inaccessible cardinal

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Universes
+-- {: .hide}
[[!include universe - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

An **inaccessible cardinal** is a [[cardinal number]] $\kappa$ which cannot be "accessed" from smaller cardinals using only the basic operations on cardinals.  It follows that the collection of sets smaller than $\kappa$ satisfies the axioms of set theory.

## Definition

The discussion here makes sense in the context of the [[axiom of choice]], since only then is the [[collection]] of [[cardinal numbers]] $Card$ [[well-ordered]] and indeed indexed by the collection of [[ordinal numbers]] $Ord$. In particular, we assume the [[law of excluded middle]] and thus use $2$ (instead of $\Omega$ or $Prop$) for the set of [[truth values]] in the definition.

An **inaccessible cardinal** is a [[regular cardinal|regular]] strong [[limit cardinal]].  Here, $\kappa$ is _regular_ if every sum of $\lt\kappa$ cardinals, each of which is $\lt\kappa$, is itself $\lt\kappa$; $\kappa$ is a _strong limit_ if $\lambda\lt \kappa$ implies $2^\lambda\lt\kappa$. In other words, the class of sets of cardinality $\lt\kappa$ is closed under the operations of indexed [[union]]s and taking [[power set]]s.

By this definition, $0$ (the cardinality of the [[empty set]]) and $\aleph_0$ (the cardinality of the set of [[natural number]]s) are both inaccessible.  Usually one explicitly requires inaccessible cardinals to be [[uncountable set|uncountable]], so as to exclude these cases.  One can also justify excluding $0$ by interpreting the requirement that $1 \lt \kappa$ as the nullary part of a requirement whose binary part is closure under indexed unions.

A __weakly inaccessible cardinal__ is a regular weak [[limit cardinal]]; sometimes inaccessible cardinals are called _strongly inaccessible_ in contrast.  Here, $\kappa$ is a _weak limit_ if $\lambda\lt\kappa$ implies $\lambda^+\lt\kappa$, where $\lambda^+$ is the smallest cardinal number $\gt\lambda$.  Every strongly inaccessible cardinal is also weakly inaccessible, while the converse is true assuming the generalized [[continuum hypothesis]].

## Properties

An (uncountable) cardinal $\kappa$ is inaccessible precisely when the $\kappa$th level $V_\kappa$ of the [[von Neumann hierarchy]] is a [[Grothendieck universe]]  ([Williams](#Williams)), and hence in particular itself a model of axiomatic [[set theory]].  For this reason, the existence of inaccessible cardinals cannot be proven in ordinary axiomatic set theory such as [[ZFC]].  The axiom asserting that there exists an inaccessible (which amounts to the existence of a Grothendieck universe) is thus the beginning of the study of [[large cardinal]]s.

If one thinks of $\aleph_0$ as already an inaccessible cardinal, then the [[axiom of infinity]] may be seen as itself the first large cardinal axiom.

## Related concepts

* [[regular cardinal]]

* [[product-regular cardinal]]

* [[Grothendieck universe]]

* [[large cardinal]]

## References

Historical review with comprehensive references:

* [[Ralf Krömer]], §6.4.5 in: *Tool and object: A history and philosophy of category theory*, Science Networks. Historical Studies **32** (2007) [[doi:10.1007/978-3-7643-7524-9](https://doi.org/10.1007/978-3-7643-7524-9)]

See also:

* [[Tom Leinster]], *Large Sets 9*, [web](https://golem.ph.utexas.edu/category/2021/07/large_sets_9.html)

The proof that a Tarski-[[Grothendieck universe]] is equivalently a set of $\kappa$-small sets for $\kappa$ an inaccessible cardinal is in

* N. H. Williams, _On Grothendieck universes_, Compositio Mathematica, 21:1 (1969) ([numdam]( http://www.numdam.org/item?id=CM_1969__21_1_1_0))
 {#Williams}

* Andreas Blass, Ioanna M. Dimitriou, Benedikt Löwe, _Inaccessible cardinals without the axiom of choice_, Fund. Math. 194 (2007) 179-189 [pdf](http://journals.impan.gov.pl/cgi-bin/fm/pdf?fm194-2-03)

> Abstract: We consider four notions of strong inaccessibility that are equivalent in ZFC and show that they are not equivalent in ZF.

[[!redirects inaccessible cardinals]]

[[!redirects strongly inaccessible cardinal]]
[[!redirects strongly inaccessible cardinals]]
[[!redirects weakly inaccessible cardinal]]
[[!redirects weakly inaccessible cardinals]]


inbound citations > history


incarnations of rational equivariant topological K-theory -- table

**Incarnations of [[rational equivariant K-theory]]:**

|  | [[cohomology]] theory | definition/equivalence due to  |
|--|--|--|
| $\simeq K_G^0\big(X; \mathbb{C} \big) $ |  **[[rational equivariant K-theory]]** |  |
| $ \simeq H^{ev}\Big( \big(\underset{g \in G}{\coprod} X^g\big)/G; \mathbb{C}  \Big) $ | [[delocalized equivariant cohomology]] | [Baum-Connes 89, Thm. 1.19](delocalized+equivariant+cohomology#ConnesBaum89) |
| $\simeq H^{ev}_{CR}\Big( \prec \big( X \!\sslash\! G\big);\, \mathbb{C}  \Big)$ | [[Chen-Ruan cohomology]]
 of [[global quotient orbifold]] | [Chen-Ruan 00, Sec. 3.1](Chen-Ruan+cohomology#ChenRuan00)  |
| $\simeq H^{ev}_G\Big( X; \, \big(G/H \mapsto \mathbb{C} \otimes Rep(H)\big)  \Big)$ | [[Bredon cohomology]]
 with [[coefficients]] in [[representation ring]] | [Ho88 6.5](traditional+orbifold+cohomology+--+references#Honkasalo88)+[Ho90 5.5](traditional+orbifold+cohomology+--+references#Honkasalu90)+[Mo02 p. 18](traditional+orbifold+cohomology+--+references#Moerdijk02),
 [Mislin-Valette 03, Thm. 6.1](equivariant+Chern+character#MislinValette03),
 [Szabo-Valentino 07, Sec. 4.2](traditiona+orbifold+cohomology#SzaboValentino07)|
| $\simeq K_G^0\big(X; \mathbb{C} \big) $ |  **[[rational equivariant K-theory]]** | [Lück-Oliver 01, Thm. 5.5](equivariant+K-theory#LueckOliver01),
 [Mislin-Valette 03, Thm. 6.1](equivariant+Chern+character#MislinValette03) |


incidence geometry

see at _[[synthetic geometry]]_

## References

* Wikipedia, _[Incidence geometry](http://en.wikipedia.org/wiki/Incidence_geometry)_

[[!redirects incidence geometries]]


inclusion function

Given a [[set]] $X$ and a [[subset]] $A$ of $X$, the __inclusion function__ of $A$ in $X$ is the [[function]] $i_A\colon A \to X$ given by
$$ i_A(n) = n $$
for every [[element]] $n$ of $X$ that belongs to $A$.

Every inclusion function is an [[injection]], and every injection is [[isomorphism|isomorphic]] (in the [[slice category]] $Set/X$) to an inclusion function.  In appropriate categories, it is common to analogously interpret [[monomorphisms]] or [[regular monomorphisms]] as inclusions of [[subobjects]].  One then speaks of __inclusion morphisms__.

The inclusion function of $A$ in $X$ is the [[restriction]] to $A$ of the [[identity function]] on $X$.  Conversely, the restriction to $A$ of any function $f\colon X \to Y$ is the [[composite]] of $f$ after the inclusion function:
$$ i_A = {{\id_X}|_A} ,$$
$$ {{f}|_A} = f \circ i_A .$$

[[!redirects inclusion function]]
[[!redirects inclusion functions]]
[[!redirects inclusion map]]
[[!redirects inclusion maps]]
[[!redirects inclusion morphism]]
[[!redirects inclusion morphisms]]


inclusion functor > history

see at _[[subcategory]]_


Inclusion Sandbox

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inclusion-exclusion

# The inclusion-exclusion principle
* table of contents
{: toc}

## Idea

The principle of inclusion and exclusion is essentially a [[sheaf]] condition for [[measures]]; it allows one to calculate the total measure of a space in terms of a [[cover]].  It applies also to [[cardinality]], that is to [[counting measure]].

## Statement

Let $X$ be a [[measure space]] with measure $\mu$.  Typical examples are a [[probability space]] or subprobability space, or an arbitrary [[set]] with [[counting measure]].  (In the latter case, we may let the measure take values in all [[cardinal numbers]], so that the measure of a subset is its cardinality.)

Now let $U$ be a measurable [[finite cover]] of $X$.  (That is, we have a [[finite set|finite]] index set $I$ and a function mapping an index $j$ in $I$ to a [[measurable subspace]] $U_j$ of $X$, such that the [[union]] $\bigcup_j U_j$ of all of these subspaces is all of $X$.)  Then given any [[natural number]] $k$ and any $k$-combination $\sigma$ from $I$ (a [[subset]] of $I$ of [[cardinality]] $k$), let the __measure__ of $\sigma$ be $\mu(\bigcap_{j \in \sigma} U_j)$, the measure of the [[intersection]] of the subspaces assigned to the indexes in $\sigma$.  We will call $k$ the __size__ of the $k$-combination $\sigma$.  Note that for $k = 0$ (so $\sigma = \empty$), $\bigcap_{j \in \empty} U_j = X$.

+-- {: .num_theorem}
###### Theorem

The sum of the measures of the combinations of even size equals the sum of the measures of the combinations of odd size.
=--

If we drop the requirement that $U$ be a cover of $X$ (that the union of $U$ be $X$), we simply have that $U$ is a cover of the actual union of $U$; the only difference is that the measure of the $0$-combination is now the measure of this union instead of the measure of all of $X$.  Then the principle for small values of $n$ looks like this:

*  $\mu(\empty) = 0$,
*  $\mu(A) = \mu(A)$,
*  $\mu(A \cup B) + \mu(A \cap B) = \mu(A) + \mu(B)$ (\*),
*  $\mu(A \cup B \cup C) + \mu(A \cap B) + \mu(A \cap C) + \mu(B \cap C) = \mu(A) + \mu(B) + \mu(C) + \mu(A \cap B \cap C)$,
*  etc.

If subtraction is available (as when $\mu$ is a [[finite measure]]), we may solve for the first term on the left (the term from the $0$-combination, that is the measure of the union):

*  $\mu(\empty) = 0$,
*  $\mu(A) = \mu(A)$,
*  $\mu(A \cup B) = \mu(A) + \mu(B) - \mu(A \cap B)$,
*  $\mu(A \cup B \cup C) = \mu(A) + \mu(B) + \mu(C) - \mu(A \cap B) - \mu(A \cap C) - \mu(B \cap C) + \mu(A \cap B \cap C)$,
*  etc.

