grace2u writes:
quote:
How could the standard of an infinite entity be measured by the error prone standard of the finite.
Because logic applies to infinite things, too. In fact, the Incompleteness Theorems were famously and primarily applied specifically to show that a question about infinities cannot be resolved using current axioms of set theory.
What is the size of the continuum?
Well, that's a hard question. "It's infinite" isn't an answer because some infinite things are bigger than others.
Let's start with just the integers. There are an infinite number of them, but let's get a handle on how many there are.
Suppose you had a hotel room with an infinite number of rooms, all numbered: 1, 2, 3, .... Plus, suppose that every single room is occupied. Someone shows up. Can you get a room for the new guest?
Yep. Simply tell everybody in the hotel to move down one room. The guest in Room #1 moves to Room #2, the guest in Room #2 moves to Room #3, and so on. This leaves Room #1 empty and the new guest can move in.
This process works for any finite number of people who show up. Ten people show up? Just have everybody move down ten rooms. This leaves Rooms #1-10 open and they move in. A hundred? A quintillion? No problem.
But suppose you had an infinite number of people show up? Could you fit them in?
Yep. Simply tell everybody in the hotel to move to the room number that is twice as large as the one currently occupied. The guest in Room #1 moves to Room #2, Room #2 moves to Room #4, Room #3 goes to Room #6, and so on.
Everybody that is currently in the hotel has a room since for every guest you can name, I can give you a room number to find that guest.
But look what we did: We just opened up every single odd numbered room. This means that an infinite number of guests is equivalent to an infinite number of guests only occupying even-numbered rooms. Since there are just as many odd numbers as even numbers, that must mean that the new guests can simply move into the odd-numbered rooms and everybody is satisfied.
Now, that's just integers (and positive integers at that). From this, we can tell that one way to see if something is the same size as the integers is to pair them up and see if they're the same size. If you can get this one-to-one correspondance, then they're the same size.
It shouldn't be too difficult to see how to add the negative integers into this (you can easily put the negative integers into one-to-one correspondance with the positive.) But for fractions (and by "fractions," I really mean the "rationals" which means any number that can be written as a fraction of two integers), things get weird.
Without getting too technical, let's just say that Cantor developed an ingenious proof that showed that the set of fractions is the exact same size as the integers (in short, it involves writing down fractions in a matrix method, which can guarantee that we get all the fractions and then, by traversing it diagonally, you can show that it is in one-to-one correspondance to the set of integers, thus the fractions are the same size as the integers.)
But what about the real numbers? Real numbers include numbers that cannot be expressed as a fraction such as pi, e, and the square root of 2 (as well as all the rationals). Because these numbers form a continuous spectrum of numbers between any two you care to name, it is often called "the continuum." What about them? If we include all these other numbers, can we put them into one-to-one correspondance with the set of integers?
It turns out we can't.
First, some prep work. I need to show that a terminating decimal has an equivalent infinite expansion. That is, I need to show that 0.999... = 1.
Let x = 0.999...
Then 10x = 9.999...
Subtracting:
10x - x = 9.999... - 0.999...
Reducing:
9x = 9
Therefore, x = 1. But we started with the statement that x = 0.999.... Therefore 1 = 0.999....
Now to the proof. Suppose you had a list of all possible decimal expansions between 0 and 1. To avoid duplicates, we will make each expression unique by making it the infinite repeating version. That is, instead of 0.5, we will use 0.49999... which is the same thing.
Now, this means we have a list of decimal expansions:
x
1 = 0.a
1b
1c
1d
1...
x
2 = 0.a
2b
2c
2d
2...
x
3 = 0.a
3b
3c
3d
3...
Let us now create a new number,
p:
p = 0.a
1b
2c
3d
4...
However, we shall alter the numbers of p as follows:
If p
i = 3, then new-p
i = 2
If p
i != 3, then new-p
i = 3
Thus, we have a number that is necessarily between 0 and 1 and yet differs from every single number in our list at the
ith decimal place. No matter what our list is, we can construct a number that isn't on the list and should be.
Therefore, the real numbers cannot be put into one-to-one correspondance with the integers. While the size of the integers is infinite, the size of the reals is larger.
[And if you really want something that will bake your noodle, consider this: The rationals (all numbers that can be expressed as a fraction of two integers) are "dense" in the reals. That is, between any two real numbers you care to name, no matter how close together they are, there is at least one rational number. Similarly, the reals are "dense" in the rationals...between any two rationals, there is at least one real. And yet, as we just showed, there are more reals than rationals. Where do we find the space to fit them all?]
It turns out that there are a whole hierarchy of infinities. Cantor showed that one can construct a power set of a set by taking every single subset of a set. This power set, he proved, is always larger than the original set. In short, if a set has size
n, then the power set has size 2
n. And for infinite sets, that means that the power set is larger still.
He assigned the size of the integers the symbol aleph-null. Thus, the power set of the integers would be of size 2
aleph-null and he called this new number aleph-one. The power set of aleph-one is aleph-two and so one. Each aleph is bigger than the one below it, even though a set of that size is infinite.
So the question is now: We know that the size of the reals is larger than aleph-null, but what is it? Is it aleph-one? Aleph-two? Or perhaps there is some other sort infinity between the alephs just as there are numbers between the integers.
It turns out that while we know that the size of the continuum (often called "c") cannot be greater than aleph-one, we don't know if it actually is aleph-one. And furthermore, we have come down to this extremely frustrating set of events:
If we assume the c is aleph-one, we do not get a contradiction. For a long time, people thought this proved the point. No contradiction? Then it must be.
Ah, but that isn't good enough. It turns out, after other research was done, that if we assume that c is not aleph-one, we also do not get a contradiction.
And as it turns out, this particular question can never be answered given the current axioms of set theory. It's a very basic question, but given the foundations of our mathematics, we can never answer it.
quote:
Because the man described laws of logic appear to be inconsistent in some cases or incomplete in others(or rather in their aplication to mathematics at times) certainly doesn't imply that the universal laws are similar.
Ah, you're a Platonist. A Platonist would look at the problem of the Continuum Hypothesis and say that the real numbers do have an actual size...we just don't know what it is (and alas, can never know).
However, this doesn't really apply to god. Since when is god an axiomatic set theory sufficiently complex to model simple arithmetic?
quote:
We do suppose however, and evidence suggests I might add(I think even most logicians would agree on this) that there is some "absolute-truth" that can be obtained or reached.
Eh, not really. More accurately, mathematicians claim that some things, given certain assumptions, can be declared to be true but that a lot of things, if the system is sufficiently complex, can never be known.
Some things simply cannot be discovered no matter how perfect your logic is.
quote:
Else all scientific work would be futile.
Not at all! You seem to be dissatisfied with "accurate enough for all known examples." Why is that?
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Rrhain
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