Veblen function - The Veblen hierarchy

In the special case when φ 0 (α)=ω α this family of functions is known as the Veblen hierarchy. The function φ 1 is the same as the ε function: φ 1 (α)= ε α. If then From this and the fact that φ β is strictly increasing we get the ordering: if and only if either ( and ) or ( and ) or ( and ).

Veblen function - Fundamental sequences for the Veblen hierarchy

The fundamental sequence for an ordinal with cofinality ω is a distinguished strictly increasing ω-sequence which has the ordinal as its limit. If one has fundamental sequences for α and all smaller limit ordinals, then one can create an explicit constructive bijection between ω and α, (i.e. one not using the axiom of choice). Here we will describe fundamental sequences for the Veblen hierarchy of ordinals. The image of n under the fundamental sequence for α will be indicated by α[n].

Veblen function - Fundamental sequences for the Veblen hierarchy

A variation of Cantor normal form used in connection with the Veblen hierarchy is — every nonzero ordinal number α can be uniquely written as, where k>0 is a natural number and each term after the first is less than or equal to the previous term, and each If a fundamental sequence can be provided for the last term, then that term can be replaced by such a sequence to get

Thorstein Veblen - Veblen and political theories

Politically, Veblen was sympathetic to state ownership. Scholars mostly disagree about the extent to which Veblen's views are compatible with Marxism, socialism, or anarchism.

Ordinal notation - Veblen

The 2-variable Veblen functions can be used to give a system of ordinal notation for ordinals less than the Feferman-Schutte ordinal. The Veblen functions in a finite or transfinite number of variables give systems of ordinal notations for ordinals less than the small and large Veblen ordinals.


Veblen may refer to:

Thorstein Veblen - Leisure class

In The Theory of the Leisure Class, Veblen writes critically of conspicuous consumption and its function in social-class consumerism and social stratification. Reflecting historically, he traces said economic behaviors back to the beginnings of the division of labor, or during tribal times. Upon the start of a division of labor, high-status individuals within the community practiced hunting and war, notably less labor-intensive and less economically productive work. Low-status individuals, on the other hand, practiced activities recognized as more economically productive and more labor-intensive, such as farming and cooking. High-status individuals, as Veblen explains, could instead afford to live their lives leisurely (hence their title as the leisure class), engaging in symbolic economic participation, rather than practical economic participation. These individuals could engage in conspicuous leisure for extended periods of time, simply following pursuits that evoked a higher social status. Rather than participating in conspicuous consumption, the leisure class lived lives of conspicuous leisure as a marker of high status. The leisure class protected and reproduced their social status and control within the tribe through, for example, their participation in war-time activities, which while they were rarely needed, still rendered their lower social class counterparts dependent upon them. During modern industrial times, Veblen described the leisure class as those exempt from industrial labor. Instead, he explains, the leisure class participated in intellectual or artistic endeavors to display their freedom from the economic need to participate in economically productive manual labor. In essence, not having to perform labor-intensive activities did not mark higher social status, but rather, higher social status meant that one would not have to perform such duties.

Thorstein Veblen - Institutional economics

Thorstein Veblen laid the foundation for the perspective of institutional economics with his criticism of traditional static economic theory. As much as Veblen was an economist, he was also a sociologist who rejected his contemporaries who looked at the economy as an autonomous, stable, and static entity. Veblen disagreed with his peers, as he strongly believed that the economy was significantly embedded in social institutions. Rather than separating economics from the social sciences, Veblen viewed the relationships between the economy and social and cultural phenomena. Generally speaking, the study of institutional economics viewed economic institutions as the broader process of cultural development. While economic institutionalism never transformed into a major school of economic thought, it allowed economists to explore economic problems from a perspective that incorporated social and cultural phenomena. It also allowed economists to view the economy as an evolving entity of bounded rationale.

Thorstein Veblen - Marriages

During his time at Carleton, Veblen met his first wife, Ellen Rolfe, the niece of the college president. They married in 1888. While some scholars have attributed his womanizing tendencies to the couple's numerous separations and eventual divorce in 1911, others have speculated that the relationship's demise was rooted in Ellen's inability to bear children. Following her death in 1926, it was revealed that she had asked for her autopsy to be sent to Veblen, her ex-husband. The autopsy showed that Ellen's reproductive organs had not developed normally, and she had been unable to bear children. A book written by Veblen's stepdaughter asserted that "this explained her disinterest in a normal wifely relationship with Thorstein" and that he "treated her more like a sister, a loving sister, than a wife".

Thorstein Veblen - Trained incapacity

In sociology, trained incapacity is "that state of affairs in which one's abilities function as inadequacies or blind spots." It means that people's past experiences can lead to wrong decisions when circumstances change. Veblen coined the concept in 1933.

Oswald Veblen - Career

Veblen taught mathematics at Princeton University from 1905 to 1932. In 1926, he was named Henry B. Fine Professor of Mathematics. In 1932, he helped organize the Institute for Advanced Study in Princeton, resigning his professorship to become the first professor at the Institute that same year. He kept his professorship at the Institute until he was made emeritus in 1950.

