Is there a legitimate mathmatical use for the value of an infinite set?
112900
you must use it when you looking for free particle in quantum mechanic to normalize wave function
504653
The set of all fractions {1/N} is equinumerous(the same size) to the positive integers N. They are both infinite sets.\n\nc does not really apply as it is a value, not dealing with set theory, but it's value is infinity, meaning it's larger than any value you can think of. If you are talking about the set {lim(x->0) 1/x}, then it only has one element, making it much smaller than a & b\n\n"a) and b) are of equal size, though if you take any fractions instead of just those between 0 and 1, you will get that a) is the largest. I attended a lecture about this a while ago, hence I can't remember the arguments, but the lecturer clearly concluded that there are more fractions than integers."\n\nThis is almost correct. If you just take all the fractions between 0 & 1 it will be a larger set than b. Since 1/N is equinumerous with N+, it follows that 2/N is equinumerous as well. All the fractions between 0 & 1 would include 1/N, 2/N, 3/N, etc. making it much larger than N+
1113261
hypothetically\n\nan infinite number
18903
There is no number that will multiply by zero to not equal zero because for example 5x5 would be the quantity of five added five times, 5x1 would be the quantity of five added one time, and 5x0 would be the quantity of five added zero times meaning there is nothing there or, if you prefer, that the quantity (Five) is not addded at all.\n\nBTW- even infinity multiplied by zero equals zero\nBTW- There is NO NUMBER!!! that will multiply by zero to to equal anything but Zero. Not Infinity or undefined!! 0x0=ZERO Infinity x 0=ZERO, ETC.
385151
A transfinite number is generally associated with an ordinal or a cardinal.  From your description, it appears that you're asking a question about cardinality (i.e. where ordering is not important).\n\nNow to clarify your question, what you said about points on a line being the next smallest transfinite number is only true if you believe in the Continuum Hypothesis, which says that the powerset (set of subsets) of the Natural numbers, P(N) is the ``next smallest cardinal''.  However, this axiom is independent of standard ZFC set theory meaning that both it and its negation are consistent with ZFC so that you can't hope to prove it or its negation true in ZFC assuming ZFC is consistent.\n\nIn the case that you do believe in the continuum hypothesis (generally denoted CH), then since it can readily be shown that the powerset of the Natural numbers can be put in a 1-1 correspondence with the points on a line, that part of your assertion is correct.\n\nHowever, the infinity you claim to be the next smallest uses something even stronger than CH, mainly P(P(N)) is the next smallest cardinal after P(N).  Since P(N) is equinumerous (equipotent or equipollent) to the points on a line (that number usually being denoted c for continuum), we can then show that there is a one-one correspondence between P(P(N)) and the set of functions f: R -> R or even the set of functions f: R -> {0,1}!  Therefore the next part of your assertion becomes correct.\n\nAgain, we still can't prove anything about what the next smallest cardinal looks like, but if we assume in general, the Generalized Continuum Hypothesis (GCH), then we simply assume that for any cardinal, the powerset of that set is the ``next smallest infinity''.\n\nIn that case, the answer to your question is simply aleph_3 = cardinality of P(P(P(N))) = cardinality of set of subsets of functions of form f: R -> R.
281372
No.\n\n\nUPDATE:\n\nSome of your here don't understand the concept of infinity. You don't ever reach, approach or ever come near infinity. A googleplex raised to the power of a googleplex is as far away from infinity as is the number 1. There is also an infinity between any two numbers. For instance, there's an infinity between the numbers 1 and 2, and so on and so on.
1043703
The ... implies an endless continuation of the series.  So the answer is \n\nINFINITY.
26988
I think the odds would be 1 in infinite. That's just because you could just any number, and you only have one guess.
597619
well that last example doesnt work...they divide by zero and try to let it slip.\n\nThings like Russells paradox messed with the mathematical world for a while.  This paradox deals with sets.\n\nIt says that we have a set R, R is the set of all sets that are not members of themselves.  This means that R  is neither a member of itself nor not a member of itself.\n\nThis is pretty neat, and it was a huge problem with set theory.  It was solved by saying that all sets are members of a universe.  I still claim that its a sticky area.\n\nhope this is what you were looking for.\n\nMatttlocke
403633
If by infinite randomness you mean true randomness, then I don't think it can. All random generators are programmes that follow an algorithm to produce 'random' results. Whilst for most purposes the result is sufficiently random to be useful, they are not as random as, for example, drawing lottery numbers.
343556
Yes, there are variou sizes of infinity. These numbers are collectively known as transfinite numbers.\n\nThe set of all integers and the set of all squares are in a 1-to-1 correspondence with one another, and both are infinite., Both are aleph-null, in Cantor's terms. But there is a bigger infinity aleph-one, the set of all real numbers, for example, has aleph-one members.
356300
infinity because you could say any number  bigger than 100000000000000000000000000000000000 is infinity
497917
No, the above answer is for BigInteger. BigDecimal allows you basically an infinite number of decimal places. Eliminates computational errors with floats and doubles in the IEEE standard
472830
agree, but in case of number its not true, it is infinite
722299
Infinite? I am not sure if you meant real things like people, sand, etc. or abstract things - but here goes:\n\nthe set of integers\nthe set of even integers\nthe number of numbers (or fractions) between 0 and 1\nthe line\nan infinite loop in computer programming
114
infinity \n\nand for those people..\n\ninfinity + 1, infinity x infinity\nand infinity ^ infinity
241288
What goes wrong is the loss of information.  All _n_ divided by zero goes into the same "infinite quality".\n\nIt is possible to admit different infinite numbers; this is Robinson's approach with "nonstandard analysis", and then there are a myriad of choices as what to admit and what to disallow.  If you take that approach, it is still easier to disallow n/0 as undefined, and permit other "zeroes" where n/d is defined, where d is one of those classes of zeroes.\n\nIt is exactly equivalent, however, to the following:\nlim(d->0) n/d.  In this case the limit doesn't exist, but there are other "zero approaching denomenators" where the limit does exist.  One very important one:\n\nlim(d->0) sin(d)/d = 1