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Reformed Christianity

Reformed Christianity, also called Calvinism, is a major branch of Protestantism that began during the sixteenth-century Protestant Reformation, a schism in the Western Church. In the modern day, it is largely represented by the Continental, Presbyterian, and Congregational traditions, as well as parts of the Anglican and Baptist traditions.

A foundational event that divided the Reformed from the Lutheran tradition occurred in 1529 when reformer Huldrych Zwingli of Zürich broke with Martin Luther on the topic of the Lord's Supper. A separate Reformed tradition developed over several generations, especially in Switzerland, Scotland and the Netherlands.

In the seventeenth century, Jacobus Arminius and the Remonstrants were expelled from the Dutch Reformed Church over disputes regarding predestination and salvation, and from that time Arminians are usually considered to be a distinct tradition from the Reformed. This dispute produced the Canons of Dort, the basis for the "doctrines of grace" or "five points" of Calvinism.

Reformed theology emphasizes the authority of the Bible, the sovereignty of God, and covenant theology, a framework for understanding the Bible based on God's covenants with people. Reformed churches have emphasized simplicity in worship. Several forms of ecclesiastical polity are exercised by Reformed churches, including presbyterian, congregational, and some episcopal.

Definition and terminology
Reformed Christianity is often called Calvinism after John Calvin, influential reformer of Geneva. The term was first used by opposing Lutherans in the 1550s. Calvin did not approve of the use of this term, and scholars have argued that use of the term is misleading, inaccurate, unhelpful,   and "inherently distortive."

The definitions and boundaries of the terms Reformed Christianity and Calvinism are contested by scholars. As a historical movement, Reformed Christianity began during the Reformation with Huldrych Zwingli in Zürich, Switzerland. Following the failure of the Marburg Colloquy between Zwingli's followers and those of Martin Luther in 1529 to mediate disputes regarding the real presence of Christ in the Lord's Supper, Reformed Protestants were defined by their opposition to Lutherans. The Reformed also opposed Anabaptist radicals thus remaining within the Magisterial Reformation. During the seventeenth-century Arminian Controversy, followers of Jacobus Arminius were forcibly removed from the Dutch Reformed Church for their views regarding predestination and salvation, and thenceforth Arminians would be considered outside the pale of Reformed orthodoxy, though some use the term Reformed to include Arminians, while using the term Calvinist to exclude Arminians.

Reformed Christianity also has a complicated relationship with Anglicanism, the branch of Protestantism originating in the Church of England. The Anglican confessions are considered reformed, and leaders of the English Reformation were influenced by Reformed, rather than Lutheran theologians, but the Church of England retained elements of Catholicism such as bishops and vestments, unlike most Reformed churches, and thus was sometimes called "but halfly Reformed." Beginning in the seventeenth century, Anglicanism broadened to the extent that Reformed theology is no longer dominant in Anglicanism. Anglicanism is generally classified as a discrete tradition from Reformed.

Some scholars argue that Reformed Baptists, who hold many of the same beliefs as Reformed Christians but not infant baptism, should be considered part of Reformed Christianity, though this would not have been the view of early modern Reformed theologians. Others disagree, asserting that Baptists should be considered a separate religious tradition.

History
The first wave of Reformed theologians included Huldrych Zwingli (1484–1531), Martin Bucer (1491–1551), Wolfgang Capito (1478–1541), John Oecolampadius (1482–1531), and Guillaume Farel (1489–1565). While from diverse academic backgrounds, their work already contained key themes within Reformed theology, especially the priority of scripture as a source of authority. Scripture was also viewed as a unified whole, which led to a covenantal theology of the sacraments of baptism and the Lord's Supper as visible signs of the covenant of grace. Another shared perspective was their denial of the Real presence of Christ in the Eucharist. Each understood salvation to be by grace alone and affirmed a doctrine of unconditional election, the teaching that some people are chosen by God to be saved. Martin Luther and his successor, Philipp Melanchthon were significant influences on these theologians, and to a larger extent, those who followed. The doctrine of justification by faith alone, also known as sola fide, was a direct inheritance from Luther.

The second generation featured John Calvin (1509–1564), Heinrich Bullinger (1504–1575), Thomas Cranmer (1489–1556), Wolfgang Musculus (1497–1563), Peter Martyr Vermigli (1500–1562), Andreas Hyperius (1511–1564) and John à Lasco (1499–1560). Written between 1536 and 1539, Calvin's Institutes of the Christian Religion was one of the most influential works of the era. Toward the middle of the 16th century, these beliefs were formed into one consistent creed, which would shape the future definition of the Reformed faith. The 1549 Consensus Tigurinus unified Zwingli and Bullinger's memorialist theology of the Eucharist, which taught that it was simply a reminder of Christ's death, with Calvin's view of it as a means of grace with Christ actually present, though spiritually rather than bodily as in Catholic doctrine. The document demonstrates the diversity as well as unity in early Reformed theology, giving it a stability that enabled it to spread rapidly throughout Europe. This stands in marked contrast to the bitter controversy experienced by Lutherans prior to the 1579 Formula of Concord.

Due to Calvin's missionary work in France, his program of reform eventually reached the French-speaking provinces of the Netherlands. Calvinism was adopted in the Electorate of the Palatinate under Frederick III, which led to the formulation of the Heidelberg Catechism in 1563. This and the Belgic Confession were adopted as confessional standards in the first synod of the Dutch Reformed Church in 1571.

In 1573, William the Silent joined the Calvinist Church. Calvinism was declared the official religion of the Kingdom of Navarre by the queen regnant Jeanne d'Albret after her conversion in 1560. Leading divines, either Calvinist or those sympathetic to Calvinism, settled in England, including Martin Bucer, Peter Martyr, and John Łaski, as did John Knox in Scotland.

During the First English Civil War, English and Scots Presbyterians produced the Westminster Confession, which became the confessional standard for Presbyterians in the English-speaking world. Having established itself in Europe, the movement continued to spread to areas including North America, South Africa and Korea.

While Calvin did not live to see the foundation of his work grow into an international movement, his death allowed his ideas to spread far beyond their city of origin and their borders and to establish their own distinct character.

Spread
Although much of Calvin's work was in Geneva, his publications spread his ideas of a correctly Reformed church to many parts of Europe. In Switzerland, some cantons are still Reformed, and some are Catholic. Calvinism became the dominant doctrine within the Church of Scotland, the Dutch Republic, some communities in Flanders, and parts of Germany, especially those adjacent to the Netherlands in the Palatinate, Kassel, and Lippe, spread by Olevianus and Zacharias Ursinus among others. Protected by the local nobility, Calvinism became a significant religion in Eastern Hungary and Hungarian-speaking areas of Transylvania. there are about 3.5 million Hungarian Reformed people worldwide.

Calvinism was influential in France, Lithuania, and Poland before being mostly erased during the Counter Reformation. One of the most important Polish reformed theologists was John a Lasco, who was also involved into organising churches in East Frisia and Stranger's Church in London. Later, a faction called the Polish Brethren broke away from Calvinism on January 22, 1556, when Piotr of Goniądz, a Polish student, spoke out against the doctrine of the Trinity during the general synod of the Reformed churches of Poland held in the village of Secemin. Calvinism gained some popularity in Scandinavia, especially Sweden, but was rejected in favor of Lutheranism after the Synod of Uppsala in 1593.

Many 17th century European settlers in the Thirteen Colonies in British America were Calvinists, who emigrated because of arguments over church structure, including the Pilgrim Fathers. Others were forced into exile, including the French Huguenots. Dutch and French Calvinist settlers were also among the first European colonizers of South Africa, beginning in the 17th century, who became known as Boers or Afrikaners.

Sierra Leone was largely colonized by Calvinist settlers from Nova Scotia, many of whom were Black Loyalists who fought for the British Empire during the American War of Independence. John Marrant had organized a congregation there under the auspices of the Huntingdon Connection. Some of the largest Calvinist communions were started by 19th- and 20th-century missionaries. Especially large are those in Indonesia, Korea and Nigeria. In South Korea there are 20,000 Presbyterian congregations with about 9–10 million church members, scattered in more than 100 Presbyterian denominations. In South Korea, Presbyterianism is the largest Christian denomination.

