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)
“When Titian was mixing brown madder,
His model was posed up a ladder.
Said Titian, “That position
Calls for coition,”
So he lept up the ladder and had her.”
—Anonymous.
“Women are to be lifted up to a physical equality with man by placing upon their shoulders equal burdens of labor, equal responsibilities of state-craft; they are to be brought down from their altruistic heights by being released from all obligations of purity, loyalty, self-sacrifice, and made free of the world of passion and self-indulgence, after the model set them by men of low and materialistic ideals.”
—Caroline Fairfield Corbin (b. c. 1835–?)
“By the “mud-sill” theory it is assumed that labor and education are incompatible; and any practical combination of them impossible. According to that theory, a blind horse upon a tread-mill, is a perfect illustration of what a laborer should be—all the better for being blind, that he could not tread out of place, or kick understandingly.... Free labor insists on universal education.”
—Abraham Lincoln (1809–1865)