axiom of limitation of size造句

• It is implied by the stronger axiom of limitation of size.
• The axiom of limitation of size implies von Neumann's 1923 axiom.
• In 1930, Ernst Zermelo defined models of set theory satisfying the axiom of limitation of size.
• The only advantage of the axiom of limitation of size is that it implies the axiom of global choice.
• The axioms of replacement, global choice, and union ( with the other axioms of NBG ) imply the axiom of limitation of size.
• The axioms of NBG with the axiom of replacement replaced by the weaker axiom of separation do not imply the axiom of limitation of size.
• The axioms of NBG with the axiom of global choice replaced by the usual axiom of choice do not imply the axiom of limitation of size.
• Hence, the axiom of limitation of size implies that there is a function that maps the class of all ordinals onto the class of all sets.
• Abraham Fraenkel and Azriel Levy have stated that the axiom of limitation of size does not capture all of the " limitation of size doctrine " because it does not imply the power set axiom.
• These set theories only substituted the axiom of replacement and a form of the axiom of choice for the axiom of limitation of size because von Neumann's axiom system contains the axiom of union.
• He also adopted the axiom of regularity, and replaced the axiom of limitation of size with the axioms of replacement and von Neumann's choice axiom . ( Von Neumann's work shows that the last two changes allow Bernays'axioms to prove the axiom of limitation of size .)
• He also adopted the axiom of regularity, and replaced the axiom of limitation of size with the axioms of replacement and von Neumann's choice axiom . ( Von Neumann's work shows that the last two changes allow Bernays'axioms to prove the axiom of limitation of size .)
• Therefore, the axiom of limitation of size fails in this model . " Ord " is an example of a proper class that cannot be mapped onto " V " because ( as proved above ) if there is a function mapping " Ord " onto " V ", then " V " can be well-ordered.
• It was proved above that the axiom of limitation of size implies that there is a function F that maps Ord onto V . Also, G was defined as a subclass of F that is a one-to-one correspondence between Dom ( G ) and V . It defines a well-ordering on V \ colon \, x if G ^ {-1 } ( x ) Therefore, G is an order isomorphism from ( Dom ( G ), to ( V,
• Von Neumann identified these sets using the criterion : " A set is'too big'if and only if it is equivalent with the set of all things . " He then restricted how these sets may be used : " & in order to avoid the paradoxes those [ sets ] which are'too big'are declared to be impermissible as " elements " . " By combining this restriction with his criterion, von Neumann obtained his first version of the axiom of limitation of size, which in the language of classes states : A class is a proper class if and only if it is equinumerous with " V ".