The general formula is $\mu(\underset{1 \le i \le n}{\bigcup A_{i}}) = \underset{(I  \neq \emptyset)\subseteq [1,n]}{\sum}(-1)^{|I|+1}\mu(\underset{i \in I}{\bigcap}A_{i})$.

This explains the term 'inclusion-exclusion'; we are alternately adding (including) and subtracting (excluding) various intersections.

## Proof

The proof is for all subspaces of a given measure space, by induction on the cardinality $n$ of the index set $I$.  Although it is atypical, one may take, as one of the basic axioms of a measure, the formula (\*), that is the inclusion-exclusion formula (on all measurable subspaces) for $n = 2$; in any case, this is easy to prove.  Then the formula for $n \gt 2$ may be proved by induction.  (The formula for $n = 0$ is the axiom that the measure of the [[empty set]] is [[zero]], and the formula for $n = 1$ is trivial.)

## As a definition of measure

The principle applies perfectly well to finitely additive measures (charges) taking values in any [[abelian monoid]].  On the other hand, the statement of inclusion-exclusion makes sense even when the cover $U$ is not finite; and if we require it whenever $U$ is [[countable set|countable]], then we force the measure to be countably additive.  If we require inclusion-exclusion for arbitrary (even uncountable) $U$, then we get the definition of a [[normal measure]].

In [[constructive mathematics]], the inclusion-exclusion principle fails in that $\mu$ may satisfy $\mu(A \cup B) = \mu(A) \cup \mu(B)$ when $A$ and $B$ are [[disjoint sets|disjoint]] without necessarily satisfying (\*).  On the other hand, we may adopt (\*) as the proper clause in the *definition* of measure.  For measures on a [[Cheng space]] (which is the best established constructive theory of measure in general), inclusion-exclusion follows from finite additivity (for disjoint sets) and the principle that intersection with [[full sets]] (which are specified for a Cheng space before any measure is given) preserves measure, so in that sense it makes no difference which axiom we take.  But (\*) is not valid constructively for cardinality in general; counting measure is a Cheng measure only on sets with [[decidable equality]].

In both cases, we see that we can adopt the principle of inclusion-exclusion (which ultimately is motivated by cardinality of finite sets) as the only principle necessary to *define* what we mean by a measure (although we must already have a [[measurable space]] to define it on).

[[!redirects inclusion-exclusion]]
[[!redirects inclusion-exclusion principle]]


incompleteness theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

* table of contents
{: toc}

## Idea

In [[logic]], an incompleteness theorem expresses limitations on provability within a (consistent) formal [[theory]]. Most famously it refers to a pair of theorems due to [[Kurt Gödel]]; the first incompleteness theorem says roughly that for any [[consistent theory]] $T$ containing [[Peano arithmetic|arithmetic]] and whose [[axioms]] form a [[recursive set]], there is an arithmetic sentence which is true for the [[natural numbers]] $\mathbb{N}$ that cannot be proven in $T$. The second incompleteness theorem shows that for such theories $T$, the sentence can be taken to be a suitable arithmetization of the statement of consistency of $T$ itself, i.e., that no such theory "can demonstrate its own consistency" -- can prove an arithmetic statement that encodes the assertion of its consistency.

An incompleteness theorem can be read as an [[undecidable proposition|undecidability result]]: there are formal propositions $p$ in the theory such that neither $p$ nor its [[negation]] $\neg{p}$ have [[proofs]] in the theory. As such, it is part of a long history; for example, axiom systems for [[Euclidean geometry]] not including the [[parallel postulate]] might not decide the parallel postulate, as shown by models for both Euclidean and non-Euclidean geometry. What was novel about Gödel's results is that they worked directly at the level of syntax and applied to any (effectively generated) extension of arithmetic, producing sentences which in effect imply their unprovability.

To some extent, Gödel's incompleteness theorems have always had an air of mystery about them, or at least a reputation of being exceedingly difficult or subtle. The major insight of Gödel, that items in formal logic can be encoded within the recursive structures afforded by arithmetic ([[Gödel numbering]]), is of course a signal achievement not to be underestimated, and the detailed working out of the result that the provability predicate can be encoded as a [[recursive set]] can be said to form the technical core of this work. But the punchline, in the form of his [[diagonalization argument]] that produces a true[^fine1] but unprovable sentence, should be seen for what it is: one of many [[diagonalization arguments]] (including for example [[Cantor's theorem]]) that are essentially alike in structure, and familiar to all mathematicians.

In an interview ([Lawvere 07](William+Lawvere#Interview07)) not long after Gödel's 100th birthday, [[William Lawvere]] answered the question

> We have recently celebrated Kurt Gödel's 100th birthday. What do you think about the extra-mathematical publicity around his incompleteness theorem?

by saying (reproduced in [Lawvere 69 reprint, p. 2](#Lawvere69)):

> "In 'Diagonal arguments and Cartesian closed categories' ([Lawvere 69](#Lawvere69)) we demystified the incompleteness theorem of Gödel and the truth-definition theory of Tarski by showing that both are consequences of some very simple algebra in the Cartesian-closed setting. It was always hard for many to comprehend how Cantor's mathematical theorem could be re-christened as a "paradox" by Russell and how Gödel's theorem could be so often declared to be the most significant result of the 20th century. There was always the suspicion among scientists that such extra-mathematical publicity movements concealed an agenda for re-establishing belief as a substitute for science. Now, one hundred years after Gödel's birth, the organized attempts to harness his great mathematical work to such an agenda have become explicit.

[[Yuri Manin]] writes in [Manin02, p. 7](#Manin02):

> Baffling discoveries such as Gödel's incompleteness of arithmetic lose some of their mystery once one comes to understand their content as a statement that a certain algebraic structure simply is not finitely generated with respect to the allowed composition laws.

This diagonalization result, so often obscured, should be laid bare as the simple piece of algebra that it is. We give one approach here, based on the algebra of [[hyperdoctrines]].

## Gödel\'s incompleteness theorems

### Categorical preliminaries

We define the [[equational theory]] $\mathbf{PRA}$ ([[primitive recursive arithmetic]]) to be [[initial object|initial]] among [[Lawvere theories]] whose generator $1$ is a parametrized [[natural numbers object]]. The [[hom-set]] of morphisms $0 \to 1$ in $\mathbf{PRA}$ is the set of equivalence classes of closed [[terms]], and is identified with the set $\mathbb{N}$ of [[numerals]]. (This $\mathbb{N}$ will serve double duty in a moment, as being also the set of objects of the Lawvere theory.)

The [[first-order theory]] of [[Peano arithmetic]], PA for short, can be presented as a [[Boolean hyperdoctrine]]

$$T: \mathbf{PRA}^{op} \to BooleanAlgebra$$

taking $j \in Ob(\mathbf{PRA}) = \mathbb{N}$ to the set $T(j)$ consisting of $j$-ary [[predicates]] in the language of PA, modulo provable equivalence in PA. If $f: j \to k$ is a morphism of $\mathbf{PRA}$ and $R \in T(k)$, we let $f^\ast (R)$ denote $T(f)(R) \in T(j)$; it can be described as the result of substituting or pulling back $R$ along $f$. There is a corresponding "action" of $\mathbf{PRA}$ on $T$,

$$\mathbf{PRA}(j, k) \times T(k) \stackrel{action}{\to} T(j)$$

$$\,$$

$$(f, R) \mapsto f^\ast(R),$$

which allows us to regard $T$ as a $\mathbf{PRA}$-module.

### Syntactic terminology and notation

Let $Form(PA)$ be the set of all formulas of PA, with $\Phi(j)$ the set of formulas with $j$ [[free variables]]. There is an equivalence relation on $\Phi(j)$, where two formulas $F, F'$ belong to the same equivalence class $[F]$ if they are provably equivalent in PA. Thus $[-]: \Phi(j) \to T(j)$ is the corresponding quotient map. Let $Term(0)$ denote the set of closed terms in the logical theory $PRA$; morphisms $0 \to 1$ in the Lawvere theory $\mathbf{PRA}$ are equivalence classes of terms, and we let $ev: Term(0) \to \mathbf{PRA}(0, 1)$ be the corresponding quotient map. The map $ev$ has a section $num: \mathbf{PRA}(0, 1) \to Term(0)$ which sends each morphism $0 \stackrel{'n'}{\to} 1$ to the corresponding numeral term (the $n$-fold successor of the zero term). Finally, we have a substitution map

$$sub: Term(0) \times \Phi(1) \to \Phi(0)$$

that substitutes a closed term $\tau$ in for the free variable of $F \in \Phi(1)$, giving a closed formula $sub(\tau, F) \in \Phi(0)$.

There is a standard Gödel coding taking formulas to numerals,

$$code: Form(PA) \to \mathbf{PRA}(0, 1),$$

and we define an application pairing $\cdot$ to be the composite along the top of the following commutative diagram:

$$\array{
Form(PA) \times \Phi(1) & \stackrel{code \times 1}{\to} & \mathbf{PRA}(0, 1) \times \Phi(1) & \stackrel{num \times 1}{\to} & Term(0) \times \Phi(1) &  \stackrel{sub}{\to} & \Phi(0) \\
 & & & _\mathllap{1 \times [-]} \searrow & \downarrow _\mathrlap{ev \times [-]} & & \downarrow _\mathrlap{[-]} \\
 & & & & \mathbf{PRA}(0, 1) \times T(1) & \underset{action}{\to} & T(0).
}$$

In other words, for all formulas $F \in Form(PA)$ and $R \in \Phi(1)$, we define the pairing $F \cdot R \coloneqq sub(num (code(F)), R)$, and according to the commutative diagram, we have the equation

$$\label{eqn1} [F \cdot R] = code(F)^\ast ([R])$$

which will be used in our diagonalization result.

### Diagonalization

The following result is routine from general recursive-arithmetic properties of Gödel coding; it applies to any extension of PA (or just [[Robinson arithmetic]]) whose formulas can be recursively (computably) enumerated.

