The natural numbers are pretty familiar.
But if I asked you to tell me what natural numbers are
and how they work without using the notion of number
in your answer, could you do it?
In the late 19th century, mathematicians
were on a quest to put all of mathematics
on firm logical footing.
Kelsey talked about this in a previous video,
where she mentioned, among other things,
that the number system can be constructed
hierarchically starting from natural numbers
and working your way up.
But what about the natural numbers themselves-- zero,
one, two, three, et cetera?
Are they fundamental, or can they also
be constructed from more basic ingredients?
Before we can tackle that question,
we need a firmer handle on what exactly
it means to be a natural number.
More specifically, we need a way to characterize
the essential features of natural numbers
without referencing directly or indirectly things
that we already know about the natural numbers,
because that would be circular.
Now it turns out that you can capture
the essence of numberhood in a small set of axioms,
analogous to Euclid's axioms in geometry,
that will allow us to build a set N that behaves just
like the natural numbers without ever explicitly mentioning
numbers, or counting, or arithmetic as we do so.
These axioms were first published in 1889 more or less
in their modern form by Giuseppe Peano
building on and integrating earlier work
by Peirce and Dedekind.
I'm going to show you how this is done.
But to appreciate what a massive achievement.
It is, I want you to try doing it yourself first.
Just casually try describing numbers
without mentioning numbers or even
inherently number-ish concepts, like one, or plus, or next.
You'll find it's pretty hard to do.
First, some preliminaries.
To make this somewhat self-contained,
I need to assume your familiarity
with some basic logical infrastructure that
is not specific to numbers.
First of all, you should be familiar with some ideas
surrounding sets, namely that the members of a set
are called elements, that two sets with an identical roster
of elements are really the same set,
and with the idea of a subset of a set--
in particular, the fact that every set
is a subset of itself.
Second, we need a concept of equals.
We'll be very bare bones about this,
requiring only that whatever equality means
the relation be reflexive, symmetric, and transitive.
And we'll also demand that the set
N we're trying to build be closed under equality.
In other words, if x is a thing in N, and x equals y,
then y is also a thing in N because, of course,
x and y are the same entity.
Now that last part may seem self evident,
but when you're laying out mathematical logic,
it's best not to leave any assumptions unstated since that
makes it harder to track which parts of an argument
actually depend on which other parts.
And finally, we'll also rely on the concept of a function,
or a map-- i.e.
a rule that associates an output with one or more inputs.
Armed with these basic tools, let's
introduce Peano's axioms incrementally
so that you can see exactly why each one needs to be there.
Now warning-- some parts of the argument are subtle,
and you may need to watch this more than once to keep up
with what's going on.
Let's begin.
Axiom one-- Zelda is an N. This is basically a declaration that
set N isn't empty.
Zelda is in there.
Who or what is Zelda?
Don't worry about it.
It'll turn out to be the zero that you know and love.
But I'm trying to use nomenclature here
that's not too suggestive so that you
won't be tempted to bring in preexisting concepts.
Remember, the goal is to arrive at numbers
without referencing numbers.
Now, to populate the rest of N, we
can just keep including the next element.
But what is next?
The whole notion of next is loaded with conceptual baggage
that can be hard to disentangle from your preexisting notions
of counting and numbers.
So that's what the remaining axioms will do.
They will define next using only more
basic and non-numerical ideas.
Let's see how that works starting with axiom two.
There exists a function S whose inputs and outputs
are elements of N. In other words,
if x is an N, then S of x is also in N. Now, on it's own,
this doesn't bias anything because we haven't specified
the mechanical rule underlying S. For instance,
suppose that Zelda is in the set,
and it turns out that S of Zelda equals Zelda.
This would still satisfy axioms one and two,
but it looks nothing like the natural numbers.
So let's fix that.
Axiom three-- Zelda is never the output of S. More
precisely, for any x in N, the statement S of x
equals Zelda is always false.
Now, this is progress.
Axiom two says that S of Zelda is in N,
and axiom three says that S of Zelda
is different from Zelda itself.
But notice that having S of S of Zelda turn out to be S of Zelda
would still be consistent with the axioms so far,
and that would bring our building process to a halt.
So we need to prevent that possibility.
Enter axiom four.
If x and y are in N, and S of x equals S of y, then x equals y.
In other words, different inputs to S
will always give different outputs.
Now, finally, we are guaranteed an infinite chain.
Watch.
Zelda is in our set by axiom one.
S of Zelda is in our set axiom two,
and it's different from Zelda by axiom three.
S of S of Zelda is in our set by axiom two,
and it's different from Zelda by axiom three,
and different from S of Zelda by axiom four.
Now we keep using axioms two, three, and four over and over.
