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#bbcode on Rizon

Name: Anonymous 2007-07-23 3:05 ID:IZLI1rXe

Where did it go?

Name: Anonymous 2007-07-23 5:04 ID:t7wreF9H

Beats me

Name: Anonymous 2007-07-23 10:46 ID:06n3YxnA

[disappear]#bbcode[/disappear]

Name: Anonymous 2007-07-23 13:01 ID:Heaven

Name: Anonymous 2009-02-25 6:22

Invariants such as Balanced   binary search trees   In addition to   featuring these predicates   upon types Sage   merges the syntactic   categories of types   and terms in   the spirit of   Pure Type Systems   to express dependent.

Name: Anonymous 2010-12-08 19:50

Name: Anonymous 2013-08-31 22:46


This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.

Name: Anonymous 2013-08-31 23:32


Note that 2|X| is the cardinality of the power set of the set X and Cantor's diagonal argument shows that 2|X| > |X| for any set X. This proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2[sup]κ[/sup). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.)

Name: Anonymous 2013-09-01 0:17


It is for example presumed impossible for any body to have infinite mass or infinite energy. Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.

Name: Anonymous 2013-09-01 1:02


For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α, the set Vα is defined to consist of all pure sets with rank less than α. The entire von Neumann universe is denoted V.

Name: Anonymous 2013-09-01 1:48


The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.

Name: Anonymous 2013-09-01 2:33


Additionally, consider for instance the unit circle S, and the action on S by a group G consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of π. Here G is countable while S is uncountable.

Name: Anonymous 2013-09-01 3:18


There are several results in category theory which invoke the axiom of choice for their proof. These results might be weaker than, equivalent to, or stronger than the axiom of choice, depending on the strength of the technical foundations.

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