+-- {: .num_lemma}
###### Lemma
There is a primitive recursive function (i.e., a morphism $d: 1 \to 1$ in $\mathbf{PRA}$) such that for all $t \in \Phi(1)$, we have $d \circ code(t) = code (t \cdot t)$.[^fine2]
=--

+-- {: .num_theorem #fix}
###### Theorem
For every $R \in \Phi(1)$ there is a fixed point in $\Phi(0)$, i.e., a closed formula $s \in \Phi(0)$ such that $s \Leftrightarrow s \cdot R$ ($s$ is provably equivalent to $s \cdot R$).
=--

+-- {: .proof}
###### Proof
Pick a formula $t \in \Phi(1)$ such that $[t] = d^\ast [R]$, and then put $s = t \cdot t$. We want to show $[s] = [s \cdot R]$. We have

$$\array{
[t \cdot t] & = & code(t)^\ast ([t]) & equation\; (1) \\
 & \coloneqq & code(t)^\ast (d^\ast [R]) & \\
 & = & (d \circ code(t))^\ast ([R]) & (functoriality) \\
 & = & (code (t \cdot t))^\ast ([R]) & (by\; the\; lemma)\\
 & = & [(t \cdot t) \cdot R] & equation\; (1)
}$$

as required.
=--

+-- {: .num_remark}
###### Remark
The proof of theorem \ref{fix} was written to lay bare its provenance as a special case of [[Lawvere's fixed point theorem]], and its essential kinship with the structure of the standard fixpoint combinator in [[combinatory algebra]]. See [Yanofsky 03](#Yanofsky03).
=--

### Incompleteness

Recall there is also a Gödel coding of PA proofs (certain sequences of PA formulas), $code: Proof(PA) \to \mathbf{PRA}(0, 1)$. The following result applies in fact to any theory like PA that extends basic Robinson arithmetic and whose axioms form a recursive set.

+-- {: .num_lemma}
###### Lemma
There is a binary primitive recursive predicate $Pf$ such that $Pf(p, x)$ is true in $\mathbb{N}$ if $p$ is the code of a proof of a sentence (a closed formula) with code $x$. Thus there is an unary recursive predicate $Prov \coloneqq \exists_p Pf(p, x) \in T(1)$. Similarly, there is an unary recursive predicate $R = \neg Prov \in T(1)$.
=--

+-- {: .num_theorem}
###### Theorem
**(Gödel's First Incompleteness Theorem)**
There exists a sentence $s$ in the language of PA such that $s$ is logically equivalent to the PA-sentence $code(s)^\ast (\neg Prov)$ (whose interpretation in the model $\mathbb{N}$ is that $s$ has no PA proof).
=--

Thus, if $s$ is false in $\mathbb{N}$, then $s$ has a PA proof (so that PA proves a false statement in a model: is inconsistent). In classical logic, the only other alternative is that $s$ is true in $\mathbb{N}$, but then this true sentence (in the language of PA) has no PA proof.

+-- {: .proof}
###### Proof
Indeed, let $R$ be a formula that represents the unary predicate $\neg Prov$, and let $s$ be a fixed point in the sense $[s] = [s \cdot R]$, which is guaranteed to exist by theorem \ref{fix}. The equality here is logical equivalence, and $[s \cdot R] = code(s)^\ast (\neg Prov)$ by equation (eq:eqn1).
=--

## Related concepts

* [[Löb's theorem]]

* [[Lawvere's fixed point theorem]]

* [[Chaitin's incompleteness theorem]]

* [[Deligne completeness theorem]]

* [[foundations of mathematics]]

* [[platonism]]

* [[arithmetic universe]]

## References
 {#References}

* {#Lawvere69} [[William Lawvere]], _Diagonal Arguments and Cartesian Closed Categories_, Lecture Notes in Mathematics, 92 (1969), 134-145 ([TAC](http://tac.mta.ca/tac/reprints/articles/15/tr15abs.html))

* {#Manin02} [[Yuri Manin]], _Georg Cantor and his heritage_, [arXiv:math/0209244](http://arxiv.org/abs/math/0209244)

* {#Yanofsky03} Noson S. Yanofsky, _A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points_, arXiv:math/0305282 ([web](http://arxiv.org/abs/math/0305282))

A brief review discussion explicitly in the context of [[type theory]]/[[topos theory]] is in

* Erik Gregersen, _[Internal lanuage](http://www.britannica.com/EBchecked/topic/369221/foundations-of-mathematics/35468/Internal-language)_, Encyclopedia Britannica

A [[formal proof]] of the Gödel-Rosser incompleteness theorem in [[Coq]] is given in

* [[Russell O'Connor]], *Essential Incompleteness of Arithmetic Verified by Coq*, in: *Theorem Proving in Higher Order Logics. TPHOLs 2005*, Lecture Notes in Computer Science **3603**, Springer (2005) 245-260 [[web](http://r6.ca/Goedel/goedel1.html), [arXiv:cs/0505034](https://arxiv.org/abs/cs/0505034), [doi:10.1007/11541868_16](https://doi.org/10.1007/11541868_16)]

Formal proof of the second incompleteness theorem:

* Lawrence Paulson, _A Mechanised Proof of Gödel's Incompleteness Theorems using Nominal Isabelle_ [[pdf](https://www.cl.cam.ac.uk/~lp15/papers/Isabelle/Goedel-ar.pdf)]

Unified account of [[Löb's theorem]]/[[Gödel's second incompleteness theorem]], [[Kripke semantics]] for [[provability logic]], and [[guarded recursion]], via [[essentially algebraic theories]] which "self-internalize"::

* [[Sridhar Ramesh]], *Introspective Theories and Geminal Categories*, PhD thesis, Berkeley (2023) [[escholarship:3mn0c475](https://escholarship.org/uc/item/3mn0c475), [[Ramesh-IntrospectiveTheories.pdf:file]]]

The following contains a careful discussion of the incompleteness theorem in the context of categorical foundations using the [[free topos]]:

* [[Jim Lambek]], [[Phil Scott]], _Reflections on Categorical Foundations of Mathematics_ , pp.171-185 in  Sommaruga (ed.), _Foundational Theories of Classical and Constructive Mathematics_,  Springer New York 2011. ([draft](https://www.site.uottawa.ca/~phil/papers/LS11.final.pdf)) {#LS11}

A _bref survol technique_ of Gödel's theorem within the landscape of [[Hilbert's program]] can be found in the first chapters of

* [[Jean-Yves Girard]], _Le point aveugle I_ , Hermann Paris 2006. ([chapters 1-2](http://iml.univ-mrs.fr/~girard/cours/hermann.pdf.gz))

See also the discussion in [[list-arithmetic pretoposes]]

* [[Maria Maietti]], _Joyal's arithmetic universe as list-arithmetic pretopos_, Theory and Applications of Categories, Vol. 24, 2010, No. 3, pp 39-83. ([TAC](http://www.tac.mta.ca/tac/volumes/24/3/24-03abs.html))

* {#vanDijkOldenziel20} Joost van Dijk, Alexander Gietelink Oldenziel, _Gödel's Incompleteness after Joyal_ , arXiv:2004.10482 (2020). ([abstract](https://arxiv.org/abs/2004.10482))

[^fine1]: Assuming that the theory is consistent. If the theory is inconsistent, then it can prove falsity and thence anything, including any internal statements of consistency.

[^fine2]: This and the next lemma demonstrate the essential truth of Paul Taylor's quip (Practical Foundations of Mathematics, p. 192): "Lemmas do the work in mathematics: Theorems, like management, just take the credit. A good lemma also survives a philosophical or technological revolution."

[[!redirects incompleteness theorem]]
[[!redirects incompleteness theorems]]
[[!redirects Gödel incompleteness theorem]]
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[[!redirects Gödel's incompleteness theorem]]
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inconsistency

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A system in [[formal logic]] is called _inconsistent_ if it admits a [[proof]] of a _[[contradiction]]_ (that is, usually, a proof of [[false]], or an [[inhabitant]] of the [[empty type]]).

Accordingly an [[axiom]] is called _inconsistent_ or to _lead to an inconsistency_ if adding it to an (implicitly understood) ambient logical system makes that system inconsistent.

In most usual logical systems, it follows that an inconsistent system admits a proof of *every* proposition, by the rule [[ex falso quodlibet]] (which is just the elimination rule for the [[empty type]]).  For this reason, sometimes (especially in [[type theory]]), the adjective "inconsistent" is used to mean a system with this property instead.  If we want to distinguish, then a system which admits a proof of every proposition may be called *trivial*.

As a further complication, a [[paraconsistent logic]] is often described as 'inconsistent but not trivial'.  However, many paraconsistent logics (such as [[dual-intuitionistic logic]]) admit ex falso quadlibet and fail to prove $\bot$; their 'inconsistency' is a proof of $p \wedge \neg{p}$ (or two proofs, one of $p$ and one of $\neg{p}$), which then fails to entail $\bot$.  We thus get three distinct levels of inconsistency:
* triviality (proving everything),
* proving a statement identified as $\bot$,
* proving $p$ and $\neg{p}$ for some statement $p$ and using an operator identified as [[negation]].

## Examples

* [[Russell's paradox]]

* [[Burali-Forti's paradox]]

## Related concepts

* [[consistency]]

* [[paraconsistent logic]]

[[!redirects inconsistency]]
[[!redirects inconsistencies]]
[[!redirects inconsistent system]]
[[!redirects inconsistent systems]]
[[!redirects inconsistent axiom]]
[[!redirects inconsistent axioms]]
[[!redirects inconsistent theory]]
[[!redirects inconsistent theories]]


ind-coherent sheaves

The [[stable (infinity,1)-category]] of _ind-coherent sheaves_ on a [[derived stack]] $X$ is defined as the [[(infinity,1)-category]] of [[ind-objects in an (infinity,1)-category|ind-objects]] in the [[(infinity,1)-category]] of [[coherent sheaves]] on $X$.

## References

* [[D. Gaitsgory]], [[N. Rozenblyum]], _[[A study in derived algebraic geometry]]_.


ind-object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An **ind-object** of a [[category]] $\mathcal{C}$ is a **formal [[filtered colimit]]** of objects of $\mathcal{C}$.  Here "formal" means that the colimit is taken in the [[category of presheaves]] of $\mathcal{C}$ (the [[free cocompletion]] of $\mathcal{C}$). The category of ind-objects of $\mathcal{C}$ is written $ind$-$\mathcal{C}$ or $Ind(\mathcal{C})$.

Here, "ind" is short for "[[inductive system]]", as in the inductive systems used to define [[directed colimits]], and as contrasted with "pro" in the dual notion of [[pro-object]] corresponding to "projective system".

Recalling the nature of [[filtered colimits]], this means that in particular chains of inclusions

$$
  c_1 \hookrightarrow c_2 \hookrightarrow c_3 \hookrightarrow c_4 \hookrightarrow \cdots
$$

of objects in $\mathcal{C}$ are regarded to converge to an object in $Ind(\mathcal{C})$, even if that object does not exist in $\mathcal{C}$ itself.  Standard examples where ind-objects are relevant are categories $\mathcal{C}$ whose objects are finite in some sense, such as [[finite sets]] or [[finite dimensional vector spaces]]. Their ind-categories contain then also the infinite versions of these objects as limits of sequences of inclusions of finite objects of ever increasing size.

Moreover, ind-categories allow one to handle "big things in terms of small things" also in another important sense: many [[large category|large categories]] are actually ([[equivalence of categories|equivalent]] to) ind-categories of [[small category|small categories]]. This means that, while [[large category|large]], they are for all practical purposes controlled by a [[small category]] (see the description of the [[hom-set]] of $Ind(\mathcal{C})$ in terms of that of $\mathcal{C}$ below). Such [[large category|large categories]] equivalent to ind-categories are therefore called [[accessible category|accessible categories]].