A 2011 report of the Pew Forum on Religious and Public Life estimated that members of Presbyterian or Reformed churches make up 7% of the estimated 801 million Protestants globally, or approximately 56 million people. Though the broadly defined Reformed faith is much larger, as it constitutes Congregationalist (0.5%), most of the United and uniting churches (unions of different denominations) (7.2%) and most likely some of the other Protestant denominations (38.2%). All three are distinct categories from Presbyterian or Reformed (7%) in this report.

The Reformed family of churches is one of the largest Christian denominations. According to adherents.com the Reformed/Presbyterian/Congregational/United churches represent 75 million believers worldwide.

The World Communion of Reformed Churches, which includes some United Churches, has 80 million believers. WCRC is the third largest Christian communion in the world, after the Roman Catholic Church and the Eastern Orthodox Churches.

Many conservative Reformed churches which are strongly Calvinistic formed the World Reformed Fellowship which has about 70 member denominations. Most are not part of the World Communion of Reformed Churches because of its ecumenical attire. The International Conference of Reformed Churches is another conservative association.

Church of Tuvalu is an officially established state church in the Calvinist tradition.

Revelation and scripture
Reformed theologians believe that God communicates knowledge of himself to people through the Word of God. People are not able to know anything about God except through this self-revelation. (With the exception of general revelation of God; "His invisible attributes, His eternal power and divine nature, have been clearly seen, being understood through what has been made, so that they are without excuse" (Romans 1:20).) Speculation about anything which God has not revealed through his Word is not warranted. The knowledge people have of God is different from that which they have of anything else because God is infinite, and finite people are incapable of comprehending an infinite being. While the knowledge revealed by God to people is never incorrect, it is also never comprehensive.

According to Reformed theologians, God's self-revelation is always through his son Jesus Christ, because Christ is the only mediator between God and people. Revelation of God through Christ comes through two basic channels. The first is creation and providence, which is God's creating and continuing to work in the world. This action of God gives everyone knowledge about God, but this knowledge is only sufficient to make people culpable for their sin; it does not include knowledge of the gospel. The second channel through which God reveals himself is redemption, which is the gospel of salvation from condemnation which is punishment for sin.

In Reformed theology, the Word of God takes several forms. Jesus Christ himself is the Word Incarnate. The prophecies about him said to be found in the Old Testament and the ministry of the apostles who saw him and communicated his message are also the Word of God. Further, the preaching of ministers about God is the very Word of God because God is considered to be speaking through them. God also speaks through human writers in the Bible, which is composed of texts set apart by God for self-revelation. Reformed theologians emphasize the Bible as a uniquely important means by which God communicates with people. People gain knowledge of God from the Bible which cannot be gained in any other way.

Reformed theologians affirm that the Bible is true, but differences emerge among them over the meaning and extent of its truthfulness. Conservative followers of the Princeton theologians take the view that the Bible is true and inerrant, or incapable of error or falsehood, in every place. This view is similar to that of Catholic orthodoxy as well as modern Evangelicalism. Another view, influenced by the teaching of Karl Barth and neo-orthodoxy, is found in the Presbyterian Church (U.S.A.)'s Confession of 1967. Those who take this view believe the Bible to be the primary source of our knowledge of God, but also that some parts of the Bible may be false, not witnesses to Christ, and not normative for the church. In this view, Christ is the revelation of God, and the scriptures witness to this revelation rather than being the revelation itself.

Covenant theology
Reformed theologians use the concept of covenant to describe the way God enters into fellowship with people in history. The concept of covenant is so prominent in Reformed theology that Reformed theology as a whole is sometimes called "covenant theology". However, sixteenth- and seventeenth-century theologians developed a particular theological system called "covenant theology" or "federal theology" which many conservative Reformed churches continue to affirm. This framework orders God's life with people primarily in two covenants: the covenant of works and the covenant of grace.

The covenant of works is made with Adam and Eve in the Garden of Eden. The terms of the covenant are that God provides a blessed life in the garden on condition that Adam and Eve obey God's law perfectly. Because Adam and Eve broke the covenant by eating the forbidden fruit, they became subject to death and were banished from the garden. This sin was passed down to all mankind because all people are said to be in Adam as a covenantal or "federal" head. Federal theologians usually imply that Adam and Eve would have gained immortality had they obeyed perfectly.

A second covenant, called the covenant of grace, is said to have been made immediately following Adam and Eve's sin. In it, God graciously offers salvation from death on condition of faith in God. This covenant is administered in different ways throughout the Old and New Testaments, but retains the substance of being free of a requirement of perfect obedience.

Through the influence of Karl Barth, many contemporary Reformed theologians have discarded the covenant of works, along with other concepts of federal theology. Barth saw the covenant of works as disconnected from Christ and the gospel, and rejected the idea that God works with people in this way. Instead, Barth argued that God always interacts with people under the covenant of grace, and that the covenant of grace is free of all conditions whatsoever. Barth's theology and that which follows him has been called "mono covenantal" as opposed to the "bi-covenantal" scheme of classical federal theology. Conservative contemporary Reformed theologians, such as John Murray, have also rejected the idea of covenants based on law rather than grace. Michael Horton, however, has defended the covenant of works as combining principles of law and love.

God
For the most part, the Reformed tradition did not modify the medieval consensus on the doctrine of God. God's character is described primarily using three adjectives: eternal, infinite, and unchangeable. Reformed theologians such as Shirley Guthrie have proposed that rather than conceiving of God in terms of his attributes and freedom to do as he pleases, the doctrine of God is to be based on God's work in history and his freedom to live with and empower people.

Reformed theologians have also traditionally followed the medieval tradition going back to before the early church councils of Nicaea and Chalcedon on the doctrine of the Trinity. God is affirmed to be one God in three persons: Father, Son, and Holy Spirit. The Son (Christ) is held to be eternally begotten by the Father and the Holy Spirit eternally proceeding from the Father and Son. However, contemporary theologians have been critical of aspects of Western views here as well. Drawing on the Eastern tradition, these Reformed theologians have proposed a "social trinitarianism" where the persons of the Trinity only exist in their life together as persons-in-relationship. Contemporary Reformed confessions such as the Barmen Confession and Brief Statement of Faith of the Presbyterian Church (USA) have avoided language about the attributes of God and have emphasized his work of reconciliation and empowerment of people. Feminist theologian Letty Russell used the image of partnership for the persons of the Trinity. According to Russell, thinking this way encourages Christians to interact in terms of fellowship rather than reciprocity. Conservative Reformed theologian Michael Horton, however, has argued that social trinitarianism is untenable because it abandons the essential unity of God in favor of a community of separate beings.

Christ and atonement
Reformed theologians affirm the historic Christian belief that Christ is eternally one person with a divine and a human nature. Reformed Christians have especially emphasized that Christ truly became human so that people could be saved. Christ's human nature has been a point of contention between Reformed and Lutheran Christology. In accord with the belief that finite humans cannot comprehend infinite divinity, Reformed theologians hold that Christ's human body cannot be in multiple locations at the same time. Because Lutherans believe that Christ is bodily present in the Eucharist, they hold that Christ is bodily present in many locations simultaneously. For Reformed Christians, such a belief denies that Christ actually became human. Some contemporary Reformed theologians have moved away from the traditional language of one person in two natures, viewing it as unintelligible to contemporary people. Instead, theologians tend to emphasize Jesus's context and particularity as a first-century Jew.

John Calvin and many Reformed theologians who followed him describe Christ's work of redemption in terms of three offices: prophet, priest, and king. Christ is said to be a prophet in that he teaches perfect doctrine, a priest in that he intercedes to the Father on believers' behalf and offered himself as a sacrifice for sin, and a king in that he rules the church and fights on believers' behalf. The threefold office links the work of Christ to God's work in ancient Israel. Many, but not all, Reformed theologians continue to make use of the threefold office as a framework because of its emphasis on the connection of Christ's work to Israel. They have, however, often reinterpreted the meaning of each of the offices. For example, Karl Barth interpreted Christ's prophetic office in terms of political engagement on behalf of the poor.

