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)
“Your home is regarded as a model home, your life as a model life. But all this splendor, and you along with it ... it’s just as though it were built upon a shifting quagmire. A moment may come, a word can be spoken, and both you and all this splendor will collapse.”
—Henrik Ibsen (1828–1906)
“I set out on this ground, which I suppose to be self evident, “that the earth belongs in usufruct to the living”: that the dead have neither powers nor rights over it.”
—Thomas Jefferson (1743–1826)
“The theory of rights enables us to rise and overthrow obstacles, but not to found a strong and lasting accord between all the elements which compose the nation.”
—Giuseppe Mazzini (1805–1872)