Ordinals: Sets or Numbers?

Date: 04/16/2010 at 07:25:07
From: Mari
Subject: Ordinals: Sets or numbers?

Hi,

I'm a little bit confused about the nature of ordinals. Some books
describe them as sets; others, as numbers!

What is the correct view? What we should call ordinals? I could not find a
convincing answer.

Thanks a lot.

Date: 04/16/2010 at 12:15:00
From: Doctor Tom
Subject: Re: Ordinals: Sets or numbers?

Hi Mari,

Good question!

It turns out that in formal mathematics, what you do is take a system of
axioms and use the rules of logic to conclude various theorems with the
underlying idea being that "If this set of axioms is true, then these
theorems are true." Obviously, different sets of starting axioms yield
different sets of "true" theorems.

All that's required for a theory to be reasonable is that the axioms are
not self-contradictory. We have Euclidean geometry, which we all learned
in high school; but there are other geometries, such as Riemannian
geometry, where the parallel postulate does not hold, and different
conclusions arise. Just because a set of axioms is consistent does not
mean that the axioms (or resulting theorems) are true in the real world.
In fact, in our world, the Riemannian geometry seems to be more true than
Euclidean since we seem to live in a curved spacetime, as Einstein
discovered.

Sets of axioms and their resulting theorems can actually have mathematical
models. For example, if I wrote down the usual arithmetic axioms (called
the axioms for a "field" by mathematicians), they'd include stuff like
commutativity of addition and multiplication, the existence of a zero and
a one, the distributive laws, and some other stuff. From these axioms, you
can conclude various theorems, but these axioms in fact have many models.

One model is the familiar real numbers.

Another is the rational numbers.

A third is the complex numbers.

There are an infinite number of others: the field consisting of only zero
and one, where one + one = zero satisfies all the field axioms. Any
theorem proved for fields in general is true of this tiny field as well.

One very useful way of thinking about mathematics is to have in mind one
(or several) models for any given system of axioms. Then you can think
about what's true in the models as well as just in terms of logic and the
axioms. For example, since we know there are a lot of different models for
the field axioms, we know it would be impossible to prove, just from the
axioms, that every field contains at least three members, since we have a
model of the axioms where that's not true.

A huge area of mathematical logic is called "model theory."

But what turns out to be an amazing thing is that formal set theory
(sometimes called "Zermelo-Fraenkel set theory," or often, just "ZF") can
be used to construct models for mathematical systems. This includes
systems as simple as the two-element field above, or the integers, natural
numbers, reals, mathematical groups, rings and fields, and on and on.

There is a "standard" ZF model of the ordinal and cardinal
numbers based on ZF, and depending on how you extend the ZF axioms, you'll
obtain different models of the ordinals and cardinals. And they are
essentially different systems, or different sets of ordinal/cardinal
numbers.

This is a deep topic which you can study for years; I've only scratched
the surface. If you're interested, I've written a couple of articles aimed
a bright high school students that talk about formal ZF and about the
usual construction of the ordinal numbers. Take a look at these:

  http://www.geometer.org/mathcircles/nothing.pdf 
   http://www.geometer.org/mathcircles/Infinity.pdf

I hope this helps!

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/18/2010 at 03:33:49
From: Mari
Subject: Ordinals: Sets or numbers?

So,

Correct me if I'm mistaken, but using a special model based on some ZFC
axioms leads to constructing ordinal sets.

I actually asked one professor in my department about this, and he said
that ordinal numbers and ordinal sets are two names for one thing! He
said: for example, 1-1 and bijection are two names for one concept.

But I was not convinced that they describe the same thing. The concept of
number and set are totally different. What do you think?

Date: 04/18/2010 at 09:33:04
From: Doctor Tom
Subject: Re: Ordinals: Sets or numbers?

Hi Mari,

The usual construction of the ordinals from ZF or ZFC involves
constructing sets that correspond to each ordinal, in the senses that we
can tell from the sets which ordinal it amounts to, and that by using
set-theoretic operations we can perform ordinal operations like
"successor," "sum," and "comparison."

The way it's usually done is by assigning the empty set to be 0, the set
containing 0 to be 1, the set containing 0 and 1 to be 2, and so on. Thus:

   5 = {0, 1, 2, 3, 4}

And each ordinal (and finite cardinal) set contains the number of objects
you'd associate with that ordinal (or cardinal).

There are other ways to do it, and if different constructions yield the
same results except for how you choose to model them, then they're
equivalent to a mathematician.

I don't know if you've ever tried to solve Sudoku puzzles, but imagine
changing one so that -- instead of writing the digits 1 through 9 -- you
substituted A for 1, B for 2, et cetera. To solve this A-through-I puzzle,
you could use exactly the same logic, since no arithmetic is involved, and
almost everyone would say the two puzzles are equivalent --- or, in
mathematical terms, "isomorphic." There are an infinite number of models
for the ordinals as well, but one that is incredibly useful and easy to
work with is the one I mentioned above (and is the one that's described in
much more detail in one of the article references I sent you in my
previous note).

Perhaps others think about it differently, but I think of systems like
"the ordinals," "the natural numbers," et cetera, not as a specific model,
but as a generic concept that includes all the things that all the models
have in common. Then you're free to think about different models if that
helps you do your work. So the system of ordinals, for me, is more like
the system of axioms that describes them. A particular model (which
happens to be the one that is almost always used) is the one where we use
some version of ZF to build a set corresponding to each ordinal and
specify certain functions that operate on those sets to perform operations
like "successor," et cetera.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/20/2010 at 21:58:51
From: Mari
Subject: Thank you (Ordinals: Sets or numbers?)

Thanks a lot your answer. That makes a good sense.

Best wishes and cheers.