Digital immortality

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Digital_immortality 

 

Digital immortality (or "virtual immortality") is the hypothetical concept of storing (or transferring) a person's personality in more durable media, i.e., a computer. The result might look like an avatar behaving, reacting, and thinking like a person on the basis of that person's digital archive. After the death of the individual, this avatar could remain static or continue to learn and develop autonomously.

A considerable portion of transhumanists and singularitarians place great hope into the belief that they may eventually become immortal by creating one or many non-biological functional copies of their brains, thereby leaving their "biological shell". These copies may then "live eternally" in a version of digital "heaven" or paradise.

The realism of the concept

The National Science Foundation has awarded a half-million-dollar grant to the universities of Central Florida at Orlando and Illinois at Chicago to explore how researchers might use artificial intelligence, archiving, and computer imaging to create convincing, digital versions of real people, a possible first step toward virtual immortality.

The Digital Immortality Institute explores three factors necessary for digital immortality. First, at whatever level of implementation, avatars require guaranteed Internet accessibility. Next, avatars must be what users specify, and they must remain so. Finally, future representations must be secured before the living users are no more.

The aim of Dmitry Itskov's 2045 Initiative is to "create technologies enabling the transfer of an individual’s personality to a non-biological carrier, and extending existence, including to the point of immortality".

Method

Reaching digital immortality is a two-step process:

1. archiving and digitizing people,
2. making the avatar live

Archiving and digitizing people

According to Gordon Bell and Jim Gray from Microsoft Research, retaining every conversation that a person has ever heard is already realistic: it needs less than a terabyte of storage (for adequate quality). The speech or text recognition technologies are one of the biggest challenges of the concept. 

A second possibility would be to archive and analyze social Internet use to map the personality of people. By analyzing social Internet use during 50 years, it would be possible to model a society's culture, a society's way of thinking, and a society's interests.

Rothblatt envisions the creation of "mindfiles" – collections of data from all kinds of sources, including the photos we upload to Facebook, the discussions and opinions we share on forums or blogs, and other social media interactions that reflect our life experiences and our unique self.

Richard Grandmorin summarized the concept of digital immortality by the following equation: "semantic analysis + social internet use + Artificial Intelligence = immortality". 

Some find that photos, videos, soundclips, social media posts and other data of oneself could already be regarded as such an archiving.

Susanne Asche states:

As a hopefully minimalistic definition then, digital immortality can be roughly considered as involving a person-centric repository containing a copy of everything that a person sees, hears, says, or engenders over his or her lifespan, including photographs, videos, audio recordings, movies, television shows, music albums/CDs, newspapers, documents, diaries and journals, interviews, meetings, love letters, notes, papers, art pieces, and so on, and so on; and if not everything, then at least as much as the person has and takes the time and trouble to include. The person’s personality, emotion profiles, thoughts, beliefs, and appearance are also captured and integrated into an artificially intelligent, interactive, con-versational agent/avatar. This avatar is placed in charge of (and perhaps "equated" with) the collected material in the repository so that the agent can present the illusion of having the factual memories, thoughts, and beliefs of the person him/herself.

— Susanne Asche, Kulturelles Gedächtnis im 21. Jahrhundert: Tagungsband des internationalen Symposiums, Digital Immortality & Runaway Technology

Making the avatar alive

Defining the avatar to be alive allows it to communicate with the future in the sense that it continues to learn, evolve and interact with people, if they still exist. Technically, the operation exists to implement an artificial intelligence system to the avatar. This artificial intelligence system is then assumed to think and will react on the base of the archive. 

Rothblatt proposes the term "mindware" for software that is being developed with the goal of generating conscious AIs. Such software would read a person's "mindfile" to generate a "mindclone." Rothblatt also proposes a certain level of governmental approval for mindware, like an FDA certification, to ensure that the resulting mindclones are well made.

Calibration process

During the calibration process, the biological people are living at the same time as their artifact in silicon. The artifact in silicon is calibrated to be as close as possible to the person in question. During this process ongoing updates, synchronization, and interaction between the two minds would maintain the twin minds as one.

