Classes in Formal Set Theories
ZF set theory does not formalize the notion of classes. They can instead be described in the metalanguage, as equivalence classes of logical formulas. For example, if is a structure interpreting ZF, then the metalanguage expression is interpreted in by the collection of all the elements from the domain of ; that is, all the sets in . So we can identify the "class of all sets" with the predicate x=x or any equivalent predicate.
Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an inaccessible cardinal κ is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe), and its subsets can be thought of as "classes".
Another approach is taken by the von Neumann–Bernays–Gödel axioms (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the set existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be a conservative extension of ZF.
Morse–Kelley set theory admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its set existence axioms. This causes MK to be strictly stronger than both NBG and ZF.
In other set theories, such as New Foundations or the theory of semisets, the concept of "proper class" still makes sense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theory with a universal set has proper classes which are subclasses of sets.
Read more about this topic: Class (set Theory)
Famous quotes containing the words classes in, classes, formal, set and/or theories:
“Between richer and poorer classes in a free country a mutually respecting antagonism is much healthier than pity on the one hand and dependence on the other, as is, perhaps, the next best thing to fraternal feeling.”
—Charles Horton Cooley (1864–1929)
“What’s the greatest enemy of Christianity to-day? Frozen meat. In the past only members of the upper classes were thoroughly sceptical, despairing, negative. Why? Among other reasons, because they were the only people who could afford to eat too much meat. Now there’s cheap Canterbury lamb and Argentine chilled beef. Even the poor can afford to poison themselves into complete scepticism and despair.”
—Aldous Huxley (1894–1963)
“There must be a profound recognition that parents are the first teachers and that education begins before formal schooling and is deeply rooted in the values, traditions, and norms of family and culture.”
—Sara Lawrence Lightfoot (20th century)
“Well, most men have bound their eyes with one or another handkerchief, and attached themselves to some of these communities of opinion. This conformity makes them not false in a few particulars, authors of a few lies, but false in all particulars. Their every truth is not quite true. Their two is not the real two, their four not the real four; so that every word they say chagrins us and we know not where to set them right.”
—Ralph Waldo Emerson (1803–1882)
“The egoism which enters into our theories does not affect their sincerity; rather, the more our egoism is satisfied, the more robust is our belief.”
—George Eliot [Mary Ann (or Marian)