing for a place as a mathematics professor at the University of Pavia (it was the Harvard of the time). To get this special place, you had to best the current professor in a geek-off of mathematical problems. To win, Cardano invented a whole new system of numbers so he could solve 'previously impossible' cubic equations- imaginary numbers.

Anyway, my point is that this development led us to understand so much more about 'normal numbers' (0.7, 1, 5/2, etc.) that we would be decades behind today if we hadn't. Imagine if we'd started off with complex numbers, and used different systems to describe them- what world would we be living in now? Logic is surprisingly subjective.