In fiction

• In the TV series Caprica a digital copy of a person is created and outlives its real counterpart after the person dies in a terrorist attack.
• In Greg Egan's Permutation City people can achieve quasi digital immortality by mind uploading a digital copy of themselves into a simulated reality.
• Memories with Maya is a novel on the concept of digital immortality.
• The Silicon Man describes Cryonics as a precursor to digital immortality.
• In the 1998 novel Vast by Linda Nagata "ghosts" are recorded memories and personalities that can be transferred to another body or kept in electronic storage, granting a limited form of immortality.
• In the TV series Captain Power and the Soldiers of the Future, Overmind and Lord Dread planned to digitize all human beings to be able to create a new world.
• In the TV series Black Mirror it commonly features the themes and ethics of digital humans, called "cookies," across multiple episodes. In San Junipero, for example, people's consciences are uploaded to the cloud.
• In the novel / Netflix series Altered Carbon, a person's memories and consciousness can be stored in a disk-shaped device called a cortical stack, which is implanted into the cervical vertebrae.
• In Frictional Games' SOMA, the story revolves around the problem of existing as a digital personality scan taken from a physical person.

How to Be an Antiracist

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/How_to_Be_an_Antiracist 

 

How to Be an Antiracist

Front cover
Author	Ibram X. Kendi
Subject	Civil rights
Publisher	Random House
Publication date	August 13, 2019
Pages	320
ISBN	9780525509288

How to Be an Antiracist is a 2019 non-fiction book by American author and historian Ibram X. Kendi. The book discusses concepts of racism and Kendi's proposals for anti-racist individual actions and systemic changes. It received positive critical reception. 

Background

At the time of authorship, Ibram X. Kendi was an assistant professor of African-American History at the University of Florida. He previously worked at the American University, where he founded the Antiracist Research and Policy Center. He wrote a 2016 book titled Stamped from the Beginning, about the origins of racism in America.

Synopsis

Kendi describes concepts of racism such as scientific racism, colorism and their intersection with demographics including gender, class and sexuality. He summarizes historical eras such as the scientific proposals of polygenism in Europe in the 1600s and racial segregation in the United States. The book also covers contemporary history such as the O. J. Simpson robbery case and 2000 United States presidential election. He also details experiences from his own life, including his change in beliefs over time, and observations from classes he has taught. Kendi comments on internalized racism and disputes the prejudice plus power model of racism. He suggests models for anti-racist individual actions and systemic changes.

Reception

The book was published in August 2019 to mixed, but generally positive reviews. In June 2020, following protests in the wake of the killing of George Floyd, sales of How to Be an Antiracist surged. The book was listed eighth and fifth in Publishers Weekly's hardcover non-fiction list on May 30 and June 6, respectively. It was listed third in USA Today's Best-Selling Books List of June 10. The book topped The New York Times Bestseller List in Hardcover Nonfiction list for sales in the week ending June 6. It has spent a total of 18 weeks on the list, as of the July 19 edition of the list.

Critical reception

Ayesha Pande praised the book in a starred review for Publishers Weekly, describing the prose as "thoughtful, sincere and polished" and the ideas as "boldly articulated" and "historically informed". Pande summarized, "This powerful book will spark many conversations". A starred review for Kirkus Reviews found it to be "not an easy read but an essential one". Jeffrey C. Stewart of The New York Times lauded it as the "most courageous book to date on the problem of race in the Western mind". Ericka Taylor of NPR praised the book as "clear and compelling", saying that it is "accessible" and "exemplifies a commitment to clarity".

In a mostly positive review by The Guardian's Afua Hirsch, the author received praise for "honesty in linking his personal struggles" to the book's subject, which Hirsch described as "brilliantly simple" and "dogmatic", but criticism for personal anecdotes that seem incomplete and for a style resembling a textbook too much. It was the Book of the Day in a review for The Observer in which Colin Grant found that the book "encourages self-reflection" and praised the writing style as "calm" but "insightful".

Coleman Hughes critiqued the book as "poorly argued, sloppily researched, insufficiently fact-checked, and occasionally self-contradictory". In the Washington Post, Randall Kennedy, praised Kendi's book for its candor, independence, and self-criticalness, but also critiqued it as having major flaws—especially being internally contradictory and poorly reasoned. Andrew Sullivan criticized the book as having the character of religious tract with overly simplistic distinctions between good and evil that cannot be falsified, and being sparse on practical suggestions.