Christians believe Jesus' death and resurrection make it possible for believers to receive forgiveness for sin and reconciliation with God through the atonement. Reformed Protestants generally subscribe to a particular view of the atonement called penal substitutionary atonement, which explains Christ's death as a sacrificial payment for sin. Christ is believed to have died in place of the believer, who is accounted righteous as a result of this sacrificial payment.

Sin
In Christian theology, people are created good and in the image of God but have become corrupted by sin, which causes them to be imperfect and overly self-interested. Reformed Christians, following the tradition of Augustine of Hippo, believe that this corruption of human nature was brought on by Adam and Eve's first sin, a doctrine called original sin.

Although earlier Christian authors taught the elements of physical death, moral weakness, and a sin propensity within original sin, Augustine was the first Christian to add the concept of inherited guilt (reatus) from Adam whereby every infant is born eternally damned and humans lack any residual ability to respond to God. Reformed theologians emphasize that this sinfulness affects all of a person's nature, including their will. This view, that sin so dominates people that they are unable to avoid sin, has been called total depravity. As a consequence, every one of their descendants inherited a stain of corruption and depravity. This condition, innate to all humans, is known in Christian theology as original sin.

Calvin thought original sin was "a hereditary corruption and depravity of our nature, extending to all the parts of the soul." Calvin asserted people were so warped by original sin that "everything which our mind conceives, meditates, plans, and resolves, is always evil." The depraved condition of every human being is not the result of sins people commit during their lives. Instead, before we are born, while we are in our mother's womb, "we are in God's sight defiled and polluted." Calvin thought people were justly condemned to hell because their corrupted state is "naturally hateful to God."

In colloquial English, the term "total depravity" can be easily misunderstood to mean that people are absent of any goodness or unable to do any good. However the Reformed teaching is actually that while people continue to bear God's image and may do things that appear outwardly good, their sinful intentions affect all of their nature and actions so that they are not pleasing to God.

Some contemporary theologians in the Reformed tradition, such as those associated with the Presbyterian Church (USA)'s Confession of 1967, have emphasized the social character of human sinfulness. These theologians have sought to bring attention to issues of environmental, economic, and political justice as areas of human life that have been affected by sin.

Salvation
Reformed theologians, along with other Protestants, believe salvation from punishment for sin is to be given to all those who have faith in Christ. Faith is not purely intellectual, but involves trust in God's promise to save. Protestants do not hold there to be any other requirement for salvation, but that faith alone is sufficient.

Justification is the part of salvation where God pardons the sin of those who believe in Christ. It is historically held by Protestants to be the most important article of Christian faith, though more recently it is sometimes given less importance out of ecumenical concerns. People are not on their own able to fully repent of their sin or prepare themselves to repent because of their sinfulness. Therefore, justification is held to arise solely from God's free and gracious act.

Sanctification is the part of salvation in which God makes believers holy, by enabling them to exercise greater love for God and for other people. The good works accomplished by believers as they are sanctified are considered to be the necessary outworking of the believer's salvation, though they do not cause the believer to be saved. Sanctification, like justification, is by faith, because doing good works is simply living as the child of God one has become.

Predestination
Reformed theologians teach that sin so affects human nature that they are unable even to exercise faith in Christ by their own will. While people are said to retain free will, in that they willfully sin, they are unable not to sin because of the corruption of their nature due to original sin. Reformed Christians believe that God predestined some people to be saved and others were predestined to eternal damnation. This choice by God to save some is held to be unconditional and not based on any characteristic or action on the part of the person chosen. This view is opposed to the Arminian view that God's choice of whom to save is conditional or based on his foreknowledge of who would respond positively to God.

Karl Barth reinterpreted the Reformed doctrine of predestination to apply only to Christ. Individual people are only said to be elected through their being in Christ. Reformed theologians who followed Barth, including Jürgen Moltmann, David Migliore, and Shirley Guthrie, have argued that the traditional Reformed concept of predestination is speculative and have proposed alternative models. These theologians claim that a properly trinitarian doctrine emphasizes God's freedom to love all people, rather than choosing some for salvation and others for damnation. God's justice towards and condemnation of sinful people is spoken of by these theologians as out of his love for them and a desire to reconcile them to himself.

Five Points of Calvinism
Much attention surrounding Calvinism focuses on the "Five Points of Calvinism" (also called the doctrines of grace). The five points have been summarized under the acrostic TULIP. The five points are popularly said to summarize the Canons of Dort; however, there is no historical relationship between them, and some scholars argue that their language distorts the meaning of the Canons, Calvin's theology, and the theology of 17th-century Calvinistic orthodoxy, particularly in the language of total depravity and limited atonement. The five points were more recently popularized in the 1963 booklet The Five Points of Calvinism Defined, Defended, Documented by David N. Steele and Curtis C. Thomas. The origins of the five points and the acrostic are uncertain, but they appear to be outlined in the Counter Remonstrance of 1611, a lesser-known Reformed reply to the Arminians, which was written prior to the Canons of Dort. The acrostic was used by Cleland Boyd McAfee as early as circa 1905. An early printed appearance of the acrostic can be found in Loraine Boettner's 1932 book, The Reformed Doctrine of Predestination.

Church
Reformed Christians see the Christian Church as the community with which God has made the covenant of grace, a promise of eternal life and relationship with God. This covenant extends to those under the "old covenant" whom God chose, beginning with Abraham and Sarah. The church is conceived of as both invisible and visible. The invisible church is the body of all believers, known only to God. The visible church is the institutional body which contains both members of the invisible church as well as those who appear to have faith in Christ, but are not truly part of God's elect.

In order to identify the visible church, Reformed theologians have spoken of certain marks of the Church. For some, the only mark is the pure preaching of the gospel of Christ. Others, including John Calvin, also include the right administration of the sacraments. Others, such as those following the Scots Confession, include a third mark of rightly administered church discipline, or exercise of censure against unrepentant sinners. These marks allowed the Reformed to identify the church based on its conformity to the Bible rather than the Magisterium or church tradition.

Regulative principle of worship
The regulative principle of worship is a teaching shared by some Calvinists and Anabaptists on how the Bible orders public worship. The substance of the doctrine regarding worship is that God institutes in the Scriptures everything he requires for worship in the Church and that everything else is prohibited. As the regulative principle is reflected in Calvin's own thought, it is driven by his evident antipathy toward the Roman Catholic Church and its worship practices, and it associates musical instruments with icons, which he considered violations of the Ten Commandments' prohibition of graven images.

On this basis, many early Calvinists also eschewed musical instruments and advocated a cappella exclusive psalmody in worship, though Calvin himself allowed other scriptural songs as well as psalms, and this practice typified Presbyterian worship and the worship of other Reformed churches for some time. The original Lord's Day service designed by John Calvin was a highly liturgical service with the Creed, Alms, Confession and Absolution, the Lord's supper, Doxologies, prayers, Psalms being sung, the Lords prayer being sung, and Benedictions.

Since the 19th century, however, some of the Reformed churches have modified their understanding of the regulative principle and make use of musical instruments, believing that Calvin and his early followers went beyond the biblical requirements and that such things are circumstances of worship requiring biblically rooted wisdom, rather than an explicit command. Despite the protestations of those who hold to a strict view of the regulative principle, today hymns and musical instruments are in common use, as are contemporary worship music styles with elements such as worship bands.

Sacraments
The Westminster Confession of Faith limits the sacraments to baptism and the Lord's Supper. Sacraments are denoted "signs and seals of the covenant of grace." Westminster speaks of "a sacramental relation, or a sacramental union, between the sign and the thing signified; whence it comes to pass that the names and effects of the one are attributed to the other." Baptism is for infant children of believers as well as believers, as it is for all the Reformed except Baptists and some Congregationalists. Baptism admits the baptized into the visible church, and in it all the benefits of Christ are offered to the baptized. On the Lord's supper, the Westminster Confession takes a position between Lutheran sacramental union and Zwinglian memorialism: "the Lord's supper really and indeed, yet not carnally and corporally, but spiritually, receive and feed upon Christ crucified, and all benefits of his death: the body and blood of Christ being then not corporally or carnally in, with, or under the bread and wine; yet, as really, but spiritually, present to the faith of believers in that ordinance as the elements themselves are to their outward senses."

