This sort of stuff is some of the most interesting and beautiful mathematics which I really recommend looking into.
I'm surprised at how many people tried to call this a fallacy though after being shown 2 valid proofs. I remember learning about this in school when I was 10 years old...
It's actually pretty intuitive; 0.999... is 'infinitely close' to 1, and hence, it is 1.
Technically this follows from the 'Archimedean property', which states that there are no 'infinitesimally small' numbers (and hence there can't be an infinitesimally small gap between the two; this answers the question about 0.00...1: it doesn't exist), and it can be proven, although it's a bit technical.
It follows that all terminating decimals, for instance
4.3238912,
actually have two expressions in decimal notation,
4.3238912
and
4.3238911999...
which turns out to be more of a slight annoyance than anything.
Originally Posted by Dianeva
My STAT professor mentioned a few days ago that of the two types of infinity:
All numbers between two numbers. Ex: [0, 1] which includes (0, 0.0000001, 0.00025, 0.589234, etc.)
and
All integers going into infinity, like [5, inf) which includes (5, 6, 7, 8, 9, ...)
the first type has been mathematically proven to contain more numbers, to be a greater infinity, than the second type. This seems it must be false to me. Infinite is infinite. How can there be a higher type of infinity? But this interests me, and I'm planning on being proven wrong when I look it up, maybe tomorrow.
Philosopher Stoned is basically correct about what he said, although there are some important distinctions to be made.
In mathematics there are actually two totally different uses of infinity.
The first is the infinity of analysis. This is used for when something is acting 'without bound' in some fashion or other.
For example, one can say that the limit of 1/x^2 as x tends to 0 'is infinity'. However, this is not really rigorous as infinity isn't a number in this context. We therefore reformulate it in the following way: "for any number [no matter how high], you can give a range around 0 so that every value of 1/x^2 in that range is greater than said number". This removes all reference to infinity.
Or, as another example, one can say that the limit of 1/x as x tends to 'infinity' is 0. This can be reformulated as "for any number [no matter how small], you can give a place on the x axis so that for any number to the right of that place, the distance between 1/x and 0 is less than said number". Again, this removes any reference to an 'infinite number'.
The second infinity is the infinity of counting. This is the one you are referring to, and as PS correctly said, technically counting refers to creating a onetoone correspondence between the elements of one set and another. He is also correct that Cantor proved that there are more real numbers (any decimal you like) and natural numbers (positive whole numbers). The proof is awesome and if you want I'll explain it.
Your stats teacher is also correct; this actually follows from above. The reason is that you can set up a onetoone correspondence between any interval of the real numbers ([0,1] in this instance) and the entire real number line.
For example, the interval (1, 1) can be sent to (inf, inf) via the function
f(x) = 1/x
If you think about it, you'll find the assertion that every element in the first set now has a corresponding element in the second set is true, and hence, according to our notion of counting, there are the same number of elements in both sets. The intuitive way to think of this is that the real number line can be 'stretched' to form any other part of the real number line; that's why it's called continuous.
[5, inf) can easily be sent to [1, inf) (the naturals) via the function f(n) = n  4, and then it easily follows from the fact that there are more reals than naturals that what your teacher said was correct.
You are right that this sets up a 'hierarchy of different infinities', and yes this is very strange, but I hope you can see that it's also welldefined. There is one mistake with what PS said, though, when he said that aleph1 (i.e. the next biggest infinity after aleph0, which represents the naturals) represents the infinity of the reals. This statement is referred to as 'the continuum hypothesis'. The amazing thing which was discovered last century is that this is actually unprovable, and so is its converse. Hence it is totally arbitrary whether it is true or not; you either assume it or you don't (so it's a kind of axiom). This has created two different 'mathematicses', each with their own different set of truths!
There are many more weird and wonderful things in this area and I've only really skimmed some of the surface.


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