Set (mathematics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Set_(mathematics) 

A set of polygons in an Euler diagram

 

In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. The arrangement of the objects in the set does not matter. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written as {2, 4, 6}, which could also be written as {2, 6, 4}.

The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived.

Etymology

The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite.

Definition

Passage with a translation of the original set definition of Georg Cantor. The German word Menge for set is translated with aggregate here.

 

A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the set's elements or members) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, one of the founders of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.

For technical reasons, Cantor's definition turned out to be inadequate; today, in contexts where more rigor is required, one can use axiomatic set theory, in which the notion of a "set" is taken as a primitive notion and the properties of sets are defined by a collection of axioms. The most basic properties are that a set can have elements, and that two sets are equal (one and the same) if and only if every element of each set is an element of the other; this property is called the extensionality of sets.

Set notation

Main article: Set notation

There are two common ways of describing, or specifying the members of, a set: roster notation and set builder notation. These are examples of extensional and intensional definitions of sets, respectively.

Roster notation

The Roster notation (or enumeration notation) method of defining a set consist of listing each member of the set. More specifically, in roster notation (an example of extensional definition), the set is denoted by enclosing the list of members in curly brackets:

A = {4, 2, 1, 3}

B = {blue, white, red}.

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified in roster notation as

{1, 2, 3, ..., 1000},

where the ellipsis ("...") indicates that the list continues in according to the demonstrated pattern.

In roster notation, listing a member repeatedly does not change the set, for example, the set {11, 6, 6} is identical to the set {11, 6}. Moreover, the order in which the elements of a set are listed is irrelevant (unlike for a sequence or tuple), so {6, 11} is yet again the same set.

Set-builder notation

In set-builder notation, the set is specified as a subset of a larger set, where the subset is determined by a statement or condition involving the elements. For example, a set F can be specified as follows:

${\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.}$

In this notation, the vertical bar ("|") means "such that", and the description can be interpreted as "F is the set of all numbers n, such that n is an integer in the range from 0 to 19 inclusive". Sometimes the colon (":") is used instead of the vertical bar.

Set-builder notation is an example of intensional definition.

Other ways of defining sets

Another method is by using a rule or semantic description:

A is the set whose members are the first four positive integers.

B is the set of colors of the French flag.

This is another example of intensional definition.

Membership

If B is a set and x is one of the objects of B, this is denoted as x ∈ B, and is read as "x is an element of B", as "x belongs to B", or "x is in B". If y is not a member of B then this is written as y ∉ B, read as "y is not an element of B", or "y is not in B".

For example, with respect to the sets A = {1, 2, 3, 4}, B = {blue, white, red}, and F = {n | n is an integer, and 0 ≤ n ≤ 19},

4 ∈ A and 12 ∈ F; and

20 ∉ F and green ∉ B.

Subsets

If every element of set A is also in B, then A is said to be a subset of B, written A ⊆ B (pronounced A is contained in B). Equivalently, one can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B.

If A is a subset of B, but not equal to B, then A is called a proper subset of B, written A ⊊ B, or simply A ⊂ B (A is a proper subset of B), or B ⊋ A (B is a proper superset of A, B ⊃ A).

The expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B (respectively B ⊇ A), whereas others use them to mean the same as A ⊊ B (respectively B ⊋ A). 

A is a subset of B

Examples:

• The set of all humans is a proper subset of the set of all mammals.
• {1, 3} ⊆ {1, 2, 3, 4}.
• {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

There is a unique set with no members, called the empty set (or the null set), which is denoted by the symbol ∅ (other notations are used; see empty set). The empty set is a subset of every set, and every set is a subset of itself:

• ∅ ⊆ A.
• A ⊆ A.

Partitions

A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S.

Power sets

The power set of a set S is the set of all subsets of S. The power set contains S itself and the empty set because these are both subsets of S. For example, the power set of the set {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The power set of a set S is usually written as P(S).

The power set of a finite set with n elements has 2ⁿ elements. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 2³ = 8 elements.

The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, the power set of a set is always strictly "bigger" than the original set in the sense that there is no way to pair every element of S with exactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)

Cardinality

The cardinality of a set S, denoted |S|, is the number of members of S. For example, if B = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted, so |{blue, white, red, blue, white}| = 3, too. 

The cardinality of the empty set is zero.

Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

Special sets

The natural numbers ℕ are contained in the integers ℤ, which are contained in the rational numbers ℚ, which are contained in the real numbers ℝ, which are contained in the complex numbers ℂ

There are some sets or kinds of sets that hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the empty set, denoted { } or ∅. A set with exactly one element, x, is a unit set, or singleton, {x}.

Many of these sets are represented using bold (e.g. P) or blackboard bold (e.g. ℙ) typeface.

Special sets of numbers include

• P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.
• N or ${\displaystyle \mathbb {N} }$, denoting the set of all natural numbers: N = {0, 1, 2, 3, ...} (sometimes defined excluding 0).
• Z or ${\displaystyle \mathbb {Z} }$, denoting the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.
• Q or ℚ, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b | a, b ∈ Z, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can be expressed as the fraction a/1 (Z ⊊ Q).
• R or ${\displaystyle \mathbb {R} }$, denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, algebraic numbers that cannot be rewritten as fractions such as √2, as well as transcendental numbers such as π, e).
• C or ℂ, denoting the set of all complex numbers: C = {a + bi | a, b ∈ R}. For example, 1 + 2i ∈ C.
• H or ℍ, denoting the set of all quaternions: H = {a + bi + cj + dk | a, b, c, d ∈ R}. For example, 1 + i + 2j − k ∈ H.

Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a proper subset of the sets listed below it. The primes are used less frequently than the others outside of number theory and related fields. 

Positive and negative sets are sometimes denoted by superscript plus and minus signs, respectively. For example, ℚ⁺ represents the set of positive rational numbers.

Basic operations

There are several fundamental operations for constructing new sets from given sets.

Unions

The union of A and B, denoted A ∪ B

Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things that are members of either A or B. 

Examples:

• {1, 2} ∪ {1, 2} = {1, 2}.
• {1, 2} ∪ {2, 3} = {1, 2, 3}.
• {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}

Some basic properties of unions:

• A ∪ B = B ∪ A.
• A ∪ (B ∪ C) = (A ∪ B) ∪ C.
• A ⊆ (A ∪ B).
• A ∪ A = A.
• A ∪ ∅ = A.
• A ⊆ B if and only if A ∪ B = B.

Intersections

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint. 

The intersection of A and B, denoted A ∩ B.

Examples:

• {1, 2} ∩ {1, 2} = {1, 2}.
• {1, 2} ∩ {2, 3} = {2}.
• {1, 2} ∩ {3, 4} = ∅.

Some basic properties of intersections:

• A ∩ B = B ∩ A.
• A ∩ (B ∩ C) = (A ∩ B) ∩ C.
• A ∩ B ⊆ A.
• A ∩ A = A.
• A ∩ ∅ = ∅.
• A ⊆ B if and only if A ∩ B = A.

Complements

The relative complement
of B in A

 

The complement of A in U

 

The symmetric difference of A and B

Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or A − B), is the set of all elements that are members of A but not members of B. It is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect. 

In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.

• A′ = U \ A

Examples:

• {1, 2} \ {1, 2} = ∅.
• {1, 2, 3, 4} \ {1, 3} = {2, 4}.
• If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E = E′ = O.

Some basic properties of complements:

• A \ B ≠ B \ A for A ≠ B.
• A ∪ A′ = U.
• A ∩ A′ = ∅.
• (A′)′ = A.
• ∅ \ A = ∅.
• A \ ∅ = A.
• A \ A = ∅.
• A \ U = ∅.
• A \ A′ = A and A′ \ A = A′.
• U′ = ∅ and ∅′ = U.
• A \ B = A ∩ B′.
• if A ⊆ B then A \ B = ∅.

An extension of the complement is the symmetric difference, defined for sets A, B as

${\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A).}$

For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

Cartesian product

A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B.

Examples:

• {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
• {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
• {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:

• A × ∅ = ∅.
• A × (B ∪ C) = (A × B) ∪ (A × C).
• (A ∪ B) × C = (A × C) ∪ (B × C).

Let A and B be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:

• | A × B | = | B × A | = | A | × | B |.

Applications

Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations. 