## Definition

There are several equivalent ways to define ind-objects.

### As diagrams

One definition is to define the objects of $ind$-$\mathcal{C}$ to be diagrams $F:D\to \mathcal{C}$ where $D$ is a [[small category|small]] [[filtered category|filtered]] category.
The idea is to think of these diagrams as being the placeholder for the [[colimit]] over them (possibly non-existent in $\mathcal{C}$).
We identify an ordinary object of $\mathcal{C}$ with the corresponding diagram $1\to \mathcal{C}$.  To see what the morphisms should be between $F:D\to \mathcal{C}$ and $G:E\to \mathcal{C}$, we stipulate that

1. The embedding $\mathcal{C}\to ind$-$\mathcal{C}$ should be [[full and faithful functor|full and faithful]],
1. each diagram $F:D\to \mathcal{C}$ should be the colimit of itself (considered as a diagram in $ind$-$\mathcal{C}$ via the above embedding), and
1. the objects of $\mathcal{C}$ should be [[compact object|compact]] in $ind$-$\mathcal{C}$.

Thus, we should have

$$
\begin{aligned}
  ind\text{-}\mathcal{C}(F,G) &= ind\text{-}\mathcal{C}(colim_{d\in D} F d, colim_{e\in E} G   e)\\
   &\cong lim_{d\in D}\; ind\text{-}\mathcal{C}(F d, colim_{e\in E} G e)\\
   &\cong lim_{d\in D} colim_{e\in E}\; ind\text{-}\mathcal{C}C(F d, G e)\\
   &\cong lim_{d\in D} colim_{e\in E}\; \mathcal{C}(F d, G e)
\end{aligned}$$

Here

* the first step is by assumption that each object is a suitable colimit;

* the second by the fact that the contravariant Hom sends colimits to limits (see properties of [[colimit]]);

* the third by the assumption that each object is a [[compact object]];

*  the last by the assumption that the embedding is a [[full and faithful functor]].

So then one _defines_

$$
  ind\text{-}\mathcal{C}(F,G) \coloneqq lim_{d\in D} colim_{e\in E}\; \mathcal{C}(F d, G e)
  \,.
$$

### As filtered colimits of representable presheaves

Recall the [[co-Yoneda lemma]] that every presheaf $X \in PSh(\mathcal{C})$ is a [[colimit]] over [[representable functor|representable presheaves]]:

there is a functor $\alpha : D \to \mathcal{C}$ (with $D$ possibly large) such that

$$
  X \simeq colim_{d \in D} Y(\alpha(d))
$$
(with $Y$ the [[Yoneda embedding]]).

+-- {: .num_defn}
###### Definition

Let $Ind(\mathcal{C}) \subset PSh(\mathcal{C})$ be the [[full subcategory]] of the [[presheaf category]] $PSh(\mathcal{C}) = [\mathcal{C}^{op},Set]$ on those [[functors]]/[[presheaves]] which are [[filtered colimits]] of [[representable functor|representables]], i.e. those for which
$$
  X \simeq colim_{d \in D} Y(\alpha(d))
$$

for $D$ some [[filtered category]].

Those for which $D$ may be chosen to be $\mathbb{N}^{\leq}$, i.e. those that arise as [[sequential colimits]], are also called _[[strict ind-objects]]_.

=--

+-- {: .num_remark}
###### Remark

The functors $\mathcal{C}^{op}\to Set$ belonging to $Ind(\mathcal{C})$ under this definition --- those which are filtered colimits of representables --- have an equivalent characterization as the [[flat functors]]: those which "would preserve all finite colimits if $\mathcal{C}$ had them".  In particular, if $\mathcal{C}$ has finite colimits, then $Ind(\mathcal{C})$ consists exactly of the finitely cocontinuous presheaves.

For more equivalent characterizations see at _[accessible category -- Definition](accessible+category#definition)_.

=--

+-- {: .num_remark}
###### Remark

Given that $[\mathcal{C}^{op},Set]$ is the [[free cocompletion]] of $\mathcal{C}$, $Ind(\mathcal{C})$ defined in this way is its "free cocompletion under filtered colimits."

=--

To compare with the first definition, notice that indeed the formula for the [[hom-sets]] is reproduced:

Generally we have

$$
  \begin{aligned}
     [\mathcal{C}^{op},Set](X,Y)
     & \simeq
     [\mathcal{C}^{op}, Set](colim_{d \in D} Y F d, colim_{d' \in D'} Y G d)
     \\
     & \simeq
     lim_{d \in D} [\mathcal{C}^{op}, Set]( Y F d, colim_{d' \in D'} Y G d)
  \end{aligned}
$$

by the fact that the [[hom-functor]] sends [[colimits]] to [[limits]] in its first argument (see _properties_ at _[[colimit]]_).

By the [[Yoneda lemma]] this is

$$
  \cdots \simeq
   lim_{d \in D} (colim_{d' \in D'} Y G d')(F d)
  \,.
$$

Using that [[colimits]] in $PSh(\mathcal{C})$ are computed objectwise (see again _properties_ at _[[colimit]]_) this is

$$
  \cdots \simeq
   lim_{d \in D} colim_{d' \in D'}
   \mathcal{C}(F d, G d')  \,.
$$

## Examples

* Let [[FinDimVect]] be the [[category]] of [[finite-dimensional vector spaces]] (over some [[field]]). Let $V$ be an infinite-dimensional vector space. Then $V$ can be regarded as an object of $ind-FinDimVect$ as the colimit $colim_{V' \hookrightarrow V} Y(V')$ over the [[filtered category]] whose objects are inclusions $V' \hookrightarrow V$ of finite dimensional vector spaces $V'$ into $V$ of the representables $Y(V') : FinDimVect^{op} \to Set$ ($Y$ is the [[Yoneda embedding]]).

* For $\mathcal{C}$ the category of finitely presented objects of some equationally defined structure, $ind\text{-}\mathcal{C}$ is the category of all these structures.

  * The category [[Grp]] of [[group]]s is the ind-category of the category of finitely generated groups.

    * The category [[Ab]] of [[abelian group]]s is the ind-category of the category of finitely generated abelian groups.

  * The category [[Set]] of [[sets]] is the ind-category of the category of [[finitely indexed sets]].

* A [[formal scheme]] is an ind-object in [[schemes]].

## Properties

In the following we write $\underset{\longrightarrow}{\lim}^f$ for the "formal colimits" defining ind-objects. I.e. if $\alpha \colon \mathcal{I} \to \mathcal{C}$ is a small [[diagram]] and with $i \colon Ind(\mathcal{C}) \hookrightarrow PSh(\mathcal{C})$ the defining inclusion, then

$$
  \underset{\longrightarrow}{\lim}^f (\alpha)
  \;\coloneqq\;
  \underset{\longrightarrow}{\lim} (i \circ \alpha )
  \,.
$$

### The category of ind-objects

+-- {: .num_prop}
###### Proposition

If $\mathcal{C}$ is a [[locally small category]] then so is $Ind(\mathcal{C})$.

=--

+-- {: .num_prop}
###### Proposition

$Ind(\mathcal{C})$ admits small [[filtered colimits]] and the defining inclusion $Ind(\mathcal{C}) \hookrightarrow PSh(\mathcal{C})$ commutes with these colimits.

=--

(e.g. [KashiwaraSchapira 06, theorem 6.1.8](#KashiwaraSchapira06))

The following says that morphisms between ind-objects are represented by [[natural transformation]] of the filtered diagrams that represent them, possibly up to restricting these diagrams first along a [[final functor]].

+-- {: .num_prop #MorphismsRepresentedByCofilteredSystemsOfMorphisms}
###### Proposition

Let

1. $\mathcal{I}_1$ and $\mathcal{I}_2$ be two [[small category|small]] [[filtered categories]];

1. $\alpha_1 \colon \mathcal{I}_1 \longrightarrow \mathcal{C}$ and $\alpha_2 \colon \mathcal{I}_2 \longrightarrow \mathcal{C}$ be two [[functors]];

1. $f
    \;\colon\;
  \underset{\longrightarrow}{\lim}^f \alpha_1
    \longrightarrow
  \underset{\longrightarrow}{\lim}^f \alpha_2$

  a [[morphism]] between their images in $Ind(\mathcal{C})$.

Then there exists

1. a small [[filtered category]] $K$

1. [[final functors]] $p_i \colon K \longrightarrow \mathcal{I}_i$

1. a [[natural transformation]] $\phi \;\colon\; \alpha_1 \circ p_1  \longrightarrow \alpha_2 \circ p_2$

such that the following [[commuting diagram|diagram commutes]]

$$
  \array{
    \underset{\longrightarrow}{\lim}^f(\alpha_1 \circ p_1)
      &\overset{\underset{\longrightarrow}{\lim}^f \phi}{\longrightarrow}&
    \underset{\longrightarrow}{\lim}^f(\alpha_2 \circ p_2)
    \\
    {}^{\mathllap{\simeq}}\downarrow
      &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \underset{\longrightarrow}{\lim}^f \alpha_1
      &\underset{f}{\longrightarrow}&
    \underset{\longrightarrow}{\lim}^f \alpha_2
  }
  \,.
$$

=--

(e.g. [KashiwaraSchapira 06, prop. 6.1.13](#KashiwaraSchapira06), [Artin-Mazur 69, appendix 3, prop. (3.1), corollary (3.2)](#ArtinMazur69))

+-- {: .num_cor}
###### Corollary

For each $f \colon A_1 \longrightarrow A_2$ a [[morphism]] in $Ind(\mathcal{C})$, then there exists

1. a [[small category|small]] [[filtered category]] $\mathcal{I}$;

1. [[functors]] $\alpha_i \;\colon\; \mathcal{I} \to \mathcal{C}$ ($i \in \{1,2\}$);

1. a [[natural transformation]] $\phi \colon \alpha_1 \longrightarrow \alpha_2$

such that

$$
  \array{
    A_1 &\overset{f}{\longrightarrow}& A_2
    \\
    {}^{\mathllap{\simeq}}\downarrow
      &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \underset{\longrightarrow}{\lim}^f \alpha_1
      &\underset{\underset{\longrightarrow}{\lim}^f \phi }{\longrightarrow}&
    \underset{\longrightarrow}{\lim}^f \alpha_2
  }
  \,.
$$

=--

(e.g. [KashiwaraSchapira 06, corollary 6.1.14](#KashiwaraSchapira06))

+-- {: .num_prop}
###### Proposition

The canonical inclusion $y \;\colon\; \mathcal{C} \hookrightarrow Ind(\mathcal{C})$ (factoring the [[Yoneda embedding]]) is [[exact functor|right exact]].

