There are 5 apples. you take away two. How many do you have?
There's an ambiguity there, as it's not really defined if you have all the 5 apples yourself to begin with. The answer could well be "5". (Granted, it says "there are", not "you have", but "there are" is not very informative. Could well mean "there are 5 apples in your possession".)
infinity does not exist because it would take infinite time for anything to become infinite. it doesn't matter how big a number is, it's always an infinite 'distance' away from infinity. that's why infinity can be any number. it's like the joker card. it is the number 0, but it can be anything.
since infinity does not exist, there is only one thing that can be infinite: nothing, zero. we know that zero is infinite because a circle has 0 polygons, but you can also say that it has infinite.
another proof that infinity is 'does not exist': infinity has no end, so it can't have a beginning either. everything that has a beginning has an end. 0 is both the beginning and the end, alpha and omega, creator and destroyer.
so how can the universe be infinite even though it's not nothing? because the space where it is born is nothing.
Infinity exist. even in Video games. like mortal kombat. you can be fighting for an infinite amount of time. now, some of the matches only last say 1 minute, but still :P it is possible. just that you are right, there is no end. but there is a beginning.
Joined: 4/20/2005
Posts: 2161
Location: Norrköping, Sweden
Here's something you can think about (perhaps you've heard this proof before, otherwise, it could be quite a challenge):
Prove that √x exists as a real number for all positive integers x.
√x is the least upper bound on all rational numbers whose square is less than x.
P.S. Such l.u.b. is greater than negative infinity (in other words, real) because there exists at least one rational number whose square is less than x, namely 0.
Even if real numbers are defined as a limit of a rational sequence, there is a rational sequence converging to √x since the rational numbers are dense on any interval, and the square function is continuous.
I consider infinity as a number, but it is not constant. Infinity does not have the same value as infinity.
Then, let X = Y = Infinity. Therefore, X-Y = 0. But, because X = infinity and Y = infinity, and infinity ≠ infinity, we can write X ≠ Y, and therefore X-Y ≠ 0 and 0 ≠ 0. Got any more grains of wisdom?
I consider infinity as a number, but it is not constant. Infinity does not have the same value as infinity.
Then, let X = Y = Infinity.
Actually, under that premise, no. Without constancy, we can't make the logical step from
X = infinity (1) and Y = infinity (2)
to
X = Y = infinity (since X = Y isn't necessarily true due to inconsistancy)
... in another light, we actually lose transitivity. We don't have any reason to think we've lost symmetry, so infinity = Y (3) follows from (2), and if we did have transitivity, we'd be able to make that logical step.
We believe mathematical entities to exist when we feel the need for them to exist, that they have a special meaning.
History has shown that acceptance of new concepts was slow. This includes negative numbers, rational numbers, irrational numbers, complex numbers, infinity. Yes, the concept of complex numbers was so bizarre at the time that they were (and still are, to some extent) called "imaginary numbers". Not only that, but by Cantor's logic, there are different types of infinities and one infinity can be greater than another.
P.S. About the X=Y=infinity thing, infinity minus infinity (or infinity plus negative infinity) is 'defined' not to exist (in other words, not defined). Any attempt to define it so far results in chaos (or so (I think) most mathematicians believe). Note the word 'believe'.
I consider infinity as a number, but it is not constant. Infinity does not have the same value as infinity.
Then, let X = Y = Infinity. Therefore, X-Y = 0. But, because X = infinity and Y = infinity, and infinity ≠ infinity, we can write X ≠ Y, and therefore X-Y ≠ 0 and 0 ≠ 0. Got any more grains of wisdom?
We believe mathematical entities to exist when we feel the need for them to exist, that they have a special meaning.
History has shown that acceptance of new concepts was slow. This includes negative numbers, rational numbers, irrational numbers, complex numbers, infinity. Yes, the concept of complex numbers was so bizarre at the time that they were (and still are, to some extent) called "imaginary numbers". Not only that, but by Cantor's logic, there are different types of infinities and one infinity can be greater than another.
P.S. About the X=Y=infinity thing, infinity minus infinity (or infinity plus negative infinity) is 'defined' not to exist (in other words, not defined). Any attempt to define it so far results in chaos (or so (I think) most mathematicians believe). Note the word 'believe'.
This debate about infinity reminds me of my 2nd semester. In a lecture labeled "Algebra for Computer Scientists" our professor was sick a good deal of the time, and a young doctoral candidate would fill in for him. Very motivated woman. One day, shortly before christmas, when the planned stuff about RSA encryption and it's algebra was finished, she announced that she'd show us a little off topic for the remainder of the lecture.
She talked about half an hour about set theory and cardinality, mostly basic stuff we already knew. And then she blew our mind:
What she told us was about this:
So, this is all interesting, but it gets much more complicated in infinite sets, like, for example, the integers and rational numbers. We know that both are infinite, and that integers are a subset of the rationals. Yet there exists a bijection between them, so they are of the same size, despite there being rational numbers that are not integer. It gets even worse with real numbers, which are proven to have no bijection with integers. How many more of them are there then? If we assume that the cardinality of the real numbers is exactly two to the power of the cardinality of the integers, we'll get interesting things. But if we negate this assumption, we'll get as many interesting other things, and best of all, it's free of any contradictions if we assume either.
And this to an audience who struggled to get their credits for simple ring/field algebra. It took me another 3 semesters to understand these few sentences even partially.
But if we assume the converse, we'll get as many interesting other things, and best of all, it's free of any contradictions if we assume either.
That is clearly false. If we let the cardinality of the set of integers be equal to 2 to the cardinality of the set of real numbers, then we can assign an unique integer to every subset of the real numbers, and for every real number x we can assign an unique integer to the set {x} and therefore, to x. But this has been proven impossible.
You got the order of integer and real mixed up there...
And this Wikipedia article does a far better job than me to explain what this eager person tried to tell us.
I thought that this was what she meant with "assuming the converse" - that one could switch them and still have a consistent system. Perhaps she meant to say the negation?