One of the main applications of naive set theory is constructing relations. A relation from a domain A to a codomain B is a subset of the Cartesian product A × B. For example, considering the set S = { rock, paper, scissors } of shapes in the game of the same name, the relation "beats" from S to S is the set B = { (scissors,paper), (paper,rock), (rock,scissors) }; thus x beats y in the game if the pair (x,y) is a member of B. Another example is the set F of all pairs (x, x²), where x is real. This relation is a subset of R' × R, because the set of all squares is subset of the set of all real numbers. Since for every x in R, one, and only one, pair (x,...) is found in F, it is called a function. In functional notation, this relation can be written as F(x) = x².

Axiomatic set theory

Although initially naive set theory, which defines a set merely as any well-defined collection, was well accepted, it soon ran into several obstacles. It was found that this definition spawned several paradoxes, most notably:

• Russell's paradox – It shows that the "set of all sets that do not contain themselves," i.e. the "set" {x|x is a set and x ∉ x} does not exist.
• Cantor's paradox – It shows that "the set of all sets" cannot exist.

The reason is that the phrase well-defined is not very well-defined. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid these paradoxes, set theory was axiomatized based on first-order logic, and thus axiomatic set theory was born. 

For most purposes, however, naive set theory is still useful.

Principle of inclusion and exclusion

The inclusion-exclusion principle can be used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.

 

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets, if the size of each set and the size of their intersection are known. It can be expressed symbolically as

${\displaystyle |A\cup B|=|A|+|B|-|A\cap B|.}$

A more general form of the principle can be used to find the cardinality of any finite union of sets:

${\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right)\\&{}-\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right)\\&{}+\ldots \\&{}+\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right).\end{aligned}}}$

De Morgan's laws

Augustus De Morgan stated two laws about sets. 

If A and B are any two sets then,

• (A ∪ B)′ = A′ ∩ B′

The complement of A union B equals the complement of A intersected with the complement of B.

• (A ∩ B)′ = A′ ∪ B′

The complement of A intersected with B is equal to the complement of A union to the complement of B.

Von Neumann universe

From Wikipedia, the free encyclopedia

 

 

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC.

The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set. In particular, the rank of the empty set is zero, and every ordinal has a rank equal to itself. The sets in V are divided into the transfinite hierarchy V_α , called the cumulative hierarchy, based on their rank.

Definition

The cumulative hierarchy is a collection of sets V_α indexed by the class of ordinal numbers; in particular, V_α is the set of all sets having ranks less than α. Thus there is one set V_α for each ordinal number α. V_α may be defined by transfinite recursion as follows:

• Let V₀ be the empty set:

  ${\displaystyle V_{0}:=\varnothing .}$

• For any ordinal number β, let V_β+1 be the power set of V_β:

  ${\displaystyle V_{\beta +1}:={\mathcal {P}}(V_{\beta }).}$

• For any limit ordinal λ, let V_λ be the union of all V-stages so far:

  ${\displaystyle V_{\lambda }:=\bigcup _{\beta <\lambda }V_{\beta }.}$

A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that defines "the set x is in V_α". 

The sets V_α are called stages or ranks.

An initial segment of the von Neumann universe. Ordinal multiplication is reversed from our usual convention; see Ordinal arithmetic.

 

The class V is defined to be the union of all the V-stages:

${\displaystyle V:=\bigcup _{\alpha }V_{\alpha }.}$

An equivalent definition sets

${\displaystyle V_{\alpha }:=\bigcup _{\beta <\alpha }{\mathcal {P}}(V_{\beta })}$

for each ordinal α, where ${\displaystyle {\mathcal {P}}(X)\!}$ is the powerset of ${\displaystyle X}$. 

The rank of a set S is the smallest α such that ${\displaystyle S\subseteq V_{\alpha }\,.}$ Another way to calculate rank is:

${\displaystyle \operatorname {rank} (S)=\bigcup \{\operatorname {rank} (z)+1\mid z\in S\}}$.

Finite and low cardinality stages of the hierarchy

The first five von Neumann stages V₀ to V₄ may be visualized as follows. (An empty box represents the empty set. A box containing only an empty box represents the set containing only the empty set, and so forth.)

The set V₅ contains 2¹⁶ = 65536 elements. The set V₆ contains 2⁶⁵⁵³⁶ elements, which very substantially exceeds the number of atoms in the known universe. So the finite stages of the cumulative hierarchy cannot be written down explicitly after stage 5. The set V_ω has the same cardinality as ω. The set V_ω+1 has the same cardinality as the set of real numbers.

