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From Wikipedia, the free encyclopedia

In mathematics, the Banach game is a topological game introduced by Stefan Banach in 1935 in the second addendum to problem 43 of the Scottish book as a variation of the Banach–Mazur game.[1]

Given a subset of real numbers, two players alternatively write down arbitrary (not necessarily in ) positive real numbers such that Player one wins if and only if exists and is in .[2]

One observation about the game is that if is a countable set, then either of the players can cause the final sum to avoid the set.[3] Thus in this situation the second player can win.

YouTube Encyclopedic

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  • Why Do We Play Games?


Hey, Vsauce, Michael here. Why do humans play games? Whether it's a video game, or a board game, or a physical game, like soccer, or football, I don't have to put the ball in the net to survive, and, even if I did, why would I invite a goalie and another team? Games are weird- this lead Bernard Suits to say in the 70's that a game is a voluntary attempt to overcome unnecessary obstacles. So, why do we play sports and games? How should we feel about intellectualism vs. athleticism? But, most importantly, why do Americans called this game soccer while the English, and the rest of the world, call it Football? It turns out that the word "soccer" doesn't come from the United States of America. Instead, the blame for the word goes to the British themselves, specifically Oxford, where, since 1875, it has been popular to add the suffix "-er" to the end of words. For example, calling Radcliffe Camera the "Radder," or these "fivers" and "ten-ers." We have been playing games with balls and our feet ever since ancient times, all over the world. In fact, as recently as 200 years ago, many of these games called themselves "football." A lack of standardization meant that it was difficult to all come together and agree on what you could and couldn't do with the ball or your feet. But, luckily, in 1863 the Football Association was founded in London, England. Association football is what we most commonly mean nowadays when talking about football, or soccer. But what's the connection if the Oxford "-er" was added to Association football, shouldn't we be calling the game "Association-er?" Well, let's take a closer look at the word "association." You see what's hiding in there? There she is- "soccer." But, we're getting ahead of ourselves because soccer, football, is just one type of game. Ultimately, what is a game? Well, one of my favorite ways of defining "game" comes from computer game designer Chris Crawford. Let's begin with a book- this is a great book, it's really fun, it's entertaining, but it's not a game. TV shows and movies are also not games because, fundamentally, they aren't interactive. But, as soon as something is both fun and interactive, well, now we've got ourselves a play-thing. There are two types of play-things, according to Crawford: If you can play with the object and it's fun but there's no goal or objective associated with it, it's a toy. If, however, there is an objective, something you're supposed to accomplish, well, now you're talking about a challenge. But there are two different types of challenges. If the challenge involves no other people or other agents, it's just you, for instance, playing alone with a Rubik's Cube, you've got yourself a puzzle. If, however, there are other people involved, well, now we've got ourselves a conflict. In a conflict, like a foot race, you aren't allowed to interfere with the other participants. This is what Crawford calls a "competition." If, however, you are allowed to interact with, and interfere with the other players, and they can do the same to you, well, in that case, we are talking about a full-fledged game. So, a game is interactive, goal-oriented, and involves other agents, for instance, other people who can interfere with and influence each other. Which means, technically speaking, that life is a game. I mean, real life. My life, your life, easily fit many definitions of "game." And, in life, there are games that we tend to call "sports." Now, competency at sports can divide humanity into two groups: Jocks, who are good at sports, and nerds who aren't. Jock's are literally named after the Jock Strap, which keeps your genitals supported while being athletic. But Jocks are cool, right? They're fit, attractive, they get invited to all the cool parties where, in high school, they can do dangerous, cool things like get drunk. And who are you if you don't get drunk? Well, what's the word "drunk" backwards? You are a "Knurd." But that's not the origin of the word "nerd." There's much debate about where the word comes from, but, what we do know is that it emerged as a slang term for "lame" or "square" in Detroit in the early 1950's. The first known use of the word in print came from Dr. Seuss himself, and, fundamentally, it may come from the word "nut," which meant a crazy person, and was later altered to "nert," and, finally, "nerd." It exists today as a word for un-athletic people largely because it was popularized through its frequent use on the US TV show "Happy Days." Whether your spend your time on athletic pursuits or intellectual pursuits, or both, games, in some form, are a part of your life. So, why do we play games? Humans, and many other animals, play. And, perhaps, "play" originated as a way of physically preparing our bodies for life's real challenges later on. That idea makes sense, but the evidence isn't convincing because, in the wild, "play" can often lead to a wasting of precious resources, injuries, and hardly comes close to