Suppose there are two triangles on a plane. The coordinates for each point of both triangles are the same.
It seems to me that there is nothing to differentiate these two triangles, and as such, they must be identical. Therefore, there can't be more than one such triangle on the plane.
I might describe two congruent triangles positioned such that either can rotate to overlap the other one perfectly. In that case, it seems that I could say that the two triangles are identical after the rotation.
I'm in first year, so I can only speculate about how set theory represents geometry, but I have a hunch that something would disappear from a set after the rotation, which would decrease the cardinality of the set. However, if there were only one triangle, I suspect that you could rotate it all you want without affecting the cardinality of a set. It would seem very weird if rotation could end the existence of a triangle in some circumstances, but not in others. Then again, perhaps that's just a consequence of the rules.
Nevertheless, I can imagine an instructor asking a student to 'rotate the triangle so that it exactly overlaps the other one, record the angle of rotation, dilate it by a factor of x, and then sketch the result. In this case, the instructor seems to talk about the plane as though it includes two identical triangles, even after the rotation; it also seems that the correct sketch would include two triangles, one inside the other.
Can a plane include two indiscernible triangles?
Best Answer
Let me first address how set theory represents geometry.
The same way the CPU in the device you are using to write this question interprets the HTML data from the website. Not interesting in details whatsoever. Set theory allows you to interpret the real numbers, and so on and so forth. It's really quite standard, and you can find it on this website on several threads. The point is that set theory interprets geometry in a way that ensures that whatever we prove "naively" without regarding the set theoretic underlying structure will be true in the interpretation. The same way that basic HTML is displayed correctly (modulo available fonts) on my Linux desktop boasting [an old] QuadCore processor, and my Nexus 4 with its ARM processor.
Now you have to ask yourself. What is a triangle? Is it a subset of $\Bbb R^2$? Is it a more abstract object? In the former case, congruent triangles are not the same, exactly in the same way that $[0,1]$ and $[2,3]$ are not the same, despite having the same length as line segments.
This means that when moving and rotating a triangle you consider a different triangle which has different elements as a set, but not different geometric properties.
If you consider triangles to be more abstract objects, not represented by sets on the plane, then the answer will depend on how you consider them. Are two congruent triangles the same? Is moving a triangle from one point on the plane to another gives you the same triangle? If the answer is yes, then there is only one triangle of each "type". If the answer is no, then it depends on your definition of a triangle.