Applications and interpretations

Applications of V as models for set theories

If ω is the set of natural numbers, then V_ω is the set of hereditarily finite sets, which is a model of set theory without the axiom of infinity.

V_ω+ω is the universe of "ordinary mathematics", and is a model of Zermelo set theory. A simple argument in favour of the adequacy of V_ω+ω is the observation that V_ω+1 is adequate for the integers, while V_ω+2 is adequate for the real numbers, and most other normal mathematics can be built as relations of various kinds from these sets without needing the axiom of replacement to go outside V_ω+ω.

If κ is an inaccessible cardinal, then V_κ is a model of Zermelo–Fraenkel set theory (ZFC) itself, and V_κ+1 is a model of Morse–Kelley set theory. (Note that every ZFC model is also a ZF model, and every ZF model is also a Z model.)

Interpretation of V as the "set of all sets"

V is not "the set of all sets" for two reasons. First, it is not a set; although each individual stage V_α is a set, their union V is a proper class. Second, the sets in V are only the well-founded sets. The axiom of foundation (or regularity) demands that every set be well founded and hence in V, and thus in ZFC every set is in V. But other axiom systems may omit the axiom of foundation or replace it by a strong negation (an example is Aczel's anti-foundation axiom). These non-well-founded set theories are not commonly employed, but are still possible to study.

A third objection to the "set of all sets" interpretation is that not all sets are necessarily "pure sets", which are constructed from the empty set using power sets and unions. Zermelo proposed in 1908 the inclusion of urelements, from which he constructed a transfinite recursive hierarchy in 1930. Such urelements are used extensively in model theory, particularly in Fraenkel-Mostowski models.

V and the axiom of regularity

The formula V = ⋃_αV_α is often considered to be a theorem, not a definition. Roitman states (without references) that the realization that the axiom of regularity is equivalent to the equality of the universe of ZF sets to the cumulative hierarchy is due to von Neumann.

The existential status of V

Since the class V may be considered to be the arena for most of mathematics, it is important to establish that it "exists" in some sense. Since existence is a difficult concept, one typically replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed by Gödel's incompleteness theorems, which effectively imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consistent.

The integrity of the von Neumann universe depends fundamentally on the integrity of the ordinal numbers, which act as the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbers and the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to rest upon von Neumann's 1923 and 1928 papers. The integrity of the construction of V by transfinite induction may be said to have then been established in Zermelo's 1930 paper.

History

The cumulative type hierarchy, also known as the von Neumann universe, is claimed by Gregory H. Moore (1982) to be inaccurately attributed to von Neumann. The first publication of the von Neumann universe was by Ernst Zermelo in 1930.

Existence and uniqueness of the general transfinite recursive definition of sets was demonstrated in 1928 by von Neumann for both Zermelo-Fraenkel set theory and Neumann's own set theory (which later developed into NBG set theory). In neither of these papers did he apply his transfinite recursive method to construct the universe of all sets. The presentations of the von Neumann universe by Bernays and Mendelson both give credit to von Neumann for the transfinite induction construction method, although not for its application to the construction of the universe of ordinary sets.

The notation V is not a tribute to the name of von Neumann. It was used for the universe of sets in 1889 by Peano, the letter V signifying "Verum", which he used both as a logical symbol and to denote the class of all individuals. Peano's notation V was adopted also by Whitehead and Russell for the class of all sets in 1910. The V notation (for the class of all sets) was not used by von Neumann in his 1920s papers about ordinal numbers and transfinite induction. Paul Cohen explicitly attributes his use of the letter V (for the class of all sets) to a 1940 paper by Gödel, although Gödel most likely obtained the notation from earlier sources such as Whitehead and Russell.

Philosophical perspectives

There are two approaches to understanding the relationship of the von Neumann universe V to ZFC (along with many variations of each approach, and shadings between them). Roughly, formalists will tend to view V as something that flows from the ZFC axioms (for example, ZFC proves that every set is in V). On the other hand, realists are more likely to see the von Neumann hierarchy as something directly accessible to the intuition, and the axioms of ZFC as propositions for whose truth in V we can give direct intuitive arguments in natural language. A possible middle position is that the mental picture of the von Neumann hierarchy provides the ZFC axioms with a motivation (so that they are not arbitrary), but does not necessarily describe objects with real existence.