Let's ask what I consider to be an
interesting question; this is a bit of
history of mathematics.
Does every vector space have a basis?
I'm going to state a theorem
Theorem: every
vector space has a basis
even
the infinite-dimensional
ones. That every finite-dimensional
vector space has a basis is clear. We have
a theorem that says any linearly
independent set of vectors can be
expanded into a basis, so take a single
vector, a single nonzero vector:
that's a linearly independent set, and we can expand
it into a basis.
Second statement is a lot less trivial
For the second statement
we'd have to expand our definitions, I
mean our current definition of linear
independence only really works if we
have a finite number of vectors, but
that's easily done, that's not what gave
people trouble.
What gave people would trouble
was that the argument
is non-constructive, using some thing
called the axiom
of choice.
We're not going to go into a bunch
of set theory here, but the axiom of
choice is one of the ZFC axioms, the
now-standard
axiom system for set theory; in fact it's
so important in this axiom system that
it gets a letter entirely to itself.  The 
Z of ZFC is a name, Zermelo, and the
F is also a name, Fraenkel,
but the C stands for choice. And for
several decades during the last century,
there was a major disagreement among
mathematicians about whether the axiom
of choice should be accepted as an axiom
of set theory, because the thing is that
all arguments that use the axiom of
choice are non-constructive.
What I mean by that is that the axiom
tells us that
some thing exists
but
does not
tell us what it is or how
to describe it, so you can sort of see
why this might have been a little
controversial, because what's the point
of knowing something exists if you can
say literally nothing about it?  And in
particular the axiom of choice ran
directly enough in the face
of a school of mathematical philosophy
called constructivism which says that
mathematical objects
should only be
viewed as real if they
can be explicitly
described (or "constructed," hence the name).
constructed. You can find
constructivists today, but most
mathematicians are not constructivists
in this day and age--the great David
Hilbert said once that doing
mathematics under constructivism is like
trying to box without using your fists,
and I think it's fair to say it that the
vast majority of working mathematicians
accept the axiom of choice with very
little hesitation. And once you have the
axiom of choice you have this.
I share this theorem and these historical details because I find them interesting;
I will say that in this
particular class we are only going to
look at finite dimensional vector spaces.
