In
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
and
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, an ordinal collapsing function (or projection function) is a technique for defining (
notations
''Notations'' is a book that was edited and compiled by American avant-garde composer John Cage (1912–1992) with Alison Knowles and first published in 1969 by Something Else Press. The book is made up of a large collection of graphical scores, ...
for) certain
recursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
large countable ordinal
In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of releva ...
s, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even
large cardinals
In the mathematical field of set theory, a large cardinal property is a certain kind of property of Transfinite number, transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, big ...
(though they can be replaced with
recursively large ordinals at the cost of extra technical difficulty), and then "collapse" them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an
impredicative
In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more com ...
manner of naming ordinals.
The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system "runs out of fuel" and cannot name a certain ordinal, a much larger ordinal is brought "from above" to give a name to that critical point. An example of how this works will be detailed below, for an ordinal collapsing function defining the
Bachmann–Howard ordinal
In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a large countable ordinal.
It is the ordinal analysis, proof-theoretic ordinal of several mathematical theory (logic), theories, such as ...
(i.e., defining a system of notations up to the Bachmann–Howard ordinal).
The use and definition of ordinal collapsing functions is inextricably intertwined with the theory of
ordinal analysis
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength.
If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has ...
, since the large countable ordinals defined and denoted by a given collapse are used to describe the ordinal-theoretic strength of certain
formal system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
s, typically
[Rathjen, 1995 (Bull. Symbolic Logic)][Kahle, 2002 (Synthese)] subsystems of
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
(such as those seen in the light of
reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in cont ...
), extensions of
Kripke–Platek set theory
The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek.
The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it.
Axioms
In its fo ...
,
Bishop
A bishop is an ordained clergy member who is entrusted with a position of authority and oversight in a religious institution.
In Christianity, bishops are normally responsible for the governance of dioceses. The role or office of bishop is ca ...
-style systems of
constructive mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
or
Martin-Löf-style systems of
intuitionistic type theory
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics.
Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and ph ...
.
(
psi
Psi, PSI or Ψ may refer to:
Alphabetic letters
* Psi (Greek) (Ψ, ψ), the 23rd letter of the Greek alphabet
* Psi (Cyrillic) (Ѱ, ѱ), letter of the early Cyrillic alphabet, adopted from Greek
Arts and entertainment
* "Psi" as an abbreviation ...
)(1)
Ordinal collapsing functions are typically denoted using some variation of either the Greek letter
(
psi
Psi, PSI or Ψ may refer to:
Alphabetic letters
* Psi (Greek) (Ψ, ψ), the 23rd letter of the Greek alphabet
* Psi (Cyrillic) (Ѱ, ѱ), letter of the early Cyrillic alphabet, adopted from Greek
Arts and entertainment
* "Psi" as an abbreviation ...
) or
(
theta
Theta (, ; uppercase: Θ or ; lowercase: θ or ; grc, ''thē̂ta'' ; Modern: ''thī́ta'' ) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth . In the system of Greek numerals, it has a value of 9.
Gr ...
).
An example leading up to the Bachmann–Howard ordinal
The choice of the ordinal collapsing function given as example below imitates greatly the system introduced by Buchholz
[Buchholz, 1986 (Ann. Pure Appl. Logic)] but is limited to collapsing one cardinal for clarity of exposition. More on the relation between this example and Buchholz's system will be said
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
.
Definition
Let
stand for the
first uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. Whe ...
, or, in fact, any ordinal which is an
-number and guaranteed to be greater than all the
countable ordinal
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
s which will be constructed (for example, the
Church–Kleene ordinal
In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using ordinal collapsing functions.
The Church–Kleene ordinal and variant ...
is adequate for our purposes; but we will work with
because it allows the convenient use of the word ''countable'' in the definitions).
We define a function
(which will be
non-decreasing and
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
), taking an arbitrary ordinal
to a countable ordinal
, recursively on
, as follows:
:Assume
has been defined for all
, and we wish to define
.
:Let
be the set of ordinals generated starting from
,
,
and
by recursively applying the following functions: ordinal
addition, multiplication and exponentiation and the function
, i.e., the restriction of
to ordinals
. (Formally, we define
and inductively
for all natural numbers
and we let
be the union of the
for all
.)