The 1689 London Baptist Confession of Faith does not use the term sacrament, but describes baptism and the Lord's supper as ordinances, as do most Baptists, Calvinist or otherwise. Baptism is only for those who "actually profess repentance towards God", and not for the children of believers. Baptists also insist on immersion or dipping, in contradistinction to other Reformed Christians. The Baptist Confession describes the Lord's supper as "the body and blood of Christ being then not corporally or carnally, but spiritually present to the faith of believers in that ordinance", similarly to the Westminster Confession. There is significant latitude in Baptist congregations regarding the Lord's supper, and many hold the Zwinglian view.

Logical order of God's decree
There are two schools of thought regarding the logical order of God's decree to ordain the fall of man: supralapsarianism (from the Latin: supra, "above", here meaning "before" + lapsus, "fall") and infralapsarianism (from the Latin: infra, "beneath", here meaning "after" + lapsus, "fall"). The former view, sometimes called "high Calvinism", argues that the Fall occurred partly to facilitate God's purpose to choose some individuals for salvation and some for damnation. Infralapsarianism, sometimes called "low Calvinism", is the position that, while the Fall was indeed planned, it was not planned with reference to who would be saved.

Supralapsarianism is based on the belief that God chose which individuals to save logically prior to the decision to allow the race to fall and that the Fall serves as the means of realization of that prior decision to send some individuals to hell and others to heaven (that is, it provides the grounds of condemnation in the reprobate and the need for salvation in the elect). In contrast, infralapsarians hold that God planned the race to fall logically prior to the decision to save or damn any individuals because, it is argued, in order to be "saved", one must first need to be saved from something and therefore the decree of the Fall must precede predestination to salvation or damnation.

These two views vied with each other at the Synod of Dort, an international body representing Calvinist Christian churches from around Europe, and the judgments that came out of that council sided with infralapsarianism (Canons of Dort, First Point of Doctrine, Article 7). The Westminster Confession of Faith also teaches (in Hodge's words "clearly impl[ies]") the infralapsarian view, but is sensitive to those holding to supralapsarianism. The Lapsarian controversy has a few vocal proponents on each side today, but overall it does not receive much attention among modern Calvinists.

Branches
The Reformed tradition is historically represented by the Continental, Presbyterian, Reformed Anglican, Congregationalist, and Reformed Baptist denominational families.

Reformed churches practice several forms of church government, primarily presbyterian and congregational, but some adhere to episcopal polity. The largest interdenominational association is the World Communion of Reformed Churches with more than 100 million members in 211 member denominations around the world. Smaller, conservative Reformed associations include the World Reformed Fellowship and the International Conference of Reformed Churches.

Continental
"Continental" Reformed churches originate in continental Europe, a term used by English speakers to distinguish them from traditions from the British Isles. Many uphold the Helvetic Confessions and Heidelberg Catechism, which were adopted in Zurich and Heidelberg, respectively. In the United States, immigrants belonging to the continental Reformed churches joined the Dutch Reformed Church there, as well as the Anglican Church.

Presbyterian
Presbyterian churches are named for their order of government by assemblies of elders, or presbyters. They are especially influenced by John Knox, who brought Reformed theology and polity to the Church of Scotland after spending time on the continent in Calvin's Geneva. Presbyterians historically uphold the Westminster Confession of Faith.

Congregational
Congregationalism originates in Puritanism, a sixteenth-century movement to reform the Church of England. Unlike the Presbyterians, Congregationalists consider the local church to be rightfully self-ruled by their own officers, not higher ecclesiastical courts. The Savoy Declaration, a revision of Westminster, is the primary confession of historic Congregationalism. Evangelical Congregationalists are internationally represented by the World Evangelical Congregational Fellowship. Christian denominations in the Congregationalist tradition include the Conservative Congregational Christian Conference in the United States, Evangelical Congregational Church in Argentina and Evangelical Fellowship of Congregational Churches in the United Kingdom, among others.

Baptist
Reformed or Calvinistic Baptists, unlike other reformed traditions, exclusively practice believer's baptism. They observe congregational polity like the Congregationalists. Their primary confession is the 1689 Baptist Confession of Faith, a revision of the Congregationalists' Savoy, but other Baptist Confessions are also used. Not all Baptists are reformed. Some Reformed Baptists accept reformed theology, especially soteriology, but do not hold to a specific confession or to covenant theology.

Anglican
Though Anglicanism today is often described as a separate branch from the Reformed, historic Anglicanism is a part of the wider Reformed tradition. The foundational documents of the Anglican church "express a theology in keeping with the Reformed theology of the Swiss and South German Reformation." The Most Rev. Peter Robinson, presiding bishop of the United Episcopal Church of North America, writes: "Cranmer's personal journey of faith left its mark on the Church of England in the form of a Liturgy that remains to this day more closely allied to Lutheran practice, but that liturgy is couple to a doctrinal stance that is broadly, but decidedly Reformed. ... The 42 Articles of 1552 and the 39 Articles of 1563, both commit the Church of England to the fundamentals of the Reformed Faith. Both sets of Articles affirm the centrality of Scripture, and take a monergist position on Justification. Both sets of Articles affirm that the Church of England accepts the doctrine of predestination and election as a 'comfort to the faithful' but warn against over much speculation concerning that doctrine. Indeed a casual reading of the Wurttemburg Confession of 1551, the Second Helvetic Confession, the Scots Confession of 1560, and the XXXIX Articles of Religion reveal them to be cut from the same bolt of cloth."

Amyraldism

Amyraldism (or sometimes Amyraldianism, also known as the School of Saumur, hypothetical universalism, post redemptionism, moderate Calvinism, or four-point Calvinism) is the belief that God, prior to his decree of election, decreed Christ's atonement for all alike if they believe, but seeing that none would believe on their own, he then elected those whom he will bring to faith in Christ, thereby preserving the Calvinist doctrine of unconditional election. The efficacy of the atonement remains limited to those who believe.

Named after its formulator Moses Amyraut, this doctrine is still viewed as a variety of Calvinism in that it maintains the particularity of sovereign grace in the application of the atonement. However, detractors like B. B. Warfield have termed it "an inconsistent and therefore unstable form of Calvinism."

Hyper-Calvinism
Hyper-Calvinism first referred to a view that appeared among the early English Particular Baptists in the 18th century. Their system denied that the call of the gospel to "repent and believe" is directed to every single person and that it is the duty of every person to trust in Christ for salvation. The term also occasionally appears in both theological and secular controversial contexts, where it usually connotes a negative opinion about some variety of theological determinism, predestination, or a version of Evangelical Christianity or Calvinism that is deemed by the critic to be unenlightened, harsh, or extreme.

The Westminster Confession of Faith says that the gospel is to be freely offered to sinners, and the Larger Catechism makes clear that the gospel is offered to the non-elect.

Neo-Calvinism
Beginning in the 1880s, Neo-Calvinism, a form of Dutch Calvinism, is the movement initiated by the theologian and former Dutch prime minister Abraham Kuyper. James Bratt has identified a number of different types of Dutch Calvinism: The Seceders—split into the Reformed Church "West" and the Confessionalists; and the Neo-Calvinists—the Positives and the Antithetical Calvinists. The Seceders were largely infralapsarian and the Neo-Calvinists usually supralapsarian.

Kuyper wanted to awaken the church from what he viewed as its pietistic slumber. He declared:

"No single piece of our mental world is to be sealed off from the rest and there is not a square inch in the whole domain of human existence over which Christ, who is sovereign over all, does not cry: 'Mine!'"

This refrain has become something of a rallying call for Neo-Calvinists.

Christian Reconstructionism
Christian Reconstructionism is a fundamentalist Calvinist theonomic movement that has remained rather obscure. Founded by R. J. Rushdoony, the movement has had an important influence on the Christian Right in the United States. The movement peaked in the 1990s. However, it lives on in small denominations such as the Reformed Presbyterian Church in the United States and as a minority position in other denominations. Christian Reconstructionists are usually postmillennialists and followers of the presuppositional apologetics of Cornelius Van Til. They tend to support a decentralized political order resulting in laissez-faire capitalism.