=--

(e.g. [KashiwaraSchapira 06, corollary 6.1.6](#KashiwaraSchapira06))

+-- {: .num_prop}
###### Proposition

Let $\mathcal{C}$ have all [[finite limits]] or all [[small limits]]. Then also $Ind(\mathcal{C})$ has all finite or small limits, respectively, and the canonical functor $y \;\colon\; \mathcal{C} \longrightarrow Ind(\mathcal{C})$ preserves these, respectively.

=--

(e.g. [KashiwaraSchapira 06, corollary 6.1.17](#KashiwaraSchapira06))

+-- {: .num_prop}
###### Proposition

If $\mathcal{C}$ has [[cokernels]], then so does $Ind(\mathcal{C})$.

If $\mathcal{C}$ has [[finite colimit]] [[coproducts]], then $Ind(\mathcal{C})$ has small [[coproducts]].

If $\mathcal{C}$ has all [[finite colimits]], then $Ind(\mathcal{C})$ has all small [[colimits]].

=--

(e.g. [KashiwaraSchapira 06, prop. 6.1.18](#KashiwaraSchapira06))

### Recognition of Ind-objects

+-- {: .num_prop}
###### Proposition

A [[functor]] $F \colon \mathcal{C}^{op} \to Set$ is in $Ind(\mathcal{C})$ (i.e. is a [[filtered colimit]] of [[representable functor|representables]]) precisely if the [[comma category]] $(Y,const_F)$ (with $Y$ the [[Yoneda embedding]]) is [[filtered category|filtered]] and [[cofinally small category|cofinally small]].

=--

+-- {: .num_prop}
###### Proposition

If $\mathcal{C}$ admits finite [[colimits]], then $Ind(\mathcal{C})$ is the [[full subcategory]] of the [[presheaf]] category $PSh(\mathcal{C})$ consisting of those functors $F \colon \mathcal{C}^{op} \to Set$ such that $F$ is [[exact functor|left exact]] and the [[comma category]] $(Y,F)$ (with $Y$ the [[Yoneda embedding]]) is [[cofinally small category|cofinally small]].

=--

### Functoriality

Ind-cocompletion is functorial -- in fact an underlying 2-functor of a [[lax-idempotent 2-monad]] (KZ-monad). More in detail:

+-- {: .num_prop #FunctorialityOfInd}
###### Proposition

Let $F \colon \mathcal{C}_1 \longrightarrow \mathcal{C}$ be a [[functor]]. Then there is a unique [[extension]] $Ind(F)$ of this functor to ind-objects, i.e. a [[commuting diagram]]

$$
  \array{
    \mathcal{C}_1 &\overset{F}{\longrightarrow}& \mathcal{C}_2
    \\
    \downarrow
      &&
    \downarrow
    \\
    Ind(\mathcal{C}_1)
      &\underset{Ind(F)}{\longrightarrow}&
    Ind(\mathcal{C}_2)
  }
  \,,
$$

such that

$$
  Ind(F)( \underset{\longrightarrow}{\lim}^f \alpha )
  \simeq
  \underset{\longrightarrow}{\lim}^f ( F \circ \alpha )
  \,.
$$

Moreover,

1. this construction respects [[composition]] in that

   $$
     Ind(G \circ F ) \simeq Ind(G) \circ Ind(F)
   $$

1. if $F$ is a [[faithful functor]] or [[fully faithful functor]], then so is $Ind(F)$, respectively.

=--

(e.g. [KashiwaraSchapira 06, prop.6.1.9-6.1.11](#KashiwaraSchapira06))

+-- {: .num_prop}
###### Proposition

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two [[small categories]]. By prop. \ref{FunctorialityOfInd} the two [[projections]] out of their [[product category]] induce a functor of the form

$$
  Ind(\mathcal{C}_1 \times \mathcal{C}_2)
    \longrightarrow
  Ind(\mathcal{C}_1) \times Ind(\mathcal{C}_2)
  \,.
$$

This is an [[equivalence of categories]].

=--

([KashiwaraSchapira 06, prop. 6.1.12](#KashiwaraSchapira06))

+-- {: .num_prop}
###### Proposition

Let $\mathcal{C}$ be a [[small category]] that has [[finite colimits]] and let $\mathcal{I}$ be a small category. Then the canonical functor

$$Ind(\mathcal{C}^{\mathcal{I}}) \to Ind(\mathcal{C})^{\mathcal{I}}$$

is an [[equivalence of categories]].

=--

([Henry 23](#Henry23))

### The case that $\mathcal{C}$ already admits filtered colimits

([KashiwaraSchapira 06, chapter 6.3](#KashiwaraSchapira06))

+-- {: .num_prop #ReflectionToYonedaEmbedding}
###### Proposition

Let $\mathcal{C}$ be a [[category]] which has all small [[filtered colimits]]. Then the canonical functor $\mathcal{C} \longrightarrow Ind(\mathcal{C})$ defines a [[reflective subcategory]], i.e. it is a [[fully faithful functor]] with a [[left adjoint]] $L$

$$
  \mathcal{C}
    \underoverset
     {\underset{}{\longrightarrow}}
     {\overset{L}{\longleftarrow}}
     {\bot}
  Ind(\mathcal{C})
$$

which takes formal filtered colimits to actual filtered colimits in $\mathcal{C}$:

$$
  L
    \;\colon\;
  \underset{\longrightarrow}{\lim}^f \alpha
    \mapsto
  \underset{\longrightarrow}{\lim} \alpha
$$

=--

([KashiwaraSchapira 06, prop. 6.3.1](#KashiwaraSchapira06))

+-- {: .num_prop #JFIsFullyFaithful}
###### Proposition

Let $F \colon \mathcal{J} \longrightarrow \mathcal{C}$ be a [[functor]] such that

1. $F$ is a [[fully faithful functor]];

1. $\mathcal{C}$ has all small [[filtered colimits]];

1. for every object $J \in \mathcal{J}$ its image $F(J) \in \mathcal{C}$ is [[compact object|compact]].

Then the composite

$$
  Ind(\mathcal{J})
    \overset{Ind(F)}{\longrightarrow}
  Ind(\mathcal{C})
    \overset{L}{\longrightarrow}
  \mathcal{C}
$$

(with $Ind(F)$ from prop. \ref{FunctorialityOfInd} and $L$ from prop. \ref{ReflectionToYonedaEmbedding}) is a [[fully faithful functor]].

=--

([KashiwaraSchapira 06, prop. 6.3.4](#KashiwaraSchapira06))

+-- {: .num_prop}
###### Proposition

If $\mathcal{C}$ is a [[category]] such that

1. every object of $\mathcal{C}$ is a [[filtered colimit]] of [[compact objects]];

1. $\mathcal{C}$ has all small [[filtered colimits]]

then the composite functor

$$
  Ind(\mathcal{C}_{cpt})
    \longrightarrow
  Ind(\mathcal{C})
    \overset{L}{\longrightarrow}
  \mathcal{C}
$$

(from prop. \ref{JFIsFullyFaithful}, where $\mathcal{C}_{cpt} \hookrightarrow \mathcal{C}$ is the [[full subcategory]] of [[compact objects]]) is an [[equivalence of categories]].

=--

([KashiwaraSchapira 06, corollary 6.3.5](#KashiwaraSchapira06))

## Applications

* One important use of categories of ind-objects is in [[abelian sheaf]]-theory: for every [[small category|small]] [[abelian category]] $\mathcal{C}$ the category $ind\text{-}\mathcal{C}$ is a [[Grothendieck category]] and hence a good coefficient object for [[abelian sheaf cohomology]].

## In higher category theory

### In $(\infty,1)$-categories

There is a notion of [[ind-object in an (∞,1)-category]].

With regard to the third of the properties listed above, notice that the [[comma category]] $(Y,const_F)$ is the [[category of elements]] of $F$, i.e. the [[pullback]] of the [[generalized universal bundle|universal Set-bundle]] $U : Set_* \to Set$ along $F : \mathcal{C}^{op} \to Set$. This means that the [[stuff, structure, property|forgetful functor]]
$
  (Y,const_F) \to \mathcal{C}
$
is the fibration classified by $F$.

This is the starting point for the definition at [[ind-object in an (∞,1)-category]].

## Related concepts

* **ind-object** / [[ind-object in an (∞,1)-category]]

* [[pro-object]] / [[pro-object in an (∞,1)-category]]

* [[sind-object]]

* [[ind-pro-object]]

* [[strict ind-object]]

* [[flat functor]]

* [[KZ-monad]]

* [[continuous category]]

## References

Ind-categories were introduced in

* [[Alexander Grothendieck]], [[Jean-Louis Verdier]] in [[SGA4]] Exp. 1 [pdf file](http://sage.math.washington.edu/home/wstein/www/home/craigcitro/sga4/Grothendieck/SGA4/sga41.pdf)

and the dual notion of [[pro-object]] in

* A. Grothendieck, _Techniques de déscente et théorèmes d'existence en géométrie algébrique, II: le théorème d'existence en théorie formelle des modules_, Seminaire Bourbaki __195__, 1960, [(pdf)](http://archive.numdam.org/ARCHIVE/SB/SB_1958-1960__5_/SB_1958-1960__5__369_0/SB_1958-1960__5__369_0.pdf).

Ind-objects are discussed in

* {#ArtinMazur69} [[Michael Artin]], [[Barry Mazur]], appendix of _Étale homotopy theory_, Lecture Notes in Maths. 100, Springer-Verlag, Berlin 1969.

(in their dual guise as [[pro-objects]])

* {#KashiwaraSchapira06} [[Masaki Kashiwara]], [[Pierre Schapira]], section 6 of _[[Categories and Sheaves]]_ , Grundlehren der mathematischen Wissenschaften 332 (2006).

The relation between the Ind-completion and the ideal completion in order theory is discussed in section 1 of

* [[Peter Johnstone]], [[André Joyal]], _Continuous categories and exponentiable toposes_ ,  JPAA **25** (1982) pp.255-296.

See also

* [[Peter Johnstone]], section VI.1 of _[[Stone Spaces]]_

They are discussed in relation to generalisations in

* [[Jiří Adámek]], [[Francis Borceux]], [[Stephen Lack]], [[Jiri Rosicky|Jiří Rosický]], _A classification of accessible categories,_ Journal of Pure and Applied Algebra 175:7-30, 2002. [abstract](http://maths.mq.edu.au/~slack/papers/acc.html)

See also the remarks at the beginning of section 5.3 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

Some relevant results are contained in:

* {#Henry23} [[Simon Henry]] [MathOverflow answer](https://mathoverflow.net/a/442827)

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ind-object (infinity,1)-category > history

< [[ind-object (infinity,1)-category]]
[[!redirects ind-object (infinity,1)-category -- history]]


ind-object in an (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _ind-object_ and _ind-category_ in an [[(∞,1)-category]] is the straightforward generalization of the notion of [[ind-object]] in an ordinary category. See there for idea and motivation.

