Escape the Plane

Question

One Sunday morning, you awake to find yourself completely alone on an infinite, flat plane. You don't remember much about the night before, other than that you may have pissed off a wizard. Next to you, you find a palette with countably infinite colors, and a note, commanding you thus:

You must paint every point on this plane, such that I will never be able to find a triangle with vertices of the same color and rational area.

If you can manage this task, the wizard will let you go free - fail, and you're trapped forever. You don't doubt the wizard's abilities, so no cheap tricks here. Considering the problem, you get to work - and an uncountably infinite amount of time later, the wizard stands beside you, admiring your handiwork.

Does the wizard set you free?

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EDIT: To eliminate lateral thinking answers based on the framing of the question, here's a formal mathematical statement of the puzzle:

Does there exist a coloring of $\mathbb{R}^2$ such that it is impossible to find a triangle with vertices of the same color and rational area?

Answer 1

Very interesting supertask.

In a 2D plane, any three noncolinear points make a triangle, so only use 2 points of each color. Because you have infinite colors, you will never run out of colors. However, this doesn't save us from our demise, as this task would take an uncountable amount of time, leaving us stuck in the plane. So, we have to approach this like a supertask. Paint the first dot in 1 minute, paint the second dot in half the time, paint the third in half of that of the second, etc. In just two minutes, and however long the wizard needs to check, you will be free from the plane!

Edit:

The above solution runs into the problem that you run out of colors because there is an uncountably infinite number of points on $\mathbb{R}^2$ and there is a countably infinite number of colors. I can get a bit closer by increasing my number colors. Instead of thinking of colors as the discrete paint splotches the wizard has given me, I'll now consider the wavelength of light that the pigment reflects (totally disregarding how paint mixing works here). Now, in each step of the supertask, mix the paints such that you get a new color (e.g. in step 1 you use paint with $700nm$, in step 2 you use paint with $700.\bar01nm$, etc.). Now you have an uncountably infinite number of paint colors. However, I feel as though the plane is full of infinite 2 dimensional points $\mathbb{R}^2$, while I only have paints with $\mathbb{R}>0$, so I still don't have nearly enough colors.

Answer 2

Surely this is simple

You have an infinity of colours so you use each colour only once. You haven't specified whether the 'points' are true points. If they are true points on a plane then they have no dimension so you can't paint them. Not even 1 molecule of paint can be used.

or

If you can manage this task, the wizard will let you go free - fail, and you're trapped forever.

Given that the task will take forever, you are trapped forever doing the task, no the wizard will not set you free.

or

You paint infinitely-long parallel single-colour straight lines. There will be no triangles having three vertices of the same colour.