New Calvinism
New Calvinism is a growing perspective within conservative Evangelicalism that embraces the fundamentals of 16th century Calvinism while also trying to be relevant in the present day world. In March 2009, Time magazine described the New Calvinism as one of the "10 ideas changing the world". Some of the major figures who have been associated with the New Calvinism are John Piper, Mark Driscoll, Al Mohler, Mark Dever, C. J. Mahaney, and Tim Keller. New Calvinists have been criticized for blending Calvinist soteriology with popular Evangelical positions on the sacraments and continuationism and for rejecting tenets seen as crucial to the Reformed faith such as confessionalism and covenant theology.

Social and economic influences
Calvin expressed himself on usury in a 1545 letter to a friend, Claude de Sachin, in which he criticized the use of certain passages of scripture invoked by people opposed to the charging of interest. He reinterpreted some of these passages, and suggested that others of them had been rendered irrelevant by changed conditions. He also dismissed the argument (based upon the writings of Aristotle) that it is wrong to charge interest for money because money itself is barren. He said that the walls and the roof of a house are barren, too, but it is permissible to charge someone for allowing him to use them. In the same way, money can be made fruitful.

He qualified his view, however, by saying that money should be lent to people in dire need without hope of interest, while a modest interest rate of 5% should be permitted in relation to other borrowers.

In The Protestant Ethic and the Spirit of Capitalism, Max Weber wrote that capitalism in Northern Europe evolved when the Protestant (particularly Calvinist) ethic influenced large numbers of people to engage in work in the secular world, developing their own enterprises and engaging in trade and the accumulation of wealth for investment. In other words, the Protestant work ethic was an important force behind the unplanned and uncoordinated emergence of modern capitalism.

Expert researchers and authors have referred to the United States as a "Protestant nation" or "founded on Protestant principles," specifically emphasizing its Calvinist heritage.

Politics and society
Calvin's concepts of God and man led to ideas which were gradually put into practice after his death, in particular in the fields of politics and society. After their fight for independence from Spain (1579), the Netherlands, under Calvinist leadership, granted asylum to religious minorities, including French Huguenots, English Independents (Congregationalists), and Jews from Spain and Portugal. The ancestors of the philosopher Baruch Spinoza were Portuguese Jews. Aware of the trial against Galileo, René Descartes lived in the Netherlands, out of reach of the Inquisition, from 1628 to 1649. Pierre Bayle, a Reformed Frenchman, also felt safer in the Netherlands than in his home country. He was the first prominent philosopher who demanded tolerance for atheists. Hugo Grotius (1583–1645) was able to publish a rather liberal interpretation of the Bible and his ideas about natural law in the Netherlands. Moreover, the Calvinist Dutch authorities allowed the printing of books that could not be published elsewhere, such as Galileo's Discorsi (1638).

Alongside the liberal development of the Netherlands came the rise of modern democracy in England and North America. In the Middle Ages, state and church had been closely connected. Martin Luther's doctrine of the two kingdoms separated state and church in principle. His doctrine of the priesthood of all believers raised the laity to the same level as the clergy. Going one step further, Calvin included elected laymen (church elders, presbyters) in his concept of church government. The Huguenots added synods whose members were also elected by the congregations. The other Reformed churches took over this system of church self-government, which was essentially a representative democracy. Baptists, Quakers, and Methodists are organized in a similar way. These denominations and the Anglican Church were influenced by Calvin's theology in varying degrees.

In another factor in the rise of democracy in the Anglo-American world, Calvin favored a mixture of democracy and aristocracy as the best form of government (mixed government). He appreciated the advantages of democracy. His political thought aimed to safeguard the rights and freedoms of ordinary men and women. In order to minimize the misuse of political power he suggested dividing it among several institutions in a system of checks and balances (separation of powers). Finally, Calvin taught that if worldly rulers rise up against God they should be put down. In this way, he and his followers stood in the vanguard of resistance to political absolutism and furthered the cause of democracy. The Congregationalists who founded Plymouth Colony (1620) and Massachusetts Bay Colony (1628) were convinced that the democratic form of government was the will of God. Enjoying self-rule, they practiced separation of powers. Rhode Island, Connecticut, and Pennsylvania, founded by Roger Williams, Thomas Hooker, and William Penn, respectively, combined democratic government with a limited freedom of religion that did not extend to Catholics (Congregationalism being the established, tax-supported religion in Connecticut). These colonies became safe havens for persecuted religious minorities, including Jews.

In England, Baptists Thomas Helwys (c. 1575–c.  1616), and John Smyth (c.  1554–c. 1612) influenced the liberal political thought of the Presbyterian poet and politician John Milton (1608–1674) and of the philosopher John Locke (1632–1704), who in turn had both a strong impact on the political development in their home country (English Civil War of 1642–1651, Glorious Revolution of 1688) as well as in North America. The ideological basis of the American Revolution was largely provided by the radical Whigs, who had been inspired by Milton, Locke, James Harrington (1611–1677), Algernon Sidney (1623–1683), and other thinkers. The Whigs' "perceptions of politics attracted widespread support in America because they revived the traditional concerns of a Protestantism that had always verged on Puritanism". The United States Declaration of Independence, the United States Constitution and (American) Bill of Rights initiated a tradition of human and civil rights that continued in the French Declaration of the Rights of Man and of the Citizen and the constitutions of numerous countries around the world, e.g. Latin America, Japan, India, Germany, and other European countries. It is also echoed in the United Nations Charter and the Universal Declaration of Human Rights.

In the 19th century, churches based on or influenced by Calvin's theology became deeply involved in social reforms, e.g. the abolition of slavery (William Wilberforce, Harriet Beecher Stowe, Abraham Lincoln, and others), women suffrage, and prison reforms. Members of these churches formed co-operatives to help the impoverished masses. The founders of the Red Cross Movement, including Henry Dunant, were Reformed Christians. Their movement also initiated the Geneva Conventions.

Others view Calvinist influence as not always being solely positive. The Boers and Afrikaner Calvinists combined ideas from Calvinism and Kuyperian theology to justify apartheid in South Africa. As late as 1974 the majority of the Dutch Reformed Church in South Africa was convinced that their theological stances (including the story of the Tower of Babel) could justify apartheid. In 1990 the Dutch Reformed Church document Church and Society maintained that although they were changing their stance on apartheid, they believed that within apartheid and under God's sovereign guidance, "...everything was not without significance, but was of service to the Kingdom of God." These views were not universal and were condemned by many Calvinists outside South Africa. Pressure from both outside and inside the Dutch Reformed Calvinist church helped reverse apartheid in South Africa.

Throughout the world, the Reformed churches operate hospitals, homes for handicapped or elderly people, and educational institutions on all levels. For example, American Congregationalists founded Harvard University (1636), Yale University (1701), and about a dozen other colleges. A particular stream of influence of Calvinism concerns art. Visual art cemented society in the first modern nation state, the Netherlands, and also Neo-Calvinism put much weight on this aspect of life. Hans Rookmaaker is the most prolific example. In literature one can think of Marilynne Robinson. In her non-fiction she powerfully demonstrates the modernity of Calvin's thinking, calling him a humanist scholar (p. 174, The Death of Adam).

Doctrine

 * Common grace
 * Reformed confessions of faith

Related

 * Boer Calvinists: Boere-Afrikaners that hold to Reformed theology
 * Continental Reformed church: Calvinist churches originating in continental Europe
 * Augustinian Calvinism: a term used to emphasize the origin of John Calvin's theology within Augustine of Hippo's theology
 * Huguenots: followers of Calvinism in France, originating in the 16th and 17th century
 * Pilgrims: English Separatists who left Europe for America in search of religious toleration, eventually settling in New England
 * Presbyterians: Calvinists in countries worldwide
 * Puritans: English Protestants who wanted to purify the Church of England
 * Waldensians: Italian Protestants, preceded Calvinism but today identify with Reformed theology

Opposing views

 * Arminianism
 * Catholicism
 * Augustinianism
 * Christian universalism
 * Eastern Orthodoxy
 * Palamism
 * Free Grace theology
 * Open theism
 * Lutheranism
 * Molinism
 * Socinianism

title: Reformed Christianity


text:
Countable set

In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.