:Then
is defined as the smallest ordinal not belonging to
.
In a more concise (although more obscure) way:
:
is the smallest ordinal which cannot be expressed from
,
,
and
using sums, products, exponentials, and the
function itself (to previously constructed ordinals less than
).
Here is an attempt to explain the motivation for the definition of
in intuitive terms: since the usual operations of addition, multiplication and exponentiation are not sufficient to designate ordinals very far, we attempt to systematically create new names for ordinals by taking the first one which does not have a name yet, and whenever we run out of names, rather than invent them in an ''ad hoc'' fashion or using
diagonal schemes, we seek them in the ordinals far beyond the ones we are constructing (beyond
, that is); so we give names to uncountable ordinals and, since in the end the list of names is necessarily countable,
will "collapse" them to countable ordinals.
Computation of values of ''ψ''
To clarify how the function
is able to produce notations for certain ordinals, we now compute its first values.
Predicative start
First consider
. It contains ordinals
and so on. It also contains such ordinals as
. The first ordinal which it does not contain is
(which is the limit of
,
,
and so on — less than
by assumption). The upper bound of the ordinals it contains is
(the limit of
,
,
and so on), but that is not so important. This shows that
.
Similarly,
contains the ordinals which can be formed from
,
,
,
and this time also
, using addition, multiplication and exponentiation. This contains all the ordinals up to
but not the latter, so
. In this manner, we prove that
inductively on
: the proof works, however, only as long as
. We therefore have:
:
for all
, where
is the smallest fixed point of
.
(Here, the
functions are the
Veblen function
In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φ ...
s defined starting with
.)
Now
but
is no larger, since
cannot be constructed using finite applications of
and thus never belongs to a
set for
, and the function
remains "stuck" at
for some time:
:
for all
.
First impredicative values
Again,
. However, when we come to computing
, something has changed: since
was ("artificially") added to all the
, we are permitted to take the value
in the process. So
contains all ordinals which can be built from
,
,
,
, the
function ''up to
'' and this time also
itself, using addition, multiplication and exponentiation. The smallest ordinal not in
is
(the smallest
-number after
).
We say that the definition
and the next values of the function
such as
are
impredicative
In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more com ...
because they use ordinals (here,
) greater than the ones which are being defined (here,
).
Values of ''ψ'' up to the Feferman–Schütte ordinal
The fact that
remains true for all
(note, in particular, that
: but since now the ordinal
has been constructed there is nothing to prevent from going beyond this). However, at
(the first fixed point of
beyond
), the construction stops again, because
cannot be constructed from smaller ordinals and
by finitely applying the
function. So we have
.
The same reasoning shows that
for all
, where
enumerates the fixed points of
and
is the first fixed point of
. We then have
.
Again, we can see that
for some time: this remains true until the first fixed point
of
, which is the
Feferman–Schütte ordinal
In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal.
It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion.
It is named after Solomon Feferman and Kurt Schütte, ...
. Thus,
is the Feferman–Schütte ordinal.
Beyond the Feferman–Schütte ordinal
We have
for all
where
is the next fixed point of
. So, if
enumerates the fixed points in question (which can also be noted
using the many-valued Veblen functions) we have
, until the first fixed point
of the
itself, which will be
(and the first fixed point
of the
functions will be
). In this manner:
*
is the
Ackermann ordinal
In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal.
Unfortunately there is ...
(the range of the notation
defined predicatively),
*
is the
"small" Veblen ordinal (the range of the notations
predicatively using finitely many variables),
*
is the
"large" Veblen ordinal (the range of the notations
predicatively using transfinitely-but-predicatively-many variables),
* the limit
of
,
,
, etc., is the
Bachmann–Howard ordinal
In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a large countable ordinal.
It is the ordinal analysis, proof-theoretic ordinal of several mathematical theory (logic), theories, such as ...
: after this our function
is constant, and we can go no further with the definition we have given.
Ordinal notations up to the Bachmann–Howard ordinal
We now explain more systematically how the
function defines notations for ordinals up to the Bachmann–Howard ordinal.
