---
Video Title: Is Zero a Natural Number?
Description: Theory of Computation
https://uvatoc.github.io

2.2: Is Zero a Natural Number?
- What Makes a Good Definition?
- Definitions of the Natural Numbers

David Evans and Nathan Brunelle
University of Virginia
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What I want to talk about that will drive
a lot of the rest of today is this

question that was on the registration
survey, is zero a natural number?

This turns out to be a
very controversial question,

and it's a pretty
interesting one to look at.

Based on your surveys, these are the
answers.

So we definitely had a split opinion.

We had a majority saying no, but not a
veto-proof majority.

So we don't want to decide mathematical
questions by non-veto-proof majorities.

And I think both Nathan and I have vetoes,
and we'll veto the no answer here.

It doesn't necessarily mean it's wrong.

How do we actually answer this question?

The first thing I'm going to say,
I'm glad more than 10% give the huh

answer, which I'm
interpreting as the answer

to this question follows
from your definition.

To answer whether
zero is a natural number,

you need a definition
of a natural number.

And we haven't given you one yet.

So you can't answer the question.

The huh answer is actually the best one,
I think.

And especially for a term like natural
number that is disputed.

Many terms in mathematics there is near
universal agreement on.

Natural number that you'll find different
books that use different definitions of it

that would lead to different answers to
this question.

If we want to answer questions like this
and know that we have the right answer,

for them to be meaningful, we need to have
real definitions.

We have to have definitions that are
precise.

What is it that we need for a definition
to be good?

Oh, I see why it has 11 apples.

Okay.

Apply.

I think you meant apply.

It's a typo.

Okay.

It's probably an auto-correct problem.

It's a good answer even without the
apples.

Okay.

Good.

Makes sense to its audience.

Okay.

I hope I've noticed.

I don't know what UWE means, even though
it's got 12 votes.

But should.

I know what UWE means?

I don't know.

Uh-oh.

I don't put anything
offensive, especially

if it's something that I
wouldn't understand.

Okay.

Good.

Okay.

So I think we've got a bunch.

Hopefully I didn't miss any of the big
ideas that are there.

Let me ask questions about a few of these.

When we say a good definition should be
precise, what does that mean?

What would make a definition precise
versus one that's imprecise?

A table is an apple.

Are you defining a table or an apple?

Well, either, really.

So a table is an apple sounds very
precise.

It doesn't match my
understanding of what a table or an

apple is, but it sounds like
a pretty precise definition.

Right?

It's saying tables and apple.

That means all apples are tables.

All tables are apples.

It's a very precise definition.

It does not correspond to our most
people's familiar notion of tables and

apples, but it's otherwise a good
definition.

Yes.

Given a set of things that
may or may not follow the

definition, a precise definition
will let you partition that.

Good.

Yeah.

Excellent.

Yes.

If we're defining something, right,
that's saying we're going to divide the

world into things that satisfy this
definition and things that don't.

If it's a precise definition,
we should be able

to do that in a way that
everyone agrees on.

And that's exactly the question about the
natural numbers.

That's exactly that kind of question.

Right?

It's saying if we have a
good definition of natural

numbers, anything anyone
gives us, we can say yes or no.

Does it satisfy that definition?

Is it a natural number?

Right?

And that's where zero is one of those
things.

I think things, something like seven,
I think you would have all said yes and

agreed that was a natural number without
needing a more precise definition.

What about a good definition is a good
definition?

Do we like that one?

That's certainly precise.

If it's a precise mathematical definition,
it should be something that we can,

for any element of our
universe, answer yes

or no, whether it
satisfies that definition.

So that does make it a boolean.

A lot of real-world definitions,
like, is it an apple?

Well, there are things that some people
might consider apples and some people

might not, and it's really
hard to have a precise enough

definition of an apple
that we would all agree on.

Yeah.

But this is not a good
definition because it's

recursively defined but
with no determination, right?

Okay, good.

Yeah, so you're good.

So a good definition is a good definition.

It's got a big problem, which you pointed
out, right?

So this is a definition that if we already
knew what a good definition was,

this would help us, but it
wouldn't help us because

we already know what a
good definition was, right?

It's not adding anything.

And the problem is, for a
definition to be good, it's

got to not depend on
things we don't already know.

If understanding a definition,
in this case, depends on already knowing

the thing you're trying to define,
well, this is really a circular definition.

It doesn't help us make progress.

It's not harmful.

Being not harmful is probably
better than 90% of definitions

that are out there, so
it's actually pretty good.

But it's not helpful either.