In more technical terms, assuming the axiom of countable choice, a set is countable if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite.

The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers.

A note on terminology
Although the terms "countable" and "countably infinite" as defined here are quite common, the terminology is not universal. An alternative style uses countable to mean what is here called countably infinite, and at most countable to mean what is here called countable.

The terms enumerable and denumerable may also be used, e.g. referring to countable and countably infinite respectively, definitions vary and care is needed respecting the difference with recursively enumerable.

Definition
A set $$S$$ is countable if: All of these definitions are equivalent.
 * Its cardinality $$|S|$$ is less than or equal to $$\aleph_0$$ (aleph-null), the cardinality of the set of natural numbers $$\N$$.
 * There exists an injective function from $$S$$ to $$\N$$.
 * $$S$$ is empty or there exists a surjective function from $$\N$$ to $$S$$.
 * There exists a bijective mapping between $$S$$ and a subset of $$\N$$.
 * $$S$$ is either finite ($$|S|<\aleph_0$$) or countably infinite.

A set $$S$$ is countably infinite if:
 * Its cardinality $$|S|$$ is exactly $$\aleph_0$$.
 * There is an injective and surjective (and therefore bijective) mapping between $$S$$ and $$\N$$.
 * $$S$$ has a one-to-one correspondence with $$\N$$.
 * The elements of $$S$$ can be arranged in an infinite sequence $$a_0, a_1, a_2, \ldots$$, where $$a_i$$ is distinct from $$a_j$$ for $$i\neq j$$ and every element of $$S$$ is listed.

A set is uncountable if it is not countable, i.e. its cardinality is greater than $$\aleph_0$$.

History
In 1874, in his first set theory article, Cantor proved that the set of real numbers is uncountable, thus showing that not all infinite sets are countable. In 1878, he used one-to-one correspondences to define and compare cardinalities. In 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.

Introduction
A set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements; for example, the set consisting of the integers 3, 4, and 5 may be denoted $$\{3, 4, 5\}$$, called roster form. This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element, sometimes an ellipsis ("...") is used to represent many elements between the starting element and the end element in a set, if the writer believes that the reader can easily guess what ... represents; for example, $$\{1, 2, 3, \dots, 100\}$$ presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possible to list all the elements, because the number of elements in the set is finite. If we number the elements of the set 1, 2, and so on, up to $$n$$, this gives us the usual definition of "sets of size $$n$$".

Some sets are infinite; these sets have more than $$n$$ elements where $$n$$ is any integer that can be specified. (No matter how large the specified integer $$n$$ is, such as $$n=10^{1000}$$, infinite sets have more than $$n$$ elements.) For example, the set of natural numbers, denotable by $$\{0, 1, 2, 3, 4, 5,\dots\}$$, has infinitely many elements, and we cannot use any natural number to give its size. It might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view works well for countably infinite sets and was the prevailing assumption before Georg Cantor's work. For example, there are infinitely many odd integers, infinitely many even integers, and also infinitely many integers overall. We can consider all these sets to have the same "size" because we can arrange things such that, for every integer, there is a distinct even integer: $$\ldots \, -\! 2\! \rightarrow \! - \! 4, \, -\! 1\! \rightarrow \! - \! 2, \, 0\! \rightarrow \! 0, \, 1\! \rightarrow \! 2, \, 2\! \rightarrow \! 4 \, \cdots$$ or, more generally, $$n \rightarrow 2n$$ (see picture). What we have done here is arrange the integers and the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set. This mathematical notion of "size", cardinality, is that two sets are of the same size if and only if there is a bijection between them. We call all sets that are in one-to-one correspondence with the integers countably infinite and say they have cardinality $$\aleph_0$$.

Georg Cantor showed that not all infinite sets are countably infinite. For example, the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers). The set of real numbers has a greater cardinality than the set of natural numbers and is said to be uncountable.

Formal overview
By definition, a set $$S$$ is countable if there exists a bijection between $$S$$ and a subset of the natural numbers $$\N=\{0,1,2,\dots\}$$. For example, define the correspondence $$ a \leftrightarrow 1,\ b \leftrightarrow 2,\ c \leftrightarrow 3 $$ Since every element of $$S=\{a,b,c\}$$ is paired with precisely one element of $$\{1,2,3\}$$, and vice versa, this defines a bijection, and shows that $$S$$ is countable. Similarly we can show all finite sets are countable.

As for the case of infinite sets, a set $$S$$ is countably infinite if there is a bijection between $$S$$ and all of $$\N$$. As examples, consider the sets $$A=\{1,2,3,\dots\}$$, the set of positive integers, and $$B=\{0,2,4,6,\dots\}$$, the set of even integers. We can show these sets are countably infinite by exhibiting a bijection to the natural numbers. This can be achieved using the assignments $$n \leftrightarrow n+1$$ and $$n \leftrightarrow 2n$$, so that $$\begin{matrix} 0 \leftrightarrow 1, & 1 \leftrightarrow 2, & 2 \leftrightarrow 3, & 3 \leftrightarrow 4, & 4 \leftrightarrow 5, & \ldots \\[6pt] 0 \leftrightarrow 0, & 1 \leftrightarrow 2, & 2 \leftrightarrow 4, & 3 \leftrightarrow 6, & 4 \leftrightarrow 8, & \ldots \end{matrix}$$ Every countably infinite set is countable, and every infinite countable set is countably infinite. Furthermore, any subset of the natural numbers is countable, and more generally: $$

The set of all ordered pairs of natural numbers (the Cartesian product of two sets of natural numbers, $$\N\times\N$$ is countably infinite, as can be seen by following a path like the one in the picture: The resulting mapping proceeds as follows: $$ 0 \leftrightarrow (0, 0), 1 \leftrightarrow (1, 0), 2 \leftrightarrow (0, 1), 3 \leftrightarrow (2, 0), 4 \leftrightarrow (1, 1), 5 \leftrightarrow (0, 2), 6 \leftrightarrow (3, 0), \ldots $$ This mapping covers all such ordered pairs.

This form of triangular mapping recursively generalizes to $$n$$-tuples of natural numbers, i.e., $$(a_1,a_2,a_3,\dots,a_n)$$ where $$a_i$$ and $$n$$ are natural numbers, by repeatedly mapping the first two elements of an $$n$$-tuple to a natural number. For example, $$(0, 2, 3)$$ can be written as $$((0, 2), 3)$$. Then $$(0, 2)$$ maps to 5 so $$((0, 2), 3)$$ maps to $$(5, 3)$$, then $$(5, 3)$$ maps to 39. Since a different 2-tuple, that is a pair such as $$(a,b)$$, maps to a different natural number, a difference between two n-tuples by a single element is enough to ensure the n-tuples being mapped to different natural numbers. So, an injection from the set of $$n$$-tuples to the set of natural numbers $$\N$$ is proved. For the set of $$n$$-tuples made by the Cartesian product of finitely many different sets, each element in each tuple has the correspondence to a natural number, so every tuple can be written in natural numbers then the same logic is applied to prove the theorem.

$$

The set of all integers $$\Z$$ and the set of all rational numbers $$\Q$$ may intuitively seem much bigger than $$\N$$. But looks can be deceiving. If a pair is treated as the numerator and denominator of a vulgar fraction (a fraction in the form of $$a/b$$ where $$a$$ and $$b\neq 0$$ are integers), then for every positive fraction, we can come up with a distinct natural number corresponding to it. This representation also includes the natural numbers, since every natural number $$n$$ is also a fraction $$n/1$$. So we can conclude that there are exactly as many positive rational numbers as there are positive integers. This is also true for all rational numbers, as can be seen below.

$$

In a similar manner, the set of algebraic numbers is countable.

