TO INFINITY AND BEYOND!
NORMAN PERLMUTTER
I have been interested in big numbers for as long as I can remember.
My mother tells me that whenI was a young child, I once received an
assignment from school to see how high I could count. We both thought
the assignment was silly, because I knew that I could count arbitrarily
high. By the time I was five years old, I had memorized the names of
the various large numbers – hundred, thousand, million, billion, trillion
. . . all the way up to nonillion, google, and googleplex. Sometimes, I
counted by 17s in my head when I couldn’t sleep at night. I was also
fascinated with the idea of infinity, although I didn’t really know so
much about it.
Later, when I was a teenager, I was fascinated to learn that infinity
comes in different sizes – for instance, the set of real numbers is larger
than the set of natural numbers, although both are infinite.
In my undergraduate education, I continued to study different sizes
of infinity. I learned about ordinal numbers. They were introduced
to me as follows. The ordinal ω is the infinity of natural nubmers.
The ordinal ω + 1 is longer than ω, but no larger. In other words,
the ordinals ω and ω + 1 have the same cardinality but different order
types. Furthermore, the ordinal 1 + ω equals ω — ordinal addition is
not commutative! At first, I found the ordinals bizarre, but fascinating.
I knew that I wanted to learn more about them.
As an undergrad, I also learned about the ZFC axioms of set theory, which lay the foundations for modern mathematics. ZFC stands
for Zermelo, Fraenkel, and choice. The history is complicated, but in
brief, Zermelo and Fraenkel, Choice are the three mathematicians who
invented this axiom system. Just joking! The first two are mathematicians, the third is the Axiom of Choice, an axiom that was added to
the axioms of Zermelo and Fraenkel.
ZFC is essentially a collection of nine formal sentences, or axioms,
that set theorists assume to be true without proof. Some of these axioms are actually axiom schemes, meaning frameworks for constructing
an infinite number of true sentences. Using these axioms, set theorists
have used sets to represent all of the more complicated structures studied in mathematics. In a certain technical sense, we can argue that
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every mathematical object of study is a set or a formal sentence about
sets.
But in the 1930s, Gödel showed that the ZFC axioms have a certain
weakness. This weakness is shared by every other axiom system complicated enough to model the natural numbers. An axiom system is
consistent if it cannot be used to prove a contradiction. For instance, if
we could prove from ZFC that 0 6= 0, then ZFC would be inconsistent.
G odel showed that the ZFC sytem cannot prove its own consistency.
That is to say, he showed that nobody can prove, using only the axioms
of ZFC, that it is impossible write a proof from the ZFC axioms that
0 6= 0. So while almost all mathematicians strongly believe that the
ZFC axioms are consistent, they cannot prove that this is true.
One of the ZFC axioms that is of particular interest in the study of
sizes of infinity is the power set axiom, which states that for every set
X, the collection of all subsets of X is also a set, called the power set
of X. We can show that the power set of X is always larger than X,
and so this gives a way of generating many different sizes of infinity –
infinitely many in fact! Now, instead of counting large finite numbers
“quadrillion, quintillion, septillion . . .” as I did as a child, I can instead
count out the names of ever-larger infinite numbers, in particular the
type known as infinite cardinal numbers. These infinite numbers are denoted by the Hebrew letter aleph. ℵ0 (also known as ω) is the smallest
size of infinity, then come ℵ1 , ℵ2 , ℵ3 . . . ℵquadrillion . . . ℵnonillion . . . ℵgoogleplex . . . .
Using another axiom of ZFC, the Union Axiom, we can take the
limit of all the infinities above to get ℵω , the ωth infinity. And after
that, we can continue counting – ℵω+1 , ℵω+2 , . . . ℵω+ω . . . ℵω2 . . . and so
on, forever. Thanks to the axioms of set theory, we can prove that all
these sizes of infinity exist.
Now, to a layperson, the infinities mentioned above may seem pretty
large. But to a professional set theorist, they are among the puniest
infinities. Many set theorists focus their studies on large cardinals,
that is, infinities so large that their existence cannot be proven by the
ZFC axioms. Instead, the existence of a large cardinal becomes its
own new axiom. The distinguishing feature of a large cardinal is not
its absolute size, but rather its size compared to the cardinals below
it. Comparing a large cardinal to the cardinals below it is analogous
to comparing the least infinite cardinal, ℵ0 , to the finite numbers. And
so the count of big numbers continues: “inaccessible cardinal, Mahlo
cardinal, weakly compact cardinal, Ramsey cardinal, measurable cardinal, strong cardinal, Woodin cardinal, supercompact cardinal, huge
cardinal, . . . ”
TO INFINITY AND BEYOND!
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However, there is a danger in defining ever larger cardinals – there is
no guarantee that the axiom asserting the existence of a large cardinal
will not lead to a contradiction. Gödel’s theorem shows that no large
cardinal axiom can prove its own consistency. Instead, each large cardinal axiom proves the consistency of those that come before it. For
instance, from the assumption of ZFC + there exists an inaccessible
cardinal, we can prove that ZFC is consistent, and from the assumption of ZFC + there exists a Mahlo cardinal, we can prove that ZFC +
there exists an inaccessible cardinal is consistent, and so on. Indeed,
this is perhaps the most important sense in which one large cardinal
can be described as larger than another.
Many large cardinals axioms are characterized by the existence of
a special kind of function, called an elementary embedding, mapping
from the universe of sets, V , into another model of set theory, M .
The smallest cardinal that is moved by such a function (known as
the critical point of the function) is generally a large cardinal. What
distinguishes many of the different large cardinal axioms is the extent
to which M resembles V . The greater the resemblance, the larger the
large cardinal. So, a natural question to ask is what happens when we
take M to resemble V as much as possible – namely by taking M = V ?
The critical point of such an elementary embedding would seem to be
the largest possible large cardinal. This sort of large cardinal was first
studied by Reinhardt and therefore took the name Reinhardt cardinal.
However, shortly thereafter, Kenneth Kunen proved that the existence
of a Reinhardt cardinal, along with the usual ZFC axioms, leads to a
contradiction [Kun71]. A Reinhardt cardinal is so big that it cannot
exist! This famous result is known as the Kunen inconsistency.
In my third year of graduate school, following my inclination to study
larger and larger numbers, I chose to give my oral exam on the topic of
generalizations of the Kunen inconsistency. This presentation followed
the research of my advisor Joel Hamkins and of Greg Kirmayer and
discussed fairly general conditions (of which the Kunen inconsistency
is the simplest example) under which elementary embeddings between
models of set theory could not exist.
After I presented my oral exam, I was fascinated enough with the
topic that I wanted to research it further. The result was a joint paper
between Hamkins, Kirmayer, and myself, [HKP12] which appears in
the Annals of Pure and Applied Logic.
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References
[HKP12] Joel David Hamkins, Greg Kirmayer, and Norman Lewis Perlmutter.
Generalizations of the Kunen inconsistency. Annals of Pure and Applied
Logic, 163(12):1872 – 1890, 2012.
[Kun71] Kenneth Kunen. Elementary embeddings and infinitary combinatorics.
The Journal of Symbolic Logic, 36(3):pp. 407–413, 1971.