simulating real attacks and life-threatening situations. The New York Times wrote a great article about this conundrum, pointing out that physically preparing the body may be less of a priority for "play." Instead, the point of "play" might be preparing the brain. Play is good for the brain, especially during formative, juvenile years when most of us have an instinctive urge to goof around, play, and pretend anyway. Young Rats confined to cages with adult rats who refused to play with them grew up with smaller, less developed brains. This has led to the hypothesis that games play a role in the development of certain brain structures, especially the cerebellum when we are young. But, to be clear, the evidence does not show that play is vital for the development of these regions. Other methods, like exercises or teaching, may have a similar effect. They might not be as fun, but this is known as equifinality. So there's debate as to just how vital play and games really are. Well, let's take a look at the rewards that games give us. In the first half of the 20th century, Abraham Maslow constructed a hierarchy of human needs. The concept is popular in developmental psychology as a way of thinking about human growth and what motivates us to do things, or, to not do things. In general, until the needs of a lower tier are fulfilled, an individual can't move on to fulfilling higher ones. For instance, achieving confidence, or satisfying the desire to learn and explore, aren't important to a person in fear for their life and safety. Play might be motivated by higher needs. Animals play, but as we've seen in nature, not the ones who are stressed or starving. The thing is, lower needs tend to be pretty clear cut. If I'm hungry and I eat what I need, I'm done. It's not that complicated. But, as creatures, and brains, and cultures become more complicated, so do their needs and the behaviors required to fulfill them. This brings us back to the fact that life, itself, fits Chris Crawford's definition of a game. Arguably, life is the largest and most complicated game on Earth. But playing your life usually isn't as easy as simply remembering to eat, and drink, and breath. In life, knowing what the correct next move is isn't always easy. Feedback is rarely immediate. I don't know if the choices I made were the best, most perfect choices for me until way after I've made them, if even then. Is this person, or city, or career right for me? In life, the rules are complicated, the goals are indeterminate, and the methods for achieving them are often unknown or different for every single person. Plus, the rewards, themselves, are often slow to come or non-existent. So, in the face of all of that, it's no surprise that we invented games within the larger game of life, itself, that ensure fast, easy-to-achieve, and understandable rewards. Animals play too, but the complexity of rules humans follow in their games, in many ways, reflect the complexity of the needs we find ourselves able to pursue. In life, I don't always know the right choice, I don't know the right job to apply for, how to explain something to a child, how to best help my friends, or when to call my mom. But in Bomberman I know exactly what every power-up does, every time, all-the-time. In Poker a royal flush beats two-of-a-kind, no question about it- couldn't be more clear. But, in my life, is an acquaintance or colleague really on my side? Well, in team sports, there's no unknown- everyone is color-coded. Games and sports are a phenomenal way to feel the rewards we need without all of the unknowns of life. Even watching games and sports, merely being a spectator, can fulfill some of Maslow's needs. I can feel a sense of belonging by supporting a team, and, by supporting a team, their successes can kind of become my successes. What a great way to get respect without doing a lot of work. It's known as BIRG-ing: Basking In Reflected Glory. The opposite is CORF-ing- Cutting Off Reflected Failure. If a team is disgraced, I can easily say I was not really ever that big of a fan anyway. BIRG-ing and CORF-ing extend beyond sports. We BIRG and CORF workplace projects, school projects, celebrities, election candidates- the point is, life is a game, but winning and losing are nebulous. So, we invented simpler games to provide psychological rewards faster, and more efficiently than life itself does. Which is why, at their darkest, games can lead to procrastination, or addiction. But don't fear, you have the potential to become a jock at the game of life, it's just not always that fun. And whenever you play man-made games, rest assured that it's simply because you, and all of us, are able to pursue the fulfillment of needs higher than any other creature on Earth. And, as always, thanks for watching. Oh, and if you want some soccer science, why not check my video with Copa90 where we investigate whether or not it's possible to kick a football with so much spin that it not only curves, but boomerangs back to the kicker. Ok, bye.


  1. ^ Mauldin, R. Daniel (April 1981). The Scottish Book: Mathematics from the Scottish Cafe (PDF) (1 ed.). Birkhäuser. p. 113. ISBN 978-3-7643-3045-3.[permanent dead link]
  2. ^ Telgársky, Rastislav (Spring 1987). "Topological Games: On the 50th Anniversary of the Banach–Mazur Game" (PDF). Rocky Mountain Journal of Mathematics. 17 (2): 227–276. at 242.
  3. ^ Mauldin 1981, p. 116.

Further reading

  • Moran, Gadi (September 1971). "Existence of nondetermined sets for some two person games over reals". Israel Journal of Mathematics. 9 (3): 316–329. doi:10.1007/BF02771682.
This page was last edited on 10 October 2019, at 00:11
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