Hi everyone Hope you are all doing well I want to ask about infinity. What does it mean? Is it a quantatiy over there that we can't determine and thus we called it infinity? I hope that u share me ur opinions What are the references that demonstrate this topic in details regarding mathematical and philosophical point of view?! THANKS
John Derbyshire's "Prime Obsession" has a nice section on this, he explains how every legitiment mathematical statements that involve infinity can be RE-WRITTEN otherwise, for example, by saying 'The series x adds up to infinity' this can be written as 'Given any number S, no matter how large, the sum of series x eventually exceeds S.' See? No infinity. He also states that non mathematical people often ask him 'You know math, huh? Tell me something, I've always wondered, what is infinity divided by infinity?' where as he can only reply, 'The words you have just uttered do not make sense. You spoke of 'infinity' as if it were a number. It's not. You may as well ask, 'What is truth divided by beauty?' I have no clue. I only know how to divide numbers.'
FYI, John Derbyshire is an offensively racist hack who has recently been fired by the National Review for failing to keep his virulent racism down to at least the coded, wink-and-nod PC format required of modern mainstream conservative mouthpieces. Except that "x exceeds any number S, no matter how large" is the definition of infinity. Not sure what the import of avoiding using the word "infinity" is supposed to be, if you're just going to substitute the very definition of the word in its place. The snarkier answer would be "which infinities are you referring to?" But if you're going to read a book about infinity by a writer who isn't a practicing mathematician, you should go for something more like David Foster Wallace's effort: http://en.wikipedia.org/wiki/Everything_and_More_(book)
The notion of infinity has many definitions, depending on the context. In it's most broad sense it's as already said, something defined to be larger than any element of the Reals, so it is not a Real number*. But then there's the notion of cardinalities. How many Natural numbers (ie whole positive numbers) are there? There's infinitely many, ie the cardinality of the set of Naturals is infinite. But then you get into curious and counter intuitive concepts like there being as many whole numbers are there are whole even numbers or prime numbers. But then the Reals also have infinite cardinality but there's strictly more Reals than Natural numbers. Two strictly different infinities! In fact you can do this again and again, each time finding a new, strictly larger, infinite cardinality. That's the wonderful world of Cantor for you. Cardinal numbers have odd properties like a+b isn't the same as b+a. * When I capitalise the 'r' in real I refer to the set of Reals, so as to distinguish it from the more usual concept of real meaning the thing exists in reality, like you or I do. Mathematicians don't use real and imaginary in the usual sense, despite what some laypersons think. I always capitalise the letter when referring to a number theoretic set to make this clear.
Yeah, sure. Have you really not heard about the big uproar over his racist writings? He lost his job in a very public row over this last week, and the stuff remains in the news. It's on the Wikipedia page about him and everything. Dude's been reviled as a racist ideologue by the left for a long time, but now has been disowned by the right as well for failing to keep his racism below the threshold of credible deniability.
In general infinite means the idea of something that has no end and its the same concept that is applied on mathematics means infinite doesn't have any solution.
That is not a coherent sentence, in that it doesn't use the words in ways consistent with their meanings. Solutions are to do with problems or questions. Infinity is a thing, a concept with specific meaning, always used as a noun. An equation can have a solution, a problem can have a solution, infinity cannot any more than 2 can have a solution or a derivative can have a solution. You can construct problems in terms of them but by themselves they are just mathematical objects.
Find a good book or do a search for Cantor and Transfinite numbers. I think Cantor's first name is Gregor, but I am not sure. Basically, he works with the concept of sets & the number of members of a set. Two sets have the same number of members (same cardinality) if there there is a one to one correspondence between the members. If a subset of a set has the same cardinality as the set, then the set is transfinite or infinite. Example: Consider the set of all positive integers & the subset of all even integers. You can match 1 & 2, 2 &4 . . . . n & 2*n. Hence there is a one to one corresondence between the two sets. Hence the set of all positive integers is transfinite or infinite. Note that you cannot make a one to one correspondnece between the set of all integers & the set of all real numbers in the interval 0 to one (you run out of integers): Hence the set of all reals has a greater cardinality than the set of all integers.The above is a brief summary of the type of logic used by Cantor. Infinity divided by infinity and zero divided by zero are undefined if there is no context. If you are dealing with functions, you can use limit concepts. For example: sin(x) /x as x goes to zero approaches one. Hence zero divided by zero can be equal to one if you are consider sin(x)/x at x = 0 There are similar situations for which infinity / infinity can be defined. I do not remember any examples & do not have time to figure one out or do some research.
