I must be misunderstanding bijection on uncountable sets. (As it relates to stereographic projection) At least one of these things must be false:


*

*The existence of a bijection between two sets implies that those sets have an equal number of elements.

*A bijection exists between {all points in a sphere} and {the cartesian plane + one point}

*There are as many points in {a sphere} as {a cartesian plane + one point}

*There are more points in {a cartesian plane + one point} than {a sphere}

*Two sets A and B have the same number of elements if A doesn't have more elements than B, and B doesn't have more elements than A.


Which one(s) are false?


*

*I thought this property is what makes bijective proofs possible (do bijective proofs only work for countable sets or something?)

*Stereographic projection

*Stereographic projection

*(This is a sketch)  Take any point on the surface of a sphere of diameter D with coordinates (x, y, z).  If z <= D/2, map that point to (x, y) on the plane. if z>D/2, map that point to (x + D + 1, y).  This maps all points in the sphere to the plane.  Consider also that the plane includes the point (x + 2D + 1, y), which is not mapped to the sphere. (In other words, smush the bottom of the sphere into a circle, and the top half of the sphere into a second circle.  You now have the rest of the cartesian plane free.)

*My best guess at a vaguely sufficient definition of "equal".

 A: This answer does not really cover the questions you ask, because I believe the problem to lie somewhere else.
In my opinion, the most problematic part of these questions is the notion of "number of elements". Let us take for another example the natural numbers $\mathbb{N}$ and look at the map $\mathbb{N} \rightarrow 2\mathbb{N}, x \mapsto 2x$. This is a bijection. If we could talk about the number of elements $\mathbb{N}$ like in the case of finite sets, then this would mean that $2 \mathbb{N}$ and $\mathbb{N}$ have the same number of elements, even though $2 \mathbb{N}$ is a proper subset of $\mathbb{N}$. I recall that some people even define a set to be infinite if it has a bijection onto a proper subset. So when we talk naively about the number of elements of an infinite set, we would have to come to the conclusion that every infinite set does have simultaneously the same number of elements as itself, more elements than itself and less elements than itself, which is not what we would like to have.
Instead of talking about number of elements, the "right" notion is the one of cardinality. We say two elements have the same cardinality, if there is a bijection between them. This reduces in the finite case to the idea of "same number of elements" but makes much more sense in the infinite case. One thing we have to keep in mind is that an infinite set does have proper subsets which have the same cardinality as itself. If we replace "number of elements" with cardinality then 1.,2. and 3. are true for the reasons you found, 4. would be false and 5. is, I think, in some way a formulation of the Schröder-Bernstein theorem.
A: There is a bijection between the sphere, $\mathbb{S}^2$ and the plane $\mathbb{R}^2$. This is true. There is also a bijection between the sphere and the plane union a point. The definition being confused in this case is not the definition of bijection but the definition of uncountable.
The tricky part is that there are too many points for uncountable sets. All intuition has to be thrown out of the window here. For example, there is a bijection between $\mathbb{R}$ and $[0,1]$. But how can this be?? There are infinitely many intervals in $\mathbb{R}$.
The defining characteristic that makes this possible is the uncountable cardinality. In purely set-theoretic terms, these sets all have the same cardinality but you are right, they are very different sets indeed. Clearly we cannot define sets to be equivalent based on their cardinality, as these sets may have vastly different properties as experienced in the above examples! Each branch of mathematics has a pretty unique way to classify such objects and these can be seen pretty evidently in the field of category theory. But I am slipping passed the point of the question. If you want, I would recommend further reading from "Set Theory and Metric Spaces by Irving Kaplansky".
A: As often happens, the false statement is the one you didn't state:


*The meaning of the word "more" is obvious, even for infinite sets.


You seem to be using the following definition of "more":


*

*$B$ has "more elements" than $A$ if (and only if) there's a function from $A$ to $B$ that's one-to-one but not onto.


For finite sets, this definition is very sensible, and it's equivalent to the following other sensible definitions:


*

*$B$ has "more elements" than $A$ if (and only if)...


*

*... there's no one-to-one function from $B$ to $A$.

*... there's a function from $B$ to $A$ that's onto but not one-to-one.

*... there's a one-to-one function from $A$ to $B$, but no bijection between $A$ and $B$.



For infinite sets, the situation is very different. There are many reasonable definitions of the word "more," but they don't all agree with each other, and they don't always behave the way you might expect!

For example, according to your definition, the set of natural numbers has more elements than itself, because the function $f(n) = 2n$ is one-to-one but not onto.
Using this fact, you can prove that although your 5th statement is true for finite sets, it's false for infinite sets when interpreted using your definition of "more," the standard definition of "the same," and the understanding that when you said "if," you meant "if and only if." To see why, let's first remember the standard definition of "the same":


*

*$A$ has "the same number of elements" as $B$ if (and only if) there's a bijection between $A$ and $B$.


There's clearly a bijection between the set of natural numbers, $\mathbb{N}$, and itself—namely, the identity map. Therefore, $\mathbb{N}$ has the same number of elements as itself. On the other hand, according to your definition of "more," $\mathbb{N}$ also has more elements than itself. This contradicts your 5th statement (assuming, again, that the "if" in that statement was supposed to mean "if and only if").

To get used to the subtleties of what "more" means for infinite sets, try playing with the definitions of "more" mentioned above.
First, pick one definition to work with. Using that definition, can you find sets $A$ and $B$ with the property that...


*

*... $B$ doesn't have more elements than $A$ and $A$ doesn't have more elements than $B$?

*... $B$ has more elements than $A$ and $A$ has more elements than $B$?


Next, try comparing definitions. Can you find sets $A$ and $B$ with the property that...


*

*... $B$ has more elements than $A$ according to one definition, but not according to another?

*... $B$ has more elements than $A$ according to one definition, but $A$ has more elements than $B$ according to another?

A: Define 'more', 'less' and 'equal':
$$|A|\geq|B|:\iff A\twoheadrightarrow B$$
$$|A|\leq|B|:\iff A\hookrightarrow B$$
$$|A|=|B|:\iff A\leftrightarrow B$$
By the axiom of choice it is symmetric:
$$|A|\geq|B|\iff|B|\leq|A|$$
The last statement is Schröder Bernstein:
$$|A|=|B|\iff|A|\leq|B|\land|B|\leq|A|$$
Now, the first statement is nothing but the definition of cardinality:
$$|A|=|B|\colon\iff A\leftrightarrow B$$
The second statement requires the stereographic projection:
$$\mathbb{S}^2\setminus\{N\}\leftrightarrow\mathbb{R}^2\implies\mathbb{S}^2\leftrightarrow\mathbb{R}^2\cup\{\infty\}$$
The third statement is a rephrasing of the second:
$$|\mathbb{S}^2|=|\mathbb{R}^2\cup\{\infty\}|$$
The fourth statement is wrong since:
$$\mathbb{R}^2\cup\{\infty\}\leftrightarrow\mathbb{R}^2$$
Concluding that the sphere has uncountable cardinality:
$$|\mathbb{S}^2|=|\mathbb{R}^2|$$