A note about base representations
Recall that if
is an ordinal which is a power of
(for example
itself, or
, or
), any ordinal
can be uniquely expressed in the form
, where
is a natural number,
are non-zero ordinals less than
, and
are ordinal numbers (we allow
). This "base
representation" is an obvious generalization of the
Cantor normal form
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an e ...
(which is the case
). Of course, it may quite well be that the expression is uninteresting, i.e.,
, but in any other case the
must all be less than
; it may also be the case that the expression is trivial (i.e.,
, in which case
and
).
If
is an ordinal less than
, then its base
representation has coefficients
(by definition) and exponents
(because of the assumption
): hence one can rewrite these exponents in base
and repeat the operation until the process terminates (any decreasing sequence of ordinals is finite). We call the resulting expression the ''iterated base
representation'' of
and the various coefficients involved (including as exponents) the ''pieces'' of the representation (they are all
), or, for short, the
-pieces of
.
Some properties of ''ψ''
* The function
is non-decreasing and continuous (this is more or less obvious from its definition).
* If
with
then necessarily
. Indeed, no ordinal
with
can belong to
(otherwise its image by
, which is
would belong to
— impossible); so
is closed by everything under which
is the closure, so they are equal.
* Any value
taken by
is an
-number (i.e., a fixed point of
). Indeed, if it were not, then by writing it in
Cantor normal form
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an e ...
, it could be expressed using sums, products and exponentiation from elements less than it, hence in
, so it would be in
, a contradiction.
* Lemma: Assume
is an
-number and
an ordinal such that
for all
: then the
-pieces (defined
above) of any element of
are less than
. Indeed, let
be the set of ordinals all of whose
-pieces are less than
. Then
is closed under addition, multiplication and exponentiation (because
is an
-number, so ordinals less than it are closed under addition, multiplication and exponentiation). And
also contains every
for
by assumption, and it contains
,
,
,
. So
, which was to be shown.
* Under the hypothesis of the previous lemma,
(indeed, the lemma shows that
).
* Any
-number less than some element in the range of
is itself in the range of
(that is,
omits no
-number). Indeed: if
is an
-number not greater than the range of
, let
be the least upper bound of the
such that
: then by the above we have
, but
would contradict the fact that
is the ''least'' upper bound — so
.
* Whenever
, the set
consists exactly of those ordinals
(less than
) all of whose
-pieces are less than
. Indeed, we know that all ordinals less than
, hence all ordinals (less than
) whose
-pieces are less than
, are in
. Conversely, if we assume
for all
(in other words if
is the least possible with
), the lemma gives the desired property. On the other hand, if
for some
, then we have already remarked
and we can replace
by the least possible with
.
The ordinal notation
Using the facts above, we can define a (canonical) ordinal notation for every
less than the Bachmann–Howard ordinal. We do this by induction on
.
If
is less than
, we use the iterated Cantor normal form of
. Otherwise, there exists a largest
-number
less or equal to
(this is because the set of
-numbers is closed): if
then by induction we have defined a notation for
and the base
representation of
gives one for
, so we are finished.
It remains to deal with the case where
is an
-number: we have argued that, in this case, we can write
for some (possibly uncountable) ordinal
: let
be the ''greatest'' possible such ordinal (which exists since
is continuous). We use the iterated base
representation of
: it remains to show that every piece of this representation is less than
(so we have already defined a notation for it). If this is ''not'' the case then, by the properties we have shown,
does not contain
; but then
(they are closed under the same operations, since the value of
at
can never be taken), so
, contradicting the maximality of
.
Note: Actually, we have defined canonical notations not just for ordinals below the Bachmann–Howard ordinal but also for certain uncountable ordinals, namely those whose
-pieces are less than the Bachmann–Howard ordinal (viz.: write them in iterated base
representation and use the canonical representation for every piece). This canonical notation is used for arguments of the
function (which may be uncountable).
Examples
For ordinals less than
, the canonical ordinal notation defined coincides with the iterated Cantor normal form (by definition).
