The Dedekind Cut

Dick Dedekind chilling in Deutschland
making the paper from the analysis class
handled the handle of Diffy-Q
and he knew that his crew in Göttingen had swass
but something was bugging him 'bout the fundamental theorem
after taking a proof syrum, the problem was clear to him
"this basis on geometry is totally loopy!
my proofs are all soupy like someone slipped me a roofie!
Diffy-Q don't toe the straight line!
I need a foundation that's more sublime and refined!"
his friends were all, "Diff, you're wasting your time
it's cool, just look at the line and everything's fine!"
but he declined, "the line is like hieroglyphic!
I need an explanation that's more scientific!
your assumption's got gumption, too much presumption
how can you prove it's a continuous function?
you only suppose what you say is true
and to assume makes an ass out of ich und du!"
butcher paper spread, dipping the quill
tappin' the skill mathematical
to make this analogy defensible
irrationals are like the incommensurables
will 'rithmatic comprehesible
and here's just the thing that'll fit the bill

The Dedekind Cut!

now, what is up with a Dedekind cut?
Here's a very concise explanation, but
you're gonna have to listen to a little but of math
so put on your thinking caps, class
every fraction affects a separation
of the rationals into two classes,
those less than and those greater than it
take that fraction, make it an element,
either the greatest of the former or the least of the latter,
and now you know exactly just what is the matter
you can do real analysis and know your stuff
'cause sets like that are a Dekekind cut
kvetching Kronecker, "that's a completed infinity!"
"that's exactly the structure I meant it to be!
you've got the rational numbers dividing the sets
giving birth to the reals like I was having sex
with a calculator boasting an infinite register!
hook me up on that date and I straight jump in bed with her!"
three thorough theorems proving the cuts to be
ordered like the reals, possessing continuity
which proves Dedekind's unbounded ingenuity
proving Diffy-Q is crushing all y'alls' booties