Minimal Model of Set Theory Is Countable
If there is a set which 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 which are
- subsets of M, hence countable,
- but uncountable from the point of view of M,
was seen as paradoxical in the early days of set theory, see Skolem's paradox.
The minimal standard model includes all the algebraic numbers and all effectively computable transcendental numbers, as well as many other kinds of numbers.
Read more about this topic: Countable Set
Famous quotes containing the words minimal, model, set and/or theory:
“For those parents from lower-class and minority communities ... [who] have had minimal experience in negotiating dominant, external institutions or have had negative and hostile contact with social service agencies, their initial approaches to the school are often overwhelming and difficult. Not only does the school feel like an alien environment with incomprehensible norms and structures, but the families often do not feel entitled to make demands or force disagreements.”
—Sara Lawrence Lightfoot (20th century)
“...that absolutely everything beloved and cherished of the bourgeoisie, the conservative, the cowardly, and the impotent—the State, family life, secular art and science—was consciously or unconsciously hostile to the religious idea, to the Church, whose innate tendency and permanent aim was the dissolution of all existing worldly orders, and the reconstitution of society after the model of the ideal, the communistic City of God.”
—Thomas Mann (1875–1955)
“The cunningest dissimulation is when a man pretends to be caught in the traps others set for him; and a man is never so easily over-reached as when he is contriving to over-reach others.”
—François, Duc De La Rochefoucauld (1613–1680)
“We commonly say that the rich man can speak the truth, can afford honesty, can afford independence of opinion and action;—and that is the theory of nobility. But it is the rich man in a true sense, that is to say, not the man of large income and large expenditure, but solely the man whose outlay is less than his income and is steadily kept so.”
—Ralph Waldo Emerson (1803–1882)