\int_{-\infty}^{\infty} \, f(t)\ dt \ = \infty means that the area under f(t) is infinite.
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Anonymous2013-08-31 9:00
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.
In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, a number between 0 and 1.
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Anonymous2013-08-31 10:30
Each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a same set can occur more than once as a factor; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family
Together these results establish that the axiom of choice is logically independent of ZF. The assumption that ZF is consistent is harmless because adding another axiom to an already inconsistent system cannot make the situation worse.
The classic example used is that of the infinite hotel paradox, also called Hilbert's paradox of the Grand Hotel. Suppose you are an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives. It's possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:
1 ↔ 2
2 ↔ 3
3 ↔ 4
...
n ↔ n + 1
...
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Anonymous2013-08-31 13:28
However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea (c. 490 BCE? – c. 430 BCE?), a pre-Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as "immeasurably subtle and profound".
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Anonymous2013-08-31 13:29
I was in Japan, but it didn't really seem like Japan because there were too many white people. The people I interacted with in the dream were all caucasian at least. The Asians were just the background people, like extras in a movie. Anyways I was walking through some sort of mall. The stores were interconnected so you walk through a door and you're in the next store. I seemed to have a set destination in mind, since I wasn't browsing any of the stores and was weaving through them all. I accidentally bump into a Japanese lady and mutter something and rush off, realizing I'm in Japan and don't actually know how to apologize. So, naturally I ask twitter.
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Anonymous2013-08-31 14:14
As to date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
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Anonymous2013-08-31 14:14
Dear John,
I have come to visit you for the first time in quite a while, and I must say that your board is dreadfully boring.
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Anonymous2013-08-31 14:59
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various (axiomatic) properties.
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Anonymous2013-08-31 15:00
Vodka makes me black out. I may even be blacked out whenever I read and write 4chan posts. Am I awake? Obviously Vlad the Impaler does not want it Putin the bum of a young boy. Hence the laws against homosexuality.
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Anonymous2013-08-31 15:45
The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
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Anonymous2013-08-31 16:30
The restriction to ZF renders any claim that relies on either the axiom of choice or its negation unprovable. For example, the Banach–Tarski paradox is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition.
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Anonymous2013-08-31 17:15
Every infinite game G_S in which S is a Borel subset of Baire space is determined.
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Anonymous2013-08-31 18:00
A group homomorphism between two groups "preserves the group structure" in a precise sense – it is a "process" taking one group to another, in a way that carries along information about the structure of the first group into the second group. The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms.
Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of higher-dimensional categories. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes".
Proposed in 1931, the Zermelo's navigation problem is a classic optimal control problem. The problems deals with a boat navigating on a body of water, originating from a point O to a destination point D. The boat is capable of a certain maximum speed, and we want to derive the best possible control to reach D in the least possible time.