We describe $\kappa$-ind-objects for $\kappa$ a [[regular cardinal]].

## Definition

The different equivalent definitions of ordinary [[ind-objects]] have their analog for [[(∞,1)-categories]].

Let in the following $C$ be a small [[(∞,1)-category]].

### In terms of formal colimits

The definition in terms of formal colimits is precisely analogous to the one for ordinary [[ind-object]]s, with colimits and limits replaced by the corresponding $\infty$-notion (compare [[homotopy limit]] and [[limit in quasi-categories]])

So the objects of $Ind C$ are small filtered diagrams $X : D_X \to C$ in $C$, and the morphisms are given by

$$
  Hom_{Ind C}(X,Y) := lim_{d\in D_X} colim_{d' \in D_Y} Hom_C(X(d), Y(d'))
  \,.
$$

(... should be made more precise...)

### In terms of filtered fibrations

Write $\kappa$ for a [[regular cardinal]] and write $ind_\kappa \text{-}C$ for the full sub-[[(∞,1)-category of (∞,1)-presheaves]] on those $(\infty,1)$-presheaves

$$
  F : C^{op} \to Top
$$

which classify [[right fibration]]s $\tilde C \to C$ such that $\tilde C$ is $\kappa$-[[filtered (infinity,1)-category|filtered]].

In the case $\kappa = \omega$ write $ind_\kappa\text{-}C = ind\text{-}C$.

### In terms of filtered colimits

Equivalently, an [[(∞,1)-presheaf]] is in $ind_\kappa\text{-}C$ if there exists a $\kappa$-[[filtered (∞,1)-category]] $D$ and an $(\infty,1)$-functor $W: D \to C$ such that $F$ is the colimit over $Y \circ W$, where $Y$ is the [[(∞,1)-Yoneda embedding]].

## Properties

Let $C$ a small $(\infty,1)$-category and $\kappa$ a [[regular cardinal]].

+-- {: .un_prop}
###### Proposition

$Ind_\kappa(C)$ is closed in $PSh(C)$ under $\kappa$-filtered [[(∞,1)-colimits]].

=--

This is [[Higher Topos Theory|HTT, prop. 5.3.5.3]].

+-- {: .un_prop}
###### Proposition

For any $F \in PSh(C)$ the following are equivalent:

1. $F$ is a $\kappa$-filtered colimit in $PSh(C)$ of a diagram in $C$;

1. $F$ belongs to $Ind_\kappa(C)$.

 If, furthermore, $C$ admits $\kappa$-small colimits, then the above are equivalent to

1. $F : C^{op} \to \infty Grpd$ preserves $\kappa$-small limits.

=--

This is [[Higher Topos Theory|HTT, corollary 5.3.5.4]].

+-- {: .un_prop}
###### Proposition

Every object of $C$ is a $\kappa$-[[compact object]] of $Ind_\kappa(C)$.

=--

This is [[Higher Topos Theory|HTT, prop. 5.3.5.5]].

This makes an $\infty$-category of ind-objects a [[compactly generated (∞,1)-category]].

## Related concepts

* [[ind-object]] / **ind-object in an $(\infty,1)$-category**

* [[pro-object]] / [[pro-object in an (∞,1)-category]]

* [[compactly generated (∞,1)-category]]

* [[accessible (∞,1)-category]]

## References

Section 5.3 and in particular 5.3.3 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

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[[!redirects (infinity,1)-categories of ind-objects]]


ind-pro-object

An __ind-pro-object__ in a category $C$
is a formal [[colimit]] of formal [[limit]]s of objects in category $C$. In other words, it
is an [[ind-object]] in the category $pro(C)$ of [[pro-object]]s in $C$. Ind-pro-objects form
a category $ind(pro(C))$. Similarly, there is a category
$pro(ind(C))$ of pro-ind-objects
and its iterated versions. For example,
__higher local fields__ can be considered as ring objects
in (iterated versions of) $pro(ind(C))$.

Regarding that there is a natural map from colimit of a limit to the limit of a colimit, but in general not the
other way around, every ind-pro-object provides a
pro-ind-object: there is a canonical functor
$ind(pro(C))\to pro(ind(C))$.

In some prominent cases in applications, say if $C$ is a [[Quillen exact category]], then certain
subcategory of locally compact objects $ind(pro(C))$ is defined and studied by Beilinson and Kato.

* A. A. Beĭlinson, _How to glue perverse sheaves_, in: K-theory, arithmetic and geometry (Moscow, 1984–1986), 42–51, Lecture Notes in Math. 1289, Springer 1987.
* [[Kazuya Kato]], _Existence theorem for higher local fields_, Geometry & Topology Monographs 3 (2000) 165–195 [doi](http://dx.doi.org/10.2140/gtm.2000.3.165)
* Luigi Previdi, _Locally compact objects in
exact categories_, [arxiv/0710.2509](http://arxiv.org/abs/0710.2509); _Sato Grassmannians for generalized Tate spaces_, Tohoku Math. J. (2) __64__:4 (2012), 489-538, [arxiv/1002.4863](http://arxiv.org/abs/1002.4863) [euclid](http://projecteuclid.org/euclid.tmj/1356038976); _The homotopy determinantal torsor_, poster at Newton Institute, [pdf](http://www-old.newton.ac.uk/programmes/GDO/Previdi.pdf);
_Generalized Tate spaces_, PhD thesis, Yale 2010 [record](http://gradworks.umi.com/34/15/3415298.html)

* Benjamin Hennion, _Formal loops, Tate objects and tangent Lie algebras_, [arxiv/1412.0053](http://arxiv.org/abs/1412.0053)

* Oliver Braunling, Michael Groechenig, Jesse Wolfson, _Tate objects in exact categories_ (with appendix by Jan Šťovíček and Jan Trlifaj), [arxiv/1402.4969](http://arxiv.org/abs/1402.4969)

* [[M. Kapranov]], E. Vasserot, _Vertex algebras and the formal loop space_, Publications Mathématiques de l'IHÉS __100__, 209-269 (2004) [eudml](https://eudml.org/doc/104201) [math.AG/0107143](http://arxiv.org/abs/math/0107143)

* [[M. Kapranov]], _Infinite-dimensional objects in algebraic geometry_, Pathways lectures, Keio university 2007,
real audio videos at [link](http://www.math.hc.keio.ac.jp/coe/videos/kapranov2007)
* D. Gaitsgory, D. Kazhdan, _Representations of algebraic groups over a 2-dimensional local field_,  GAFA 14 (2004), 535–574, [math.RT/0302174](http://arxiv.org/abs/math/0302174)
* D. V. Osipov, A. N. Parshin, _Harmonic analysis on local fields and adelic spaces I_, Izvestiya: Mathematics 2008, 72:5, pp. 915-976 [arxiv/0707.1766](http://arxiv.org/abs/0707.1766)

The following work of Drinfeld does not treat ind-pro-objects (only ind-schemes and some pro-modules over them), but has
influenced many of the works above on Tate objects:

* [[Vladimir Drinfeld]], _Infinite-dimensional vector bundles in algebraic geometry: an introduction_, in: The unity of mathematics, 263–304, Progr. Math., 244, Birkhäuser Boston 2006, [math.AG/0309155](http://arxiv.org/abs/math/0309155)

A category $C_1$ somewhat similar to that of Beilinson and Kato, and its higher analogues $C_n$ were studied in

* D.V. Osipov, _Adeles on $n$-dimensional schemes and categories $C_n$_, Intern. J. Math. __18__:3 (2007) 269-279, [math.AG/0509189](http://arxiv.org/abs/math/0509189)

[[!redirects ind-pro-objects]]
[[!redirects pro-ind-object]]
[[!redirects pro-ind-objects]]


ind-scheme

#Contents#
* table of contents
{:toc}

## Idea

An **ind-scheme** is an [[ind-object]] in the category of [[algebraic scheme]]s which can be presented as a filtered colimit of schemes where all morphisms are [[closed embeddings of schemes]]. Various variants of 'the' category of [[formal scheme]]s are proper subcategories of the category of ind-schemes. While many analytic varieties have a structure of [[algebraic variety|algebraic varieties]], and loops in a manifold or in an analytic variety can be treated via infinite-dimensional manifolds, their algebraic analogues are examples of ind-schemes.

## Related concepts

* [[formal scheme]]

## References

* [[Mikhail Kapranov]], Éric Vasserot, _Vertex algebras and the formal loop space_, Publications Mathématiques de l'IHÉS, 100 (2004), p. 209-269 [numdam](http://www.numdam.org/item?id=PMIHES_2004__100__209_0), [djvu](http://archive.numdam.org/article/PMIHES_2004__100__209_0.djvu), [pdf](http://archive.numdam.org/article/PMIHES_2004__100__209_0.pdf)

* [[Mikhail Kapranov|M. Kapranov]], É. Vasserot, _Formal loops II : a local Riemann–Roch theorem for determinantal gerbes_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 40 no. 1 (2007) 113--133, [numdam](http://www.numdam.org/item?id=ASENS_2007_4_40_1_113_0), [math.AG/0509646](https://arxiv.org/abs/math.AG/0509646)

* [[Alexander Beilinson|A. Beilinson]], [[Vladimir Drinfel'd]], _Quantization of Hitchin's integrable system and Hecke eigensheaves on Hitchin system_, preliminary version ([pdf](http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf))

[[!redirects ind-schemes]]


Indagationes Mathematicae

A mathematical journal by Koninlijk Wiskundig Genootschap (KWG; Royal Dutch Mathematical Society), previously by Koninklijke Akademie van Wetenschappen (KNAW; Royal Netherlands Academy).

Prior to 1990, this journal existed under different names (used in parallel) and different volume numbering schemes (also used in parallel).
See the [Wikipedia page](https://en.wikipedia.org/wiki/Indagationes_Mathematicae) for more information.

Chronology:

1895: Founded under the title of _Koninklijke Akademie van Wetenschappen: Proceedings of the Section of Sciences_.

1939: Mathematics articles published separately, with alternative title (used in parallel) _Indagationes Mathematicae_, with 41 subtracted from the volume number.

1951: Split into three journals.  Series A (Mathematical Sciences) published by North-Holland under the alternative title (used in parallel) _Indagationes Mathematicae (Proceedings)_.

1990: Published under the sole title of _Indagationes Mathematicae_.  Volume numbering reset to 1.

Digital archives:

* 1895–1950: [KNAW Digital Library](https://dwc.knaw.nl/).

* 1951–1989: [Indagationes Mathematicae (Proceedings)](https://www.sciencedirect.com/journal/indagationes-mathematicae-proceedings).