Thorstein Veblen - Education

After Veblen graduated from Carleton in 1880 he traveled east to study philosophy at Johns Hopkins University. While at Johns Hopkins he studied under Charles Sanders Peirce. When he failed to obtain a scholarship there he moved on to Yale University, where he found economic support for his studies, obtaining a Doctor of Philosophy in 1884, with a major in philosophy and a minor in social studies. His dissertation was titled "Ethical Grounds of a Doctrine of Retribution". At Yale, he studied under renowned academics such as philosopher Noah Porter and sociologist William Graham Sumner.

Veblen function - Transfinitely many variables

The smallest ordinal α such that α is greater than φ applied to any function with support in α (i.e., which cannot be reached “from below” using the Veblen function of transfinitely many variables) is sometimes known as the “large” Veblen ordinal.

Veblen function

In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in. If φ 0 is any normal function, then for any non-zero ordinal α, φ α is the function enumerating the common fixed points of φ β for β<α. These functions are all normal.

Thorstein Veblen - Theory of business enterprise

The central problem for Veblen was the friction between "business" and "industry". Veblen identified "business" as the owners and leaders whose primary goal was the profits of their companies but, in an effort to keep profits high, often made efforts to limit production. By obstructing the operation of the industrial system in that way, "business" negatively affected society as a whole (through higher rates of unemployment, for example). With that said, Veblen identified business leaders as the source of many problems in society, which he felt should be led by people such as engineers, who understood the industrial system and its operation, while also having an interest in the general welfare of society at large.

Thorstein Veblen - Academic career

After graduation from Yale in 1884, Veblen was essentially unemployed for seven years. Despite having strong letters of recommendation, he was unable to obtain a university position. It is possible that his dissertation research on "Ethical Grounds of a Doctrine of Retribution" (1884) was considered undesirable. However this possibility can no longer be researched because Veblen's dissertation has been missing from Yale since 1935. Apparently the only scholar who ever studied the dissertation was Joseph Dorfman, for his 1934 book Thorstein Veblen and His America. Dorfman says only that the dissertation, advised by evolutionary sociologist William Graham Sumner, studies such evolutionary thought as that of Herbert Spencer, as well as the moral philosophy of Kant. Some historians have also speculated that this failure to obtain employment was partially due to prejudice against Norwegians, while others attribute this to the fact that most universities and administrators considered him insufficiently educated in Christianity. Most academics at the time held divinity degrees, which Veblen did not have. Also, it did not help that Veblen openly identified as an agnostic, which was highly uncommon for the time. As a result, Veblen returned to his family farm, a stay during which he had claimed to be recovering from malaria. He spent those years recovering and reading voraciously. It is suspected that these difficulties in beginning his academic career later inspired portions of his book The Higher Learning in America (1918), in which he claimed that true academic values were sacrificed by universities in favor of their own self-interest and profitability.

Trickle-down effect - Veblen

When applied to fashion, this theory states that when the lowest social class, or simply a perceived lower social class, adopts the fashion, it is no longer desirable to the leaders in the highest social class. The theory has been associated with later trickle-down theories as importantly, Veblen also observed that the upper classes found more extravagant ways of exercising conspicuous consumption in order to differentiate themselves from the class imitating their original consumption behaviour.

Thorstein Veblen - Death

After his wife Ann's premature death in 1920, Veblen became active in the care of his stepdaughters. Becky went with him when he moved to California, looked after him there, and was with him at his death in August 1929, just a few months shy of the Great Depression, the economic crisis he had anticipated in Absentee Ownership and Business Enterprise in Recent Times. Prior to his death, Veblen had earned a comparatively high salary from the New School. Since he lived frugally, Veblen invested his money in California raisin vineyards and the stock market. Unfortunately, after returning to northern California, Veblen lost the money he had invested and was living in a town shack while earning $500 to $600 a year from royalties and was sent $500 a year from a former Chicago student.

Thorstein Veblen - Legacy

Veblen is regarded as one of the co-founders of the American school of institutional economics, alongside John R. Commons and Wesley Clair Mitchell. Economists who adhere to this school organize themselves in the Association for Institutional Economics (AFIT). The Association for Evolutionary Economics (AFEE) gives an annual Veblen-Commons award for work in Institutional Economics and publishes the Journal of Economic Issues. Some unaligned practitioners include theorists of the concept of "differential accumulation".

Oswald Veblen - Early life

Oswald went to school in Iowa City. He did his undergraduate studies at the University of Iowa, where he received an A.B. in 1898, and Harvard University, where he was awarded a second B.A. in 1900. For his graduate studies, he went to study mathematics at the University of Chicago, where he obtained a Ph.D. in 1903. His dissertation, A System of Axioms for Geometry was written under the supervision of E. H. Moore. During World War I, Veblen served first as a captain, later as a major in the United States Army.