I'm a mathematician. There are counting
numbers and we always start counting at
zero. So zero, 1, 2- in school you do
this and you have rationals and reals
and things, but we'll just put the
integers for now. These are the things
that we learn in school to add and
multiply; we notice that there's not a
biggest one. If I ask you, can you find a
bigger integer than 100? - (Brady: 101)
101, excellent. How about a million? - (Million and one!)
Pretty good! So for any n in the
natural numbers; this is this blackboard
boldface, that's its name, N. For n in N
there is, let's call it k, bigger than
n, agreed? You just added 1 to the
numbers I gave you, you can always do
that, but what about if we want a really
big k? So what about a k, bigger than n,
for all n? And so there's not anything
in this list here of natural numbers, or
our imagined list that goes on forever.
There's nothing there that's bigger than
all the others, because you can always
add 1. But, I'm a model theorist; we
don't just look at N we look at other
number systems, and there's one in which
we have a k that's bigger than all of
these. Out here, let's put it here. I'll
call it N star. I'll just say, it's
another version of N, it has a lot of the
same properties of N but we can give
ourselves a number out here which is
bigger than all of these at once. So
we're greedy or we're lazy or we say, why
not? Because we're mathematicians. Of
course it's not bigger than itself, it's
not bigger than k plus 1, but for the
ordinary natural numbers it's bigger.
(Brady: But it's still a natural number?)
It's still a natural number and so it has lots of the
properties of natural numbers. Now the- we
can do this because we're model
theorists and we use something called a
compactness theorem. But there's a cost,
we have to be careful what we can say
and what we can't say.
But we can say about these infinite
integers, and they're more, I'll get to that
in a second. We can say about these
infinite integers whether they're even
or odd; whether they're prime; whether
they're squares - all of those are
properties of ordinary natural numbers
and so you can ask that about your new
and wonderful infinite one. - (Brady: So hang on-)
(is this infinity?) - They're all
infinite, they're bigger than all of
the finite numbers. But what- you know,
when people say The Infinity that's not
what we're talking about. There are lots
of kinds of infinity that the set theorists
talk about, but even here we're
talking about just a nice listable,
countable thing with a lot of elements
in it that are bigger than all the
natural numbers. And if this were even,
for example, what does it mean to be even?
You say an integer is even if it's equal
to twice another one, yeah? Well then
what about L? L is infinite too, it sits
here somewhere.
It's less than k, because k's double it,
but it's out here, it's bigger than all
of these. - (Brady: If I start with k and start)
(counting backwards, where am I going to)
(end up?) - You do it forever.
k, k minus 1, k minus 2, k minus a million - all infinite.
(Brady: And I'll never get back to)
(that line that's got N on it?) - No, never.
Any more than you ever end here as you
go up 1. Going up one never takes you, a
step at a time, outside N. So there's
something else going on that allows us
to have k. Actually I'm going to draw the
number line now, the the real numbers
let's say or the rationals, it doesn't
matter. So let's think about all the
numbers like pi; pi should be somewhere
about here right? Square root of 2 goes
in here. So in school we learn this line
and- you know, all the real numbers. This has an
ordering too right? We have positive
elements on the right, negative on the left;
these have square roots, these don't. But
we can think about the ordering and say,
okay I want something which is bigger
than all the integers! Just mention the
integers again
and here we go with k, it's out here. We
can play the same game, starting with the
real numbers, and get an infinite real
number. And of course it's got all its
buddies, it's got a whole copy of this as
buddies and more. Because what about now,
before we divided by 2, if k was even?
But in the real numbers all the non-zero
elements, and k is certainly non-zero, all
the non-zero elements - you can take their
reciprocals, you can divide by them. So
what happens with one over k? - (Brady: So this is-)
(this is almost like 1 divided by)
(infinity almost?) - It almost is. And so 1
over k - where does it fit in this line?
Well it's not out here, actually minus k is
out here, you can think of that. It's
beyond all the negatives, the regular
ones, these are the ordinary reals - k is
here. 1 over k is positive and it fits
infinitesimally close to 0 - positive. So 1
over k is smaller than 1 over 3, 1 over 4,
1 over 5, 1 over n and so forth.
(Brady: It's just getting tighter and tighter into that zero?)
- It's an infinitesimal; which is
something 19th century mathematicians
and some current physicists like to have.
It's infinitesimal- and so around every
around zero we have what I think some of
the philosophers called a monad; which is
a whole little interval of elements
which are infinitely close.
There's the monad, if you will. So they're
all things like 1 over k, or in our
earlier line 1 over- 1 over 1/2 k.
(Brady: 1 over 2k, 1 over k-) - All of those, they're all
there. They're all in there. - (Brady: If they're all)
(infinitely close, are they occupying the same space?)
No. No,
there's room. There's room. We just put
them all in there and they're in an
order; like 1 over k is going to be
bigger than 1 over k squared, for example.
(Brady: So 1 over k-) - So the ordering out here is
just flipped when you put it here.
(So 1 over k squared is closer to the)
(zero than 1 over k?) - Although they're both
infinitely small, yes.
[Extras:] One of the things that's fun to do,
either with N or the Reals, is a sort of
thought experiment thinking, well okay
are there infinite primes? Are there
infinite integers that are the products
of exactly two primes? Are there infinite
integers that are powers of two?