* 1990–present: [Indagationes Mathematicae](https://www.sciencedirect.com/journal/indagationes-mathematicae).

category: reference

[[!redirects Koninklijke Akademie van Wetenschappen]]


indecomposable object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

# Indecomposable objects
* table of contents
{: toc}

## Definition

An object $X$ of a category $C$ is **indecomposable** if it cannot be expressed as a non-trivial [[coproduct]] of objects of $C$.  Formally, $X$ is indecomposable if given an isomorphism $X \cong \coprod_i U_i$, there is a unique index $i$ such that $X \cong U_i$ and $U_j \cong 0$ for all $j \neq i$, where $0$ is an [[initial object]].

The requirement that $i$ be unique keeps the initial object itself from being indecomposable; this is analogous to being [[too simple to be simple]].

## Remarks

If $C$ is an [[extensive category]], meaning that coproducts in $C$ are [[disjoint coproduct|disjoint]] and pullback-stable, then we have

+-- {: .un_prop}
###### Proposition
An object $X$ of $C$ is indecomposable if and only if it is [[connected object|connected]], that is if the hom functor $\hom(X,-)$ preserves coproducts.
=--

+-- {: .proof}
###### Proof
If $X$ is connected, then a morphism $k \colon X \to \coprod_i U_i$ factors uniquely as $\iota_i \circ \bar k \colon X \to U_i \to \coprod_i U_i$, where $\iota_i$ is the coprojection.  Suppose $k$ is invertible -- then the composite $k^{-1} \iota_i \bar k \colon X \to U_i \to \coprod_i U_i \to X$ is the identity.  Consider
$$
U_i \overset{\iota_i}{\to} \coprod_i U_i \overset{k^{-1}}{\to} X \overset{k}{\to} \coprod_i U_i
$$
Of course $k \circ k^{-1} = 1$, while $k = \iota_i \bar k$ as before.  So $\iota_i \circ \bar k k^{-1} \iota_i = \iota_i$.  But because $C$ is extensive the coprojections are monic, so $\bar k k^{-1} \iota_i = 1$.  Thus $\bar k$ is an isomorphism $X \cong U_i$, with inverse $k^{-1} \iota_i$.  Because coproducts in $C$ are disjoint, the pullback of distinct coprojections is $0$, and because $U_i \cong X \cong \coprod_i U_i$, the pullback of $\iota_i$ along $\iota_j$ is an isomorphism, showing that $U_j \cong 0$ for $j \neq i$.

Conversely, assume $X$ is indecomposable.  Given $k \colon X \to \coprod_i U_i$, we are to produce a unique $X \to U_i$ as above.  Because $C$ is extensive, $k$ is isomorphic (in the slice category $C/\coprod_i U_i$) to $\coprod_i k_i \colon \coprod_i X_i \to \coprod_i U_i$ for some family $\{k_i \colon X_i \to U_i\}$.  But $X \cong \coprod_i X_i$ gives us by indecomposability of $X$ an isomorphism $X \cong X_i$ for a unique $i$, which composed with $k_i$ gives a morphism $X \cong X_i \to U_i$.  Because $\coprod_i k_i = [\iota_i k_i]_i$ (where the right-hand side is the copairing of the family $\{\iota_i k_i\}_i$), and because the $X_j \cong 0$ for $j\neq i$, the composite $\iota_i k_i \colon X \cong X_i \to U_i \to \coprod_i U_i$ is equal to $k$.  Hence $X$ is connected.
=--

If $C$ is a [[presheaf category]] $[S^{op}, Set]$ (thus a [[Grothendieck topos]] and so _a fortiori_ (infinitary) extensive), then it is easy to see that the [[representable functors]] $S(-,s)$ are connected and so indecomposable.  Conversely, the objects of $[S^{op}, Set]$ that are indecomposable as well as [[projective object|projective]] are precisely the objects of the [[Cauchy completion]] of $S$.

## Lambek--Scott indecomposability

[[Introduction to Higher-Order Categorical Logic|Lambek & Scott]] give a different definition of indecomposability.  Generalizing their definition slightly, we may say that an object $X$ is **indecomposable** (in the sense of Lambek--Scott) if any [[jointly epimorphic family]] $\{U_i \to X\}_i$ of arrows into $X$ contains at least one [[epimorphism]] $U_i \twoheadrightarrow X$, and moreover the unique arrow $0 \to X$ is not epic (this to ensure that $0$ is not indecomposable).

If the epi $U_i \twoheadrightarrow X$ is required to be [[regular epimorphism|regular]], then in an _[[extensive category]]_ the Lambek--Scott definition implies that given above: if $k \colon X \cong \coprod_i U_i$, then the family $\{k^{-1} \iota_i \colon U_i \to \coprod_i U_i \cong X\}_i$ is jointly epic, so it contains a regular epi $\iota_i k^{-1}$.  But extensivity implies that $\iota_i$ is a [[monomorphism]], so the regular epi $\iota_i k^{-1}$ is also monic and hence an isomorphism.  The converse does not hold in general, but it does hold if $X$ is [[projective object|projective]].  See [this MathOverflow thread](http://mathoverflow.net/questions/35855/indecomposable-objects-in-a-category) for a discussion.

## Indecomposability vs irreducibility
{#irreducible}

An [[indecomposable representation]] is precisely an indecomposable object in an appropriate category $Rep$ of [[representations]], as one would expect.  In contrast, an [[irreducible representation]] is precisely a [[simple object]] in $Rep$.  Every irreducible representation is indecomposable, but the converse holds only in special situations (such as the category of finite-dimensional linear representations of a real semisimple Lie group).

However, one level [[decategorified]], an [[irreducible element]] of a [[poset]] $P$ is precisely an indecomposable object of $P$ when thought of as a [[thin category]].  In contrast, a simple object is analogous to an [[atomic element]], although they are not the same thing.  (One might say that atomic = $0$-simple.)  Again, every atomic element is irreducible, but the converse holds only in special situations (such as the power set of any set).

The bottom line is that 'irreducible' and 'indecomposable' sometimes mean the same thing but sometimes don\'t, and 'irreducible' doesn\'t even mean the same thing across different fields.

[[!redirects indecomposable object]]
[[!redirects indecomposable objects]]


indefinite integral

# Indefinite integrals
* table of contents
{: toc}

## Idea

An indefinite integral is something less definite than a [[definite integral]].  Whereas a definite integral is typically some kind of [[number]] or other concrete quantity, an indefinite integral is typically another variable quantity of the same type as the integrand.

The term 'indefinite integral' is itself rather indefinite, having been used for a variety of slightly different concepts.  Both _semidefinite integrals_ and _antiderivatives_ are more precise versions of indefinite integrals.  The [[fundamental theorem of calculus]] is basically the [[theorem]] that these different kinds of indefinite integral are essentially the same thing.

## Definitions and notation

To begin with, we will discuss the [[integration]] of [[real number|real]]-valued [[functions]] on the [[real line]], but much of this can be generalized to other contexts.  So let $f$ be a [[partial function]] from $\mathbb{R}$ to $\mathbb{R}$; typically, the [[domain]] of $f$ will be an [[interval]], but we do not require this.

+-- {: .num_defn #semi}
###### Definition

If $a$ is a [[real number]] (usually in the domain of $f$, or at least in the domain\'s [[topological closure|closure]]), then the __semidefinite integral__ of $f$ from $a$ (or with __initial point__ $a$) is the function
$$ x \mapsto \int_a^x f(t) \,\mathrm{d}t .$$
(If $x \lt a$, then we must define $\int_a^x$ as $-\int_x^a$.)
=--

The semidefinite integral is defined in terms of the definite integral.  We can put names such as 'Riemann' and 'Lebesgue' between 'semidefinite' and 'integral' to specify a particular kind of definite integral to be used.  Note that the domain of the semidefinite integral is an interval containing $a$ and contained in the domain of $f$ (or at least in its closure if we allow [[improper integral]]s or integrating [[almost functions]]).  If we start by defining $f$ as a [[locally integrable function]] on a closed interval $I$ containing $a$, then the semidefinite integral will also have $I$ as its domain.

The value at $x$ of the semidefinite integral from $a$ may be denoted
$$ \int_a f(x) \,\mathrm{d}x $$
for short.  Notice that this notation has no dummy variable; we only need to introduce the dummy variable $t$ to unfold the definition.  (Indeed, some writers will abuse notation, writing $\int_a^x f(x) \,\mathrm{d}x$ for the semidefinite integral.)  But as in $\mathrm{d}y/\mathrm{d}x$, the $x$ here is not a [[free variable]] either, since we cannot freely use substitution; it has to be viewed as [[variable quantity]] instead.  Rather, if you want to evaluate $\int_a f(x) \,\mathrm{d}x$ when $x$ is some number $b$, then the notation for this is $(\int_a f(x) \,\mathrm{d}x)|_{x=b}$ following the general logic of variable quantities; but unwrapping the definition of semidefinite integral, this can also be written as $\int_a^b f(x) \,\mathrm{d}x$.  (In each of these, $x$ has now become a dummy variable.)

+-- {: .num_defn #semiIV}
###### Definition

If $a$ and $C$ are real numbers (with $a$ in the domain of $f$ or its closure), then the __indefinite integral__ of $f$ from $a$ with __initial value__ $C$ is the function
$$ x \mapsto C + \int_a^x f(t) \,\mathrm{d}t .$$
We may write this value as $C + \int_a f(x) \,\mathrm{d}x$ for short.
=--

This is only one of the meanings of 'indefinite integral', but it is the only one that doesn\'t have alternative unambiguous terminology.  Note that $C$ is the value of the indefinite integral at $a$; thus, $C$ is the initial value if $a$ is the initial point.  But for authors who use this concept, there is often no need to mention either $a$ or $C$ (and hence no terminology needed for them), because they are interested only in whether some other function $F$ is an indefinite integral of $f$, where $f$ is a locally integrable function on some closed interval.

+-- {: .num_defn #antider}
###### Definition

If $F$ is a partial function from $\mathbb{R}$ to $\mathbb{R}$, then $F$ is an __antiderivative__ of $f$ (or an __antidifferential__ of $f \,\mathrm{d}x$) if $f$ is the [[derivative]] of $F$ on its domain:
$$ \forall\, x \in \dom F,\; f(x) = F'(x) .$$
A posteriori, $F$ must be [[differentiable function|differentiable]].
=--

This is the usual meaning of 'indefinite integral' in modern Calculus textbooks using the [[Riemann integral]], especially when the domain of $f$ is an interval.

+-- {: .num_defn #almostantider}
###### Definition

If $F$ is a Lebesgue-measurable partial [[almost function]] from $\mathbb{R}$ to $\mathbb{R}$, then $F$ is an __almost antiderivative__ of $f$ if $f$ is the derivative of $F$ [[almost everywhere]]:
$$ \operatorname{ess}\forall\, x \in \dom F,\; f(x) = F'(x) .$$
We are especially interested in the case where $F$ is [[absolutely continuous function|absolutely continuous]].
=--

This is not standard terminology, but it fits in well with other 'almost' terminology in [[measure theory]].  This is a common meaning of 'indefinite integral' when using the [[Lebesgue integral]].