So I think unambiguous at least fits with
our notion of precise here of saying

everyone should interpret the same way and
it should tell us exactly, yes or no,

whether this item satisfies the definition
or not.

So let me ask you about what... so it
makes sense to its audience.

And I think I copied that one right.

That's definitely a good property for a
definition to have.

Who is the audience?

Definitely four definitions that we should
come up with in this class.

I think that's a good answer, right?

The audience is people in this class.

If a definition doesn't
make sense to you in

this class, it's not
a helpful definition.

And definitely part of what we hope you'll
learn in this class is to understand

definitions that might be hard to
understand if you looked at them first.

But you'll get better at that,
yeah.

But in general, is that the audience when
people write a definition?

Who are they writing it for?

Other people in the same field?

So other people studying the same thing?

It should make sense
to people who think about

it the same way that
you're thinking about it.

Okay.

There's a lot of assumptions that go into
your definition about what the people

consuming it, what your audience already
knows and already accepts.

To make sense to that audience, it's
got to build on things they already know.

It's got to build on things
that are already well

accepted and understood
by the audience, right?

Because if it's using something that you
don't know already, right?

And the circular definition is
kind of the extreme example

of that, that you're using the
thing that you want to define.

Although we will soon
see recursive definitions

that do that in a way
that actually is okay.

Right?

That you certainly often
benefit from using the

thing you're wanting to
define in the definition.

But the broader point is you need to build
on things that you already have defined,

or that the audience already accepts and
knows what they mean.

Let's look at some definitions and see if
we think they're good.

So the first place to start is always
Wikipedia.

Wikipedia, for many things, is actually
pretty good.

This is pretty bad.

Here is its definition of a natural
number.

The natural numbers are those used in
counting and ordering.

How do we like that definition?

Okay, good.

Yeah, first of all, like
having both counting

and ordering in there
should really disturb you.

One of the worst things
about the way kids learn about

numbers is those concepts
are not really separated.

So there's ordering, when people count,
they're ordering.

So this is conflating the terms a little
bit.

When you're counting quantities,
right, this is actually cardinality.

You're measuring sizes.

All right, so this is cardinal.

Cardinal numbers are ones to measure
sizes.

In English, the labels of the cardinal and
orderly numbers are almost all the same,

or at least very similar.

And ordering is orderly numbers.

So what's a number that's different how we
say it in English, whether it's a counting

number, a cardinal number or an orderly
number?

Which is which?

So first and one are related concepts,
but they have different words.

Good.

Yeah.

So one is a quantity, like one is for
measuring how many of something you have.

And ordinal first is the first in order.

And if you interchange
those in most English

sentences, it would
sound really weird, right?

It would sound wrong to say there is first
apple in the back, right?

These are very different concepts,
right?

So the fact that Wikipedia has muddled its
definition by saying they're kind of the

same thing, and natural numbers apply to
both.

Which one of these has a zero?

Yeah, cardinal has a zero.

It's very natural to talk about,
I have zero apples.

It is not so natural to talk about the
zeroth in an ordering.

Sometimes people do, right?

There are places where
the zeroth makes sense,

but it's pretty unnatural
in most contexts.

So these are different things.

If you give a definition, like Wikipedia
does, you have not answered the question

whether zero is a natural number or not,
right?

So it's definitely not a precise enough
definition to answer this question.

It does have nice pictures of apples,
though.

So that's good.

So this is a quote from
another look at this as

saying kind of the history
of counting numbers.

Zero was definitely invented later,
right?

We had one, two, three, four, five,
six, seven, and usually both this is true

in history, as well as when children learn
about numbers.

They learn a lot of numbers before they
get to zero.

They say it's... Zero is slightly
less natural than the others.

Which, if that was where they left things,
would be a pretty unsatisfying

mathematical definition to say these
numbers are more natural than others,

but we didn't really decide
where to draw the line, but at least

in this definition, they are
including zero in the definition.

This model is sort of a definition by
example.

They're saying, I'm going to tell you what
the natural numbers are by giving you some

examples, and hoping you get a good idea
from that what they all are.

So, this is the one, two, three... This
one doesn't start at zero, you notice.

And this mysterious dot, dot, dot.

If you define your natural numbers like
that, well, then you are in this setting

of... Yeah, you just know
what they are, and they were

ordained by some power to know
what the natural numbers were.

And if you were an ancient Greek,
you choose your particular power that the

ancient Greeks thought was responsible for
making these numbers.

But the definitions in our textbook are
pretty similar to that, right?

So, this is the definition.

I'm going to use the
TCS book as shorthand for

the main textbook that
we will use for this class.

We will follow the structure of that book
fairly closely through most of the class.