Sometimes more than one mapping is useful: a set $$A$$ to be shown as countable is one-to-one mapped (injection) to another set $$B$$, then $$A$$ is proved as countable if $$B$$ is one-to-one mapped to the set of natural numbers. For example, the set of positive rational numbers can easily be one-to-one mapped to the set of natural number pairs (2-tuples) because $$p/q$$ maps to $$(p,q)$$. Since the set of natural number pairs is one-to-one mapped (actually one-to-one correspondence or bijection) to the set of natural numbers as shown above, the positive rational number set is proved as countable.

$$

With the foresight of knowing that there are uncountable sets, we can wonder whether or not this last result can be pushed any further. The answer is "yes" and "no", we can extend it, but we need to assume a new axiom to do so.

$$

For example, given countable sets $$\textbf{a},\textbf{b},\textbf{c},\dots$$, we first assign each element of each set a tuple, then we assign each tuple an index using a variant of the triangular enumeration we saw above: $$ \begin{array}{ c|c|c } \text{Index} & \text{Tuple} & \text {Element} \\ \hline 0 & (0,0) & \textbf{a}_0 \\ 1 & (0,1) & \textbf{a}_1 \\ 2 & (1,0) & \textbf{b}_0 \\ 3 & (0,2) & \textbf{a}_2 \\ 4 & (1,1) & \textbf{b}_1 \\ 5 & (2,0) & \textbf{c}_0 \\ 6 & (0,3) & \textbf{a}_3 \\ 7 & (1,2) & \textbf{b}_2 \\ 8 & (2,1) & \textbf{c}_1 \\ 9 & (3,0) & \textbf{d}_0 \\ 10 & (0,4) & \textbf{a}_4 \\ \vdots & & \end{array} $$

We need the axiom of countable choice to index all the sets $$\textbf{a},\textbf{b},\textbf{c},\dots$$ simultaneously.

$$

This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which is countable by the previous theorem.

$$

The elements of any finite subset can be ordered into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets.

$$

These follow from the definitions of countable set as injective / surjective functions.

Cantor's theorem asserts that if $$A$$ is a set and $$\mathcal{P}(A)$$ is its power set, i.e. the set of all subsets of $$A$$, then there is no surjective function from $$A$$ to $$\mathcal{P}(A)$$. A proof is given in the article Cantor's theorem. As an immediate consequence of this and the Basic Theorem above we have: $$

For an elaboration of this result see Cantor's diagonal argument.

The set of real numbers is uncountable, and so is the set of all infinite sequences of natural numbers.

Minimal model of set theory is countable
If there is a set that is a standard model (see inner model) of ZFC set theory, then there is a minimal standard model (see Constructible universe). The Löwenheim–Skolem theorem can be used to show that this minimal model is countable. The fact that the notion of "uncountability" makes sense even in this model, and in particular that this model M contains elements that are: was seen as paradoxical in the early days of set theory; see Skolem's paradox for more.
 * subsets of M, hence countable,
 * but uncountable from the point of view of M,

The minimal standard model includes all the algebraic numbers and all effectively computable transcendental numbers, as well as many other kinds of numbers.

Total orders
Countable sets can be totally ordered in various ways, for example:
 * Well-orders (see also ordinal number):
 * The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...)
 * The integers in the order (0, 1, 2, 3, ...; −1, −2, −3, ...)
 * Other (not well orders):
 * The usual order of integers (..., −3, −2, −1, 0, 1, 2, 3, ...)
 * The usual order of rational numbers (Cannot be explicitly written as an ordered list!)

In both examples of well orders here, any subset has a least element; and in both examples of non-well orders, some subsets do not have a least element. This is the key definition that determines whether a total order is also a well order.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Countable set


text:
Cahn–Ingold–Prelog priority rules

In organic chemistry, the Cahn–Ingold–Prelog (CIP) sequence rules (also the CIP priority convention; named after Robert Sidney Cahn, Christopher Kelk Ingold, and Vladimir Prelog) are a standard process to completely and unequivocally name a stereoisomer of a molecule. The purpose of the CIP system is to assign an R or S descriptor to each stereocenter and an E or Z descriptor to each double bond so that the configuration of the entire molecule can be specified uniquely by including the descriptors in its systematic name. A molecule may contain any number of stereocenters and any number of double bonds, and each usually gives rise to two possible isomers. A molecule with an integer $n$ describing the number of stereocenters will usually have $2n$ stereoisomers, and $2n−1$ diastereomers each having an associated pair of enantiomers. The CIP sequence rules contribute to the precise naming of every stereoisomer of every organic molecule with all atoms of ligancy of fewer than 4 (but including ligancy of 6 as well, this term referring to the "number of neighboring atoms" bonded to a center).

The key article setting out the CIP sequence rules was published in 1966, and was followed by further refinements, before it was incorporated into the rules of the International Union of Pure and Applied Chemistry (IUPAC), the official body that defines organic nomenclature, in 1974. The rules have since been revised, most recently in 2013, as part of the IUPAC book Nomenclature of Organic Chemistry. The IUPAC presentation of the rules constitute the official, formal standard for their use, and it notes that "the method has been developed to cover all compounds with ligancy up to 4... and… [extended to the case of] ligancy 6… [as well as] for all configurations and conformations of such compounds." Nevertheless, though the IUPAC documentation presents a thorough introduction, it includes the caution that "it is essential to study the original papers, especially the 1966 paper, before using the sequence rule for other than fairly simple cases."

A recent paper argues for changes to some of the rules (sequence rules 1b and 2) to address certain molecules for which the correct descriptors were unclear. However, a different problem remains: in rare cases, two different stereoisomers of the same molecule can have the same CIP descriptors, so the CIP system may not be able to unambiguously name a stereoisomer, and other systems may be preferable.

Steps for naming
The steps for naming molecules using the CIP system are often presented as:
 * 1) Identification of stereocenters and double bonds;
 * 2) Assignment of priorities to the groups attached to each stereocenter or double-bonded atom; and
 * 3) Assignment of R/S and E/Z descriptors.

Assignment of priorities
R/S and E/Z descriptors are assigned by using a system for ranking priority of the groups attached to each stereocenter. This procedure, often known as the sequence rules, is the heart of the CIP system. The overview in this section omits some rules that are needed only in rare cases.
 * 1) Compare the atomic number (Z) of the atoms directly attached to the stereocenter; the group having the atom of higher atomic number Z receives higher priority (i.e. number 1).
 * 2) If there is a tie, the atoms at distance 2 from the stereocenter have to be considered: a list is made for each group of further atoms bonded to the one directly attached to the stereocenter. Each list is arranged in order of decreasing atomic number Z. Then the lists are compared atom by atom; at the earliest difference, the group containing the atom of higher atomic number Z receives higher priority.
 * 3) If there is still a tie, each atom in each of the two lists is replaced with a sublist of the other atoms bonded to it (at distance 3 from the stereocenter), the sublists are arranged in decreasing order of atomic number Z, and the entire structure is again compared atom by atom. This process is repeated recursively, each time with atoms one bond farther from the stereocenter, until the tie is broken.

Isotopes
If two groups differ only in isotopes, then the larger atomic mass is used to set the priority.

Double and triple bonds
If an atom, A, is double-bonded to another atom, then atom A should be treated as though it is "connected to the same atom twice". An atom that is double-bonded has a higher priority than an atom that is single bonded. When dealing with double bonded priority groups, one is allowed to visit the same atom twice as one creates an arc.

When B is replaced with a list of attached atoms, A itself, but not its "phantom", is excluded in accordance with the general principle of not doubling back along a bond that has just been followed. A triple bond is handled the same way except that A and B are each connected to two phantom atoms of the other.

Geometrical isomers
If two substituents on an atom are geometric isomers of each other, the Z-isomer has higher priority than the E-isomer. A stereoisomer that contains two higher priority groups on the same face of the double bond (cis) is classified as "Z." The stereoisomer with two higher priority groups on opposite sides of a carbon-carbon double bond (trans) is classified as "E."

Cyclic molecules
To handle a molecule containing one or more cycles, one must first expand it into a tree (called a hierarchical digraph) by traversing bonds in all possible paths starting at the stereocenter. When the traversal encounters an atom through which the current path has already passed, a phantom atom is generated in order to keep the tree finite. A single atom of the original molecule may appear in many places (some as phantoms, some not) in the tree.