## Properties

The main property linking the different kinds of indefinite integral is the _[[fundamental theorem of calculus]]_ (FTC).  For various definitions of integral, one can prove that every semidefinite integral, or more generally any indefinite integral in the sense of Definition \ref{semiIV}, is an antiderivative; and that every antiderivative, or more generally every almost antiderivative, is an indefinite integral; possibly with technical conditions (depending on the type of integral concerned) such as differentiability or absolute continuity.  See that article for details.

Indefinite integrals provide solutions to [[differential equations]].  Of course, the definition of an antiderivative is that it is the solution to a particularly simple differential equation.  Employing the FTC, we see that the indefinite integrals are the solutions to the corresponding initial-value problems.  Specifically, the solution to
$$ \frac{\mathrm{d}y}{\mathrm{d}x} = f(x),\; y|_{x=a} = C $$
is the indefinite integral of $f$ with initial point $a$ and initial value $C$:
$$ y = C + \int_a f(x) \,\mathrm{d}x .$$
Some more general initial-value problems can be solved similarly; for example, the implicit solution to the separable IVP
$$ g(y) \frac{\mathrm{d}y}{\mathrm{d}x} = f(x),\; y|_{x=a} = C $$
is
$$ \int_C g(y) \,\mathrm{d}y = \int_a f(x) \,\mathrm{d}x .$$
Similarly, the solution to the linear IVP
$$ \frac{\mathrm{d}y}{\mathrm{d}x} + P(x) \,y = Q(x),\; y|_{x=a} = C $$
is
$$ y = \frac{C + \int_a Q(x) \,\mu(x) \,\mathrm{d}x}{\mu(x)} ,$$
where $\mu(x) = \exp \int_a P(x) \,\mathrm{d}x$.

## Generalizations

### In cartesian spaces

When $\omega$ is an exterior $1$-form on a subspace $S$ of the [[cartesian space]] $\mathbb{R}^n$, then we can define an antiderivative (or antidifferential) of $\omega$ to be any real-valued function $f$ on $S$ such that $\mathrm{d}f = \omega$.  Then when $P$ is a point in $S$ (or perhaps its closure), we can define the value of the semidefinite integral of $\omega$ with initial point $P$ to be the integral of $\omega$ along a straight line segment from $P$; the domain is a [[star-convex set]] radiating from $P$ and contained in (the closure of) $S$.  If we define an indefinite integral as a semidefinite integral plus a constant initial value, then every antiderivative of $\omega$ on a star-convex set is an indefinite integral.  Conversely, every indefinite integral is an antiderivative if $\omega$ is closed.  If $\omega$ is not closed, then it still has indefinite integrals (as long as it\'s continuous or otherwise locally integrable), but these are no longer antiderivatives (which non-closed forms never have).

This can perhaps be generalized to [[Riemannian manifolds]] by considering integrals along [[geodesics]]; although the geodesic between two points is not always unique (even when it exists), it is unique on a sufficiently small (and often quite large) neighbourhood.  (For example, on a [[sphere]], as long as $\omega$ is integrable, we can define the indefinite integral this way at every point except the one directly opposite the initial point.)

However, this straight-line definition seems rather artificial, and it might be best to use the more notion of semidefinite integral below, applicable only to closed forms but on more general manifolds.

### On manifolds

We can generalize to [[exterior differential forms]] on [[differentiable manifolds]], generalizing the FTC to the [[Stokes theorem]].  It\'s clear what an antiderivative is in this context: $\alpha$ is an __exterior antiderivative__ of $\omega$ iff $\omega$ is the [[exterior derivative]] of $\alpha$.  On a [[smooth manifold]], we know what 'almost' means and so can also define exterior almost antiderivatives.

If $\omega$ is a $1$-form on any differentiable manifold and $P$ is a point in its domain, then the semidefinite integral of $\omega$ with initial point $P$ is defined at another point $Q$ iff the integral of $\omega$ is the same along any path from $P$ to $Q$ (and then that integral is the value).  We can define an indefinite integral by adding a constant initial value.  Then every antiderivative on a [[path-connected space|path-connected]] domain is an indefinite integral, and conversely every indefinite integral is an antiderivative.  By definition, $\omega$ is exact iff an antiderivative exists, and therefore iff an indefinite integral exists on each [[path-connected component]].  Similarly, $\omega$ is closed iff it has an indefinite integral on a neighbourhood of each point; if the domain of $\omega$ is [[simply-connected space|simply connected]], then the indefinite integral can be extended to the entire domain.

It remains to consider a good notion of semidefinite and indefinite integrals for higher-rank forms.

### In algebraic limit fields

In an [[algebraic limit field]] $F$, the notion of a derivative is well defined and is represented by the [[Newton-Leibniz operator]] $\tilde{D}$. Given a [[subset]] $S \subseteq F$, and a function $f:S \to F$, the set of antiderivatives of $f$ is the [[fiber]] of the [[Newton-Leibniz operator]] at $f$:

$$\{g \in D^1(S, F) \vert \tilde{D}(g) = f\}$$

An __antiderivative__ is an element of the above subset.

## See also

* [[differentiable function]]
* [[integrable function]]

[[!redirects indefinite integral]]
[[!redirects indefinite integrals]]
[[!redirects indefinite integration]]
[[!redirects indefinite integrations]]

[[!redirects semidefinite integral]]
[[!redirects semidefinite integrals]]
[[!redirects semidefinite integration]]
[[!redirects semidefinite integrations]]

[[!redirects antiderivative]]
[[!redirects antiderivatives]]
[[!redirects antidifferential]]
[[!redirects antidifferentials]]
[[!redirects antidifferentiation]]
[[!redirects antidifferentiations]]


independent family of sets

# Contents
* table of contents
{:toc}

## Idea

To say a family of subsets of a set is independent means that there are no trivial Boolean combinations that can be formed from them except tautologically trivial ones.

## Definition

Let $X$ be a set. A family $F \subseteq P X$ (or $F \subseteq 2^X$) is *independent* if, for any finite subfamily of pairwise distinct elements of $F$, say $S_1, \ldots, S_m, T_1, \ldots, T_n$, the intersection

$$S_1 \cap \ldots \cap S_m \cap \neg T_1 \cap \ldots \cap \neg T_n$$

is [[inhabited set|inhabited]].

Another way of saying it: let $Bool(F)$ denote the [[free object|free]] [[Boolean algebra]] generated by the set $F$. An inclusion $i: F \hookrightarrow P X$ names an *independent family* if the induced Boolean algebra map $\widehat{i}: Bool(F) \to P X$ is itself [[injective map|injective]]. (To see how the alternative condition follows, consider elements of $Bool(F)$ as written in disjoint normal form, i.e., as disjoint unions of finite intersections of literals $S, \neg T$; use this to deduce $\ker(\widehat{i}) = 0$) This is likely the origin of the term "independent family"; cf. the [abstract definition of linearly independent set](/nlab/show/linearly+independent+subset#abstractly).

## Application: counting ultrafilters

Let $X$ be an infinite set, of cardinality $\kappa$.

+-- {: .num_theorem}
###### Theorem
The number of [[ultrafilters]] on $X$ is $2^{2^\kappa}$.
=--

+-- {: .proof}
###### Proof
Begin by noting that the set of ultrafilters $Bool(P X, 2)$ is a subset of $Set(P X, 2)$, a set of cardinality $2^{2^\kappa}$. So it suffices to prove $Bool(P X, 2)$ has at least $2^{2^\kappa}$ elements.

The free Boolean algebra $Bool(X)$ generated by $X$ also has cardinality $\kappa$. (Consider it as the directed union of $Bool(F)$ ranging over finite subsets $F$ of $X$, and use the fact that $Bool(F)$ is finite.) The [[Stone space]] of $Bool(X)$, i.e., the [[Zariski spectrum|spectrum]] of $Bool(X)$ consisting of Boolean ring homomorphisms $S: Bool(X) \to 2$, is the [[compactum]] $2^X$. Let $\langle -, - \rangle: Bool(X) \times 2^X \to 2$ be the associated canonical pairing.

For each $S \in 2^X$ let $A_S = \{\phi \in Bool(X): \langle \phi, S \rangle = 1\}$. We claim that the $A_S$ form an independent family, i.e., for pairwise distinct $S_1, \ldots, S_m, T_1, \ldots, T_n \in 2^X$, the finite intersection

$$A_{S_1} \cap \ldots \cap A_{S_m} \cap \neg A_{T_1} \cap \ldots \cap \neg A_{T_n}$$

is inhabited; this is equivalent to the claim that for pairwise distinct $S_1, \ldots, S_m, T_1, \ldots, T_n \in 2^X$ there exists $\phi \in Bool(X)$ such that $\langle \phi, S_i \rangle = 1$ and $\langle \phi, T_j \rangle = 0$. To prove this, note that $\phi \mapsto \langle \phi, - \rangle$ is an isomorphism $Bool(X) \to CH(2^X, 2)$ by [[Stone duality]], where $CH(2^X, 2)$ is isomorphic to the Boolean algebra of [[clopens]], and there exists a clopen separating the set of points $\{S_1,\ldots, S_m\}$ from the set of points $\{T_1, \ldots, T_n\}$ (a proof may be based on the lemma [here](/nlab/show/compact+Hausdorff+rings+are+profinite#compopen), taking $V$ to be the complement of $\{T_1, \ldots, T_n\}$).

By transport along a [[bijection]] $X \cong |Bool(X)|$, this result implies there is an independent family $\{A_S\}_{S \in 2^\kappa}$ of subsets of $X$ that has size $2^\kappa$, or in other words there is an injective Boolean algebra map $Bool(2^\kappa) \to P X$ sending $S \in 2^\kappa$ to $A_S$. We finish by observing that given a Boolean algebra injection $i: A \to B$, any Boolean algebra map $f: A \to 2$ extends to a Boolean algebra map $g: B \to 2$; this follows from the [[ultrafilter principle]] as the ultrafilter $f^{-1}(1) \subset A$ in $A$ generates a proper [[filter]] in $B$ by injectivity of $i$, which may then be extended to an ultrafilter in $B$. (See also [here](/nlab/show/injective+object#in_boolean_algebras): $2$ is an injective object.) From this observation, the induced map

$$Bool(P X, 2) \to Bool(Bool(2^\kappa), 2) \cong 2^{2^\kappa}$$

is [[surjective function|surjective]], and thus the set $Bool(P X, 2)$ of ultrafilters on $X$ has cardinality at least $2^{2^\kappa}$.
=--