But what we'll do in lecture, we'll often
be taking some small part of that and

expanding it a lot, or bringing other
things into it.

But you definitely will benefit
from reading the textbook,

and I think there's a
lot of good material in it.

Their definition of the natural numbers is
this.

Different from the previous
one, because it does

start with zero, and
it's got the dot, dot, dot.

If you look at the textbook that a few of
you used in discrete math, if you took it

with me a few years ago, their definition
is pretty similar, except for it's got

this, you know, ask your instructor for a
complete list.

That's kind of a
humorous way to say, like,

yeah, this is really not
a very good definition.

They're building on
our intuition of what the

dot, dot, dot means,
knowing that it doesn't.

But the other part of that
definition, right, this is the

real definition, is it's the
set of non-negative integers.

And that was actually part of the
definition in the TCS book, the book that

we're using for this class, that I cut out
on the previous slide, because it was with

an IE, that is to say, the non-negative
integers.

So is that a good definition?

If we define the natural numbers as the
non-negative integers, are we done and happy?

Is zero a negative integer?

Zero is not a negative integer,
but it is a non-negative integer.

What's the opposite of non-negative?

Yeah.

What's the opposite of non-negative?

Negative.

You would think so, wouldn't you?

But we have this... what's the opposite of
positive?

Non-positive.

Non-positive, okay.

So what are the non-positive numbers?

Yeah.

So if the opposite of non-negative...

So the opposite of non-negative better not
be negative.

Right?

So if the opposite of non-negative...

Actually, so the opposite of...

Okay.

So the negative integers, right?

We think those are negative one,
negative two, and negative three.

So when we talk about the opposite of a
set, what do we mean?

Is that well-defined what it means to say
the opposite?

Like, in English we say good is the
opposite of bad.

And there are lots of words that we think
we...

I know even in pre-K, my son is learning
opposites.

So people know about opposites.

Does it make sense here?

What does it mean?

Just everything that's not in the set.

Okay, good.

Yeah.

So that would be a good mathematical...

The more precise term would be complement,
right?

The complement of a set is everything in
the universe that's not in that set.

The opposite...

If we think of opposite as complement of a
set as its complement, right?

That means everything in the universe
except for what's in the set.

So what's the complement of the negative
integers?

Yeah.

Well, it depends on what our universe is.

Without defining our universe,
it doesn't make sense to

say what's the complement
of the non-negative integers.

If the universe is all the
whole numbers, then the

opposite of the non-negatives
would be the positives.

Right?

Because the whole numbers are zero,
the negatives, and the positives.

But if our universe is all numbers,
then the complement would include all the

non-integer real numbers, and could be
imaginary numbers, could have apples in it

if that's part of the universe as well,
right?

So it's not defined unless we have a
universe to talk about.

Now that we've resolved that, do we like
this definition?

It's really precise, right?

Is it for the right audience?

It's precise, right?

But it's kind of defining a simpler
concept in terms of a more complicated one.

If you are assuming you need to define the
natural numbers, well, in a lot of cases

you do, even if you have a fairly
sophisticated mathematical audience that

you assume already understands the
integers well.

Because there is this ambiguity about
zero, whether it's a member.

But if you are thinking someone doesn't
actually have a good understanding of the

natural numbers and I would claim none of
us actually have a good understanding of

the natural numbers yet, this is not a
really helpful definition, right?

It certainly doesn't help
you understand what

the natural numbers
are in a fundamental way.

It just says, if you already understand
this much more complicated concept,

if you understood that, if you didn't
understand that, well, you would look for

a definition of the integers, and you
would find one like this.

What are the integers?

The integers are zero, the positive
natural numbers, oops, the natural numbers

are the non-negative integers,
and they're additive inverses.

So we got an awful lot of complicated
concepts used to define the integers,

including a dependence
on the natural numbers,

which we defined in
terms of the integers.

At least Wikipedia's internal consistency
are the definitions, if you learned

everything you know from Wikipedia,
which sometimes is recommended,

but maybe if you want
to understand the natural

numbers, it's not going
to work out very well.

The book, this is from the TCS book,
they assume it's understood what it is,

but define it sort of with the
dot-dot-dots, right?

This is a definition by
example, and what dot-dot-dot

means, you hope people
understand in the right way.

But it's a very tricky concept,
and it does go out to infinity, right?

And what infinity means is a very tricky
concept that we will talk about.

We don't want definitions like that.

We want precise definitions.

We don't want to define something in terms
of something more complex.

We need to start our
definitions from something

that we already know
and build up from there.

We need to start out the interpretive
engineering thinking.

We need to start our pró Artist Group for
Five of Charles