The Echoes of the Mind (46) Sets

Sets

A set is a Many that allows itself to be thought of as a One. ~ Georg Cantor

Ultimately mathematics is about sets. A set is an enumerated or rule-specified collection of entities. A portion of a set is a subset.

Though sets may be of any entity, many sets have numbers as their members. Mathematical operations are the manipulation of sets, typically with 1 or more members (elements). This characterization is conventional but has been controversial.

German mathematician Georg Cantor developed set theory 1874–1884. In his work, Cantor found within sets an “infinity of infinities.”

Integers are a subset of real numbers, even as both sets have an infinite number of members. Cantor understood that there are more real numbers than integers, and so realized infinity can be relative.

The essence of mathematics is its freedom. ~ Georg Cantor

Some of Cantor’s findings relating to infinity were so counterintuitive as to shock contemporaries. Cantor posited transfinite numbers: numbers larger than finite numbers, but shy of being absolutely infinite.

German mathematician Leopold Kronecker, a contemporary of Cantor who worked in number theory and algebra, was a stern critic of Cantor, calling him a “scientific charlatan” and a “corrupter of youth.” (Socrates was similarly accused of corrupting the young with foul ideas.)

God made the integers, all else is the work of man. ~ Leopold Kronecker

Some Christian theologians considered Cantor’s work blasphemous: challenging the absolute infinity of God. Cantor, a deeply religious man, rejected the charge.

In the late 19th century, French polymath Henri Poincaré called set theory a “grave disease” infecting mathematics. Criticism was mixed with accolades. Cantor received the highest honor from The Royal Society of London in 1904 for his work on set theory. Yet this did not stop the carping. Austrian English philosopher Ludwig Wittgenstein lamented that mathematics had been “ridden through and through with the pernicious idioms of set theory,” which were “utter nonsense.”

The internal contradictions in set theory prompted Dutch mathematician and philosopher L.E.J. Brouwer to tautologically remark:

A false theory which is not stopped by a contradiction is nonetheless false.

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Sets are notated within braces. The set of seasons is stated thusly:

Seasons = {winter, spring, summer, fall}

A set may be empty (Æ or {}). Note that Æ is not the same as {Æ}. Whereas Æ (or {}) is an empty set, like an empty bag, {Æ} is a bag with an empty bag.

Members of a set are designated by Î, while nonmembers are Ï.

1 Î {1, 2, 3}

4 Ï {1, 2, 3}

Sets can be compared, or otherwise operated on in various ways. Sets may, for example, be equal (the same) or share some elements. A few simple examples illustrate, using sets A & B below.

A = {1, 2, 3}

B = {2, 3, 4}

A union (∪) of 2 sets is a set that has all elements.

A ∪ B = {1, 2, 3, 4}

The intersection (∩) of 2 sets is a set that has shared elements.

A ∩ B = {2}

To simplify conceptualization, sets are often visualized using Venn diagrams, introduced by English logician and philosopher John Venn in 1880. (If the sets A and B had no shared members, their representative Venn diagram circles would not overlap.)

A set difference (\) comprises those elements of one set not in another.

A \ B = {1}

B \ A = {4}

The most widely used sets, which are of numbers, are all infinite, even as integers and rational numbers are eminently countable. That there are sets with an infinite number of elements ({∞}) cannot be proven from first principles, and so is accepted as axiomatically true.

The foregoing is just a warm-up to the wondrous world of sets, whose scope is so vast as to envelop both mathematics and logic.

Set theory has its paradoxes. Not surprisingly, almost all revolve around the ever-troublesome notion of infinity (∞). A few other paradoxes arise in the unaccountability of relations between sets.

Most astonishing is that world of rigorous fantasy we call mathematics. ~ English anthropologist Gregory Bateson