So s of s of s of Zelda will be in the set,
and it's different from Zelda and from every element we've
already mentioned and so on and so on.
This is starting to look like an infinite list of counting
numbers.
If we introduce some shorthand names
for these elements borrowing Arabic numerals,
like 0 for Zelda, and one for S of Zelda,
and two for S of S of Zelda, and so on,
it will look exactly like the counting numbers.
Now, what about the function S?
Notice that we never specified its internal mechanics.
We just enforced some constraints
on its behavior that apparently capture the intuitive
notion of next, or successor, ergo the S.
And we used as ingredients only the abstract concept
of a function and the idea of equals.
So we're done, right?
Well, not so fast.
There's a subtle loophole in axioms one through four.
Suppose that we throw two other characters into N. Let's call
them Mario and Luigi.
And suppose it turns out that S of Mario
is Luigi and S of Luigi is Mario.
Tacking on these plumbers with this weird S behavior
doesn't violate any of the axioms so far,
but it does leave us with a set that
looks somehow bigger than the familiar natural numbers.
So axioms two through four apparently
do not quite capture the idea of next
because they would allow Mario and Luigi to be in the set,
but detached from everybody else so to speak.
To prevent this, we used Peano's fifth axiom also known
as the axiom of induction, which cleans up this mess
by fiat in a very subtle way.
Here's axiom five.
Suppose that T is any subset of N that
satisfies two properties-- first that Zelda is in T and second
that if x is in T, then S of x is also in T. Then
by declaration, T is the set N. In other words, axiom five
declares N to be the minimal set that will satisfy
axioms one through four.
Now since the infinite chain we already
have fits that bill by axiom five,
that set is N. We are done.
And adjoining Mario and Luigi along with their weird S loop
would give you something different from N.
So what does it mean to be a natural number?
It means membership in a set that
is endowed with the notion of successor,
or next, and one of whose elements
has the privileged status of not being anybody else's successor.
Now, the miracle of Peano's axioms, in my opinion,
is the revelation that next is not a fundamental concept,
that it can be reduced to more basic logical ideas.
And to me, the demonstration that this can even
be done at all is one of the most beautiful arguments
in all of mathematics.
Now, for some loose ends.
First, what about arithmetic?
It turns out that you can define both addition
and multiplication solely in terms of the successor function
S. Each of those operations, in fact, can be viewed
as a function that takes two inputs from N
and yields one output in N. And they can be defined recursively
by the following rules.
Now, I don't have time to do it here,
but it is a fun challenge to play around with these rules
in order to derive simple facts like one plus two equals three,
or that addition and multiplication are
commutative and associative, or that they
obey a distributive rule.
Super fun.
Second loose end.
We arrived at a notion of successorship,
and we built a set that contains entities related to one another
by iterated succession.
But we never specified what those entities actually
were with Zelda, or zero, or any other members of N
actually are.
We just asserted that here's a thing
and here's some other things, and they're all in this set N.
But we're leaving open the question,
what are those things?
Are the irreducible fundamental ingredients
in the mathematical pyramid?
Or can even these entities, the natural numbers,
be built out of simpler entities in the same way
that the integers get built from the naturals?
We'll answer that next time, and I'll see you then.
Some cliffhanger, huh?
Don't worry.
You're not going to have to wait a full week for the resolution.
Our next episode about the natural numbers
will be coming out in just a few days,
so be on the lookout for it.
And at the tail end of that episode,
I'll be responding to your comments
from the Telling Time on a Torus episode,
and I'll be announcing the winners of the torus clock
challenge, so tune in.
But in the meantime, I did want to give you
a brief update on the status of the metallic ratio
challenge, which I posed in this episode
that you should go watch if you don't
know what I'm talking about.
Here's the update.
First of all, a lot of you have submitted code or results
of code that shows numerical checks
for very high sided polygons and very high end metallic ratios
to see if the diagonal to side ratio on any polygon
ever comes out to be any of the metallic ratios.
I think some of you have checked up
to 100 million sided polygons and very high end metallic
ratios with no affirmative results.
That strongly suggests that there
is no metallic ratio above the silver ratio that ever
occurs in any polygon at all.
But that's not the same thing as having a proof that either
this never happens or finding that one that elusive metallic
ratio where it does.
So, so far we're not there.
Now, some of you have submitted ostensible proofs
of why no metallic ratio above the silver ratio
will ever be the diagonal to side
ratio in any regular polygon.
I'm still going through all these responses individually.
I'm pretty backlogged over here.
But once I've managed to check them all,
before I'd name names and give away any t-shirts or something,
we want to make sure these things are correct.
I'm going to look them all over.
I'm going to have Tai-Danae look them over.
And once we know what's going on, you will know.
We'll announce that in the comments
of some future Infinite Series episode.
Stay tune.