For ordinals less than
, the notation coincides with iterated base
notation (the pieces being themselves written in iterated Cantor normal form): e.g.,
will be written
, or, more accurately,
. For ordinals less than
, we similarly write in iterated base
and then write the pieces in iterated base
(and write the pieces of ''that'' in iterated Cantor normal form): so
is written
, or, more accurately,
. Thus, up to
, we always use the largest possible
-number base which gives a non-trivial representation.
Beyond this, we may need to express ordinals beyond
: this is always done in iterated
-base, and the pieces themselves need to be expressed using the largest possible
-number base which gives a non-trivial representation.
Note that while
is equal to the Bachmann–Howard ordinal, this is not a "canonical notation" in the sense we have defined (canonical notations are defined only for ordinals ''less'' than the Bachmann–Howard ordinal).
Conditions for canonicalness
The notations thus defined have the property that whenever they nest
functions, the arguments of the "inner"
function are always less than those of the "outer" one (this is a consequence of the fact that the
-pieces of
, where
is the largest possible such that
for some
-number
, are all less than
, as we have shown above). For example,
does not occur as a notation: it is a well-defined expression (and it is equal to
since
is constant between
and
), but it is not a notation produced by the inductive algorithm we have outlined.
Canonicalness can be checked recursively: an expression is canonical if and only if it is either the iterated Cantor normal form of an ordinal less than
, or an iterated base
representation all of whose pieces are canonical, for some
where
is itself written in iterated base
representation all of whose pieces are canonical and less than
. The order is checked by lexicographic verification at all levels (keeping in mind that
is greater than any expression obtained by
, and for canonical values the greater
always trumps the lesser or even arbitrary sums, products and exponentials of the lesser).
For example,
is a canonical notation for an ordinal which is less than the Feferman–Schütte ordinal: it can be written using the Veblen functions as
.
Concerning the order, one might point out that
(the Feferman–Schütte ordinal) is much more than
(because
is greater than
of anything), and
is itself much more than
(because
is greater than
, so any sum-product-or-exponential expression involving
and smaller value will remain less than
). In fact,
is already less than
.
Standard sequences for ordinal notations
To witness the fact that we have defined notations for ordinals below the Bachmann–Howard ordinal (which are all of countable
cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses the ...
), we might define standard sequences converging to any one of them (provided it is a limit ordinal, of course). Actually we will define canonical sequences for certain uncountable ordinals, too, namely the uncountable ordinals of ''countable'' cofinality (if we are to hope to define a sequence converging to them...) which are representable (that is, all of whose
-pieces are less than the Bachmann–Howard ordinal).
The following rules are more or less obvious, except for the last:
* First, get rid of the (iterated) base
representations: to define a standard sequence converging to
, where
is either
or
(or
, but see below):
** if
is zero then
and there is nothing to be done;
** if
is zero and
is successor, then
is successor and there is nothing to be done;
** if
is limit, take the standard sequence converging to
and replace
in the expression by the elements of that sequence;
** if
is successor and
is limit, rewrite the last term
as
and replace the exponent
in the last term by the elements of the fundamental sequence converging to it;
** if
is successor and
is also, rewrite the last term
as
and replace the last
in this expression by the elements of the fundamental sequence converging to it.
* If
is
, then take the obvious
as the fundamental sequence for
.
* If
then take as fundamental sequence for
the sequence
* If
then take as fundamental sequence for
the sequence
* If
where
is a limit ordinal of ''countable'' cofinality, define the standard sequence for
to be obtained by applying
to the standard sequence for
(recall that
is continuous and increasing, here).
* It remains to handle the case where
with
an ordinal of ''uncountable'' cofinality (e.g.,
itself). Obviously it doesn't make sense to define a sequence converging to
in this case; however, what we can define is a sequence converging to some
with countable cofinality and such that
is constant between
and
. This
will be the first fixed point of a certain (continuous and non-decreasing) function
. To find it, apply the same rules (from the base
representation of
) as to find the canonical sequence of
, except that whenever a sequence converging to
is called for (something which cannot exist), replace the
in question, in the expression of
, by a
(where
is a variable) and perform a repeated iteration (starting from
, say) of the function
: this gives a sequence
tending to
, and the canonical sequence for
is
,
,
... If we let the
th element (starting at
) of the fundamental sequence for
be denoted as