Infinity is a number that is extremely large but has no bounds. There infinite number of infinities. sonic
Let me stop you there - checking back through this thread will give you the definition of infinity: "x exceeds any number S, no matter how large" so infinity is not a number. Any explanation of infinity that starts "Infinity is a number ..." is bound to fail.
Not necessarily so! Im with you if by "number" is meant "Natural number" or "Real Number" , but since there are different kinds of number nothing prevents an infinity to be one of a certain kind of number! Cantor created an arithmetic for "Transfinite" numbers where the smallest transfinite number is the number of all natural numbers. http://en.wikipedia.org/wiki/Georg_Cantor Our language invites us to think there are objects and qualities, so there might be an object called "the infinity" and a quality called "to be infinite". And more objects and qualities of the same kind... Life is wonderful Please Register or Log in to view the hidden image!
I had a Cantor phase recently. Went to the library's rare book room to read and checked some texts out and bought one text so I could write in it. The continuum hypothesis just strikes me as profound for some reason. But as to infinity, aside from very narrow symbols which work basicically the same way imaginary numbers do, as a sort of a placeholder for the batch of numbers you are drawing from, i.e. an abstraction which we use as a convenience to make number theory slightly more self consistent, it really means the conceivable horizon. Big. Really big. More than you can imagine. Still more. On a scale of some sort. Which presents one set of problems when the scale is more of a ratio of one thing compared to another. Like pretty, or strong, or smart, or powerful or knowledgeable, or even perfect. You end up with are you strong enough to lift a planet? Which brings an arbitrariness of scale in. Someone might be strong enough to lift the moon if it were resting on Earth. But by the same token, that person would also be strong enough to lift Earth by standing on the moon. What constitutes a planet? Etc. Or, can god make a burrito so hot he can't eat it? Or the idea that your love is infinite for you spouse. Or the idea that there is a Greatest possible being or even a way to measure wtf that might mean or stuff like that. But even in Math it isn't a real idea that anybody uses in any way or with any more understanding than imaginary numbers. Math seems to need a pool into which we can occasionally drop bits and fish them back out later. The infinite sets are just groups with some shared characteristic which is abstracted enough that no conceivable obstacle could prevent the series from continuing. The integers or rationals or whatever continue with no obvious impediment therefore we call that lack of impediment infinity. No one has any way to say what it is like. It's is like looking into the sky? No. The boundary is just far. And we can extend that boundary with expensive equipment to a theoretical limit of just before the edge of our light cone but that still isn't infinity. I suspect that a great deal of the strife and discord humans have brought about over our history is due to people really believing that because they have a word for farther than I can imagine, that Infinity has any meaning other than as an arcane and frankly dispensable mathematical term. It is possible to frame calculus without infinities. Rather than a smooth function being 'infinitely differentiable" call it "anywhere differentiable" which is actually more accurate if you think about it because if you have an infinitely differentiable line you are supposing you have an infinity of time to operate. Anyway, anyone who tells you that they know what infinity is or that it is well understood by education variant x, are making a claim of convenience which ultimately doesn't mean anything. IOW, those are the people you need the most patience with. Please Register or Log in to view the hidden image!
Cantor was slightly religious. He created a theory and arithmetic for infinities all based on his proof of the impossibility to create a one to one function between natural and real numbers. So he got a new set (ok maybe im careless here but im trying to be simple) of numbers: Transfinite Numbers their natural order is always based on the fact that the power set of a set is larger than the set itself. But shouldnt there be something larger? Something larger than everything else? What could that be, he wondered , GOD?? So he called it: The Absolute. The problem is that you cant understand it. If you try to you will contradict yourself! Example: Suppose that Everything = The Absolute 1 Everything is the sum of everything else. 2 Everything added to everything else is more than everything!(Contradiction!) To tidy things up he introduces the Reflection Principle: Whatever is said aboute the Absolute is truthfully said about some lesser infinity. I itch to find a chink in Cantors armour but he seems invincible!
When I was a more… sedated lad, I would lull my bored existence in bed by reaching out as far as I could into the furthest penetration of an infinitely long dark universe. It use to drive me bananas because at a certain point in my excursion… I'd get overwhelmingly bored— grey numbing solid boredom seemed to deck the outer limits.
