Now post some new interesting shit, feels like we've been having the same fucking discussion for ages...
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Anonymous2013-08-31 14:22
If my rather charismatic Bard rolls 1d20+8 for a gather information check, it takes 1d4+1 HOURS to complete. So does that mean I can't do anything else for christ knows how many turns?! Surely by that time the party would have moved on!
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Anonymous2013-08-31 14:29
Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation. This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.
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Anonymous2013-08-31 15:07
There is now a 1.54 patch. Also there is a patch to update from 1.50 to 1.54 (and from 1.52 to 1.54 if you have the DL version).
This update adds a third version of the "Secret Basement", the dungeon that generates random items in your inventory each floor.
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Anonymous2013-08-31 15:14
These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo-Fraenkel set theory.
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Anonymous2013-08-31 16:00
Another equivalent axiom only considers collections X that are essentially powersets of other sets:
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Anonymous2013-08-31 16:45
The axiom of constructibility and the generalized continuum hypothesis both imply the axiom of choice, but are strictly stronger than it.
Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion. That is, every consistent set of first-order sentences can be extended to a maximal consistent set.
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Anonymous2013-08-31 18:15
Associativity: If f : a → b, g : b → c and h : c → d then h ∘ (g ∘ f) = (h ∘ g) ∘ f, and
It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described. This can be visualized using Cantor's diagonal argument; classic questions of cardinality (for instance the continuum hypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals. In more recent times mathematicians have been describing the properties of larger and larger cardinals.
The Indian mathematical text Surya Prajnapti (c. 3rd–4th century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
The IEEE floating-point standard (IEEE 754) specifies the positive and negative infinity values. These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
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Anonymous2013-08-31 22:11
For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.
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Anonymous2013-08-31 22:40
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Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard form of axiomatic set theory.
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Anonymous2013-08-31 23:41
The status of the axiom of choice varies between different varieties of constructive mathematics.
A category is itself a type of mathematical structure, so we can look for "processes" which preserve this structure in some sense; such a process is called a functor.
This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of ω-category corresponding to the ordinal number ω.
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Anonymous2013-09-01 2:42
associates to each object X \in C an object F(X) \in D,
fop(a *op b) = f(b * a) = f(b) * f(a) = fop(a) *op fop(b).
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Anonymous2013-09-01 10:30
Formally, assuming the axiom of choice, the cardinality of a set X is the least ordinal α such that there is a bijection between X and α. This definition is known as the von Neumann cardinal assignment. If the axiom of choice is not assumed we need to do something different. The oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the class [X] of all sets that are equinumerous with X. This does not work in ZFC or other related systems of axiomatic set theory because if X is non-empty, this collection is too large to be a set. In fact, for X ≠ ∅ there is an injection from the universe into [X] by mapping a set m to {m} × X and so by limitation of size, [X] is a proper class. The definition does work however in type theory and in New Foundations and related systems. However, if we restrict from this class to those equinumerous with X that have the least rank, then it will work (this is a trick due to Dana Scott: it works because the collection of objects with any given rank is a set).
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Anonymous2013-09-01 11:16
In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asaṃkhyāta ("countless, innumerable") and ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.