Stereocenters: R/S

A chiral sp3 hybridized isomer contains four different substituents. All four substituents are assigned prorites based on its atomic numbers. After the substituents of a stereocenter have been assigned their priorities, the molecule is oriented in space so that the group with the lowest priority is pointed away from the observer. If the substituents are numbered from 1 (highest priority) to 4 (lowest priority), then the sense of rotation of a curve passing through 1, 2 and 3 distinguishes the stereoisomers. In a configurational isomer, the lowest priority group (most times hydrogen) is positioned behind the plane or the hatched bond going away from the reader. The highest priority group will have an arc drawn connecting to the rest of the groups, finishing at the group of third priority. An arc drawn clockwise, has the rectus (R) assignment. An arc drawn counterclockwise, has the sinister (S) assignment. The names are derived from the Latin for 'right' and 'left', respectively. When naming an organic isomer, the abbreviation for either rectus or sinister assignment is placed in front of the name in parentheses. For example, 3-methyl-1-pentene with a rectus assignment is formatted as (R)-3-methyl-1-pentene. A practical method of determining whether an enantiomer is R or S is by using the right-hand rule: one wraps the molecule with the fingers in the direction 1 → 2 → 3. If the thumb points in the direction of the fourth substituent, the enantiomer is R; otherwise, it is S.

It is possible in rare cases that two substituents on an atom differ only in their absolute configuration (R or S). If the relative priorities of these substituents need to be established, R takes priority over S. When this happens, the descriptor of the stereocenter is a lowercase letter (r or s) instead of the uppercase letter normally used.

Double bonds: E/Z
For double bonded molecules, Cahn–Ingold–Prelog priority rules (CIP rules) are followed to determine the priority of substituents of the double bond. If both of the high priority groups are on the same side of the double bond (cis configuration), then the stereoisomer is assigned the configuration Z (zusammen, German word meaning "together"). If the high priority groups are on opposite sides of the double bond ( trans configuration ), then the stereoisomer is assigned the configuration E (entgegen, German word meaning "opposed")

Coordination compounds
In some cases where stereogenic centers are formed, the configuration must be specified. Without the presence of a non-covalent interaction, a compound is achiral. Some professionals have proposed a new rule to account for this. This rule states that "non-covalent interactions have a fictitious number between 0 and 1" when assigning priority. Compounds in which this occurs are referred to as coordination compounds.

Spiro compounds
Spiro structures contain chiral molecules with no say asymmetric center. The rings of a spiro structure lie at right angles to each other. It's important to note that the mirror images of spiro structures are non-superimposable and are enantiomers.

Examples
The following are examples of application of the nomenclature.

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!colspan="2"|R/S assignments for several compounds
 * bgcolor="#FFFFFF"| R-bromochlorofluoroiodomethane-2D.png
 * bgcolor="#FFFFFF" valign=top| The hypothetical molecule bromochlorofluoroiodomethane shown in its (R)-configuration would be a very simple chiral compound. The priorities are assigned based on atomic number (Z): iodine (Z = 53) > bromine (Z = 35) > chlorine (Z = 17) > fluorine (Z = 9). Allowing fluorine (lowest priority, number 4) to point away from the viewer the rotation is clockwise hence the R assignment.
 * bgcolor="#FFFFFF"| L-serine-skeletal.png
 * bgcolor="#FFFFFF" valign="top" | In the assignment of L -serine highest priority (i.e. number 1) is given to the nitrogen atom (Z = 7) in the amino group (NH2). Both the hydroxymethyl group (CH2OH) and the carboxylic acid group (COOH) have carbon atoms (Z = 6) but priority is given to the latter because the carbon atom in the COOH group is connected to a second oxygen (Z = 8) whereas in the CH2OH group carbon is connected to a hydrogen atom (Z = 1). Lowest priority (i.e. number 4) is given to the hydrogen atom and as this atom points away from the viewer, the counterclockwise decrease in priority over the three remaining substituents completes the assignment as S.
 * bgcolor="#FFFFFF"| (S)-Carvone.svg
 * bgcolor="#FFFFFF" valign=top| The stereocenter in (S)-carvone is connected to one hydrogen atom (not shown, priority 4) and three carbon atoms. The isopropenyl group has priority 1 (carbon atoms only), and for the two remaining carbon atoms, priority is decided with the carbon atoms two bonds removed from the stereocenter, one part of the keto group (O, O, C, priority number 2) and one part of an alkene (C, C, H, priority number 3). The resulting counterclockwise rotation results in S.
 * }
 * bgcolor="#FFFFFF"| (S)-Carvone.svg
 * bgcolor="#FFFFFF" valign=top| The stereocenter in (S)-carvone is connected to one hydrogen atom (not shown, priority 4) and three carbon atoms. The isopropenyl group has priority 1 (carbon atoms only), and for the two remaining carbon atoms, priority is decided with the carbon atoms two bonds removed from the stereocenter, one part of the keto group (O, O, C, priority number 2) and one part of an alkene (C, C, H, priority number 3). The resulting counterclockwise rotation results in S.
 * }

Describing multiple centers
If a compound has more than one chiral stereocenter, each center is denoted by either R or S. For example, ephedrine exists in (1R,2S) and (1S,2R) stereoisomers, which are distinct mirror-image forms of each other, making them enantiomers. This compound also exists as the two enantiomers written (1R,2R) and (1S,2S), which are named pseudoephedrine rather than ephedrine. All four of these isomers are named 2-methylamino-1-phenyl-1-propanol in systematic nomenclature. However, ephedrine and pseudoephedrine are diastereomers, or stereoisomers that are not enantiomers because they are not related as mirror-image copies. Pseudoephedrine and ephedrine are given different names because, as diastereomers, they have different chemical properties, even for racemic mixtures of each.

More generally, for any pair of enantiomers, all of the descriptors are opposite: (R,R) and (S,S) are enantiomers, as are (R,S) and (S,R). Diastereomers have at least one descriptor in common; for example (R,S) and (R,R) are diastereomers, as are (S,R) and (S,S). This holds true also for compounds having more than two stereocenters: if two stereoisomers have at least one descriptor in common, they are diastereomers. If all the descriptors are opposite, they are enantiomers.

A meso compound is an achiral molecule, despite having two or more stereogenic centers. A meso compound is "superimposable" on its mirror image, therefore it reduces the number of stereoisomers predicted by the 2n rule. This occurs because the molecule obtains a plane of symmetry that causes the molecule to rotate around the central carbon–carbon bond. One example is meso-tartaric acid, in which (R,S) is the same as the (S,R) form. In meso compounds the R and S stereocenters occur in symmetrically positioned pairs.

Relative configuration
The relative configuration of two stereoisomers may be denoted by the descriptors R and S with an asterisk (*). (R*,R*) means two centers having identical configurations, (R,R) or (S,S); (R*,S*) means two centers having opposite configurations, (R,S) or (S,R). To begin, the lowest-numbered (according to IUPAC systematic numbering) stereogenic center is given the R* descriptor.

To designate two anomers the relative stereodescriptors alpha (α) and beta (β) are used. In the α anomer the anomeric carbon atom and the reference atom do have opposite configurations (R,S) or (S,R), whereas in the β anomer they are the same (R,R) or (S,S).

Faces
Stereochemistry also plays a role assigning faces to trigonal molecules such as ketones. A nucleophile in a nucleophilic addition can approach the carbonyl group from two opposite sides or faces. When an achiral nucleophile attacks acetone, both faces are identical and there is only one reaction product. When the nucleophile attacks butanone, the faces are not identical (enantiotopic) and a racemic product results. When the nucleophile is a chiral molecule diastereoisomers are formed. When one face of a molecule is shielded by substituents or geometric constraints compared to the other face the faces are called diastereotopic. The same rules that determine the stereochemistry of a stereocenter (R or S) also apply when assigning the face of a molecular group. The faces are then called the Re-face and Si-face. In the example displayed on the right, the compound acetophenone is viewed from the Re-face. Hydride addition as in a reduction process from this side will form the (S)-enantiomer and attack from the opposite Si-face will give the (R)-enantiomer. However, one should note that adding a chemical group to the prochiral center from the Re-face will not always lead to an (S)-stereocenter, as the priority of the chemical group has to be taken into account. That is, the absolute stereochemistry of the product is determined on its own and not by considering which face it was attacked from. In the above-mentioned example, if chloride (Z = 17) were added to the prochiral center from the Re-face, this would result in an (R)-enantiomer.

title: Cahn–Ingold–Prelog priority rules