[evan_au]

the set of rational numbers between 0 and 1 ...is (a) infinite (b) larger than the set of integers

This didn't quite gel with me - I vaguely recall from university seeing a proof that the number of rational numbers between 0 & 1 was countably infinite. It was a geometric construction on the number plane, effectively making use of the mathematical fact that ∞²=∞.

In doing a quick search, I found a different proof that shows that members of the set of all rational numbers (from -∞ to +∞) has the same quantity as the number of positive integers.

Like a lot of things in maths, it comes down to being clever about how you count. In this case, the cleverness comes in being able to count an infinite number of things in a finite time.

But similar kinds of cleverness in counting are implicit in common mathematical operations like integration, differentiation and probability.

[Bill S]

∞2=∞.

From this it might be argued that we must be dealing with two infinities of different sizes.  However 0^2=0, so are we dealing with different sixes of zero?

[yor_on]

Yor_on.  Your last post seemed to say something interesting; I just wish I knew what it was.   []

Call it meta physics, the question of what exactly we think we're doing
As long as you just accept bases you won't move forward in your questioning. define that inside, don't accept a outside. then start to ask yourself how the unknown will fit such a proposal.
=

If you do it as me then the unknown will be what 'is', we being a symmetry break Well, possibly? and then dimensions should lose their former 'idea history', as I think then, needing to be reevaluated.

[Bill S]

I have started working through the thread picking out some bits which seem important to me, and adding some comments relevant to my understanding.  Lack of time has prevented me from getting very far, but I would really appreciate some comments.
 

Well then find the radius of the circle whose circumference is infinite.

This is a mathematical question, so mathematical infinities would be appropriate.  If the circumference is infinite, then the radius is also infinite, but they must be different sized infinities, and different infinities are acceptable in mathematics.  We cannot assign any finite value to either radius or circumference, effectively, infinite means so large that we might as well consider it infinite. 

The point here is that any system that can normally be considered as bounded cannot include an infinite component. So if the mathematics of a formerly finite system go infinite something is terribly wrong.

The first sentence is true if it refers to what we might need to call “absolute” infinity.   If we are using mathematical infinities, then it can be argued that, for example, there is an infinite number of rational numbers between any two integers.  However, Zeno’s paradox leads us to suspect that that this is a “fact” only in principle.

The second sentence must be true, as any finite thing would approach infinity infinitely, thus it would never “arrive”.     

Let me make this very clear first; {infinity} is not a number.

As far as I can tell, everyone in this thread agrees with this statement.

There is an infinite number of integers 1,2,3,.... because we can always add one more

There are rational numbers between the integers 1, 3/2, 7/4, 2, 9/4, 19/8, 3....

Indeed there is an infinite number of rational numbers between any two integers

So the number of rational numbers must be greater than the number of integers

As so often happens when talking to scientists about infinity, we return here to mathematical infinities.  Although these are boundless, their “infinite” nature can never be physically demonstrated, because it is not possible to count to infinity, nor to enumerate all the rational numbers between two integers.

IMHO if ever you bump into an infinity in physics, then something is wrong somewhere.

Nonsense. It's beginning to become clear that the universe is flat and boundless and as such goes on forever, never ending. That's what it means to be infinite.

Infinite answers are certainly not right, so they are a sign that your theory is not very good.  A theory needs to fit the data, but it also needs to make mathematical sense.

Do we lack reasonable consensus between scientists as to exactly what is meant by “infinity” in individual cases?  It seems so, but that presents a problem for non-experts.  How does one choose whom to believe? 

…… there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity.

That makes perfect sense to me, as long as we accept that mathematicians will always be able to fit mathematical infinities into finite spaces.

[Bill S]

I posted the last bit in a hurry without checking for typos, crackpottery etc, nor did I ask if anyone objected to my using initials with the quotes.   

[chiralSPO]

The only way would be able to prove something is infinite would be by abstraction. I don't think we could ever claim to have measured or experimentally determined an infinite value, countable or uncountable.

In the same way, zero is also kind of an infinity. Reciprocal and logarithmic relationships abound in physice. For instance, one cannot actually actually achieve absolute zero on the Kelvin scale (for many reasons!), only arbitrarily close to it; the distance between two particles cannot really be zero or infinite, only arbitrarily far or close. Yet there are also laws of conservation--meaning a derivative implied to be precisely zero, but verified to be zero within some small, but obviously nonzero confidence interval.


Does the fact that we cannot experimentally verify something make it untrue through?

The math predicts infinities or zeros, but we must be careful here, also to remember that our beloved equations are ALL MODELS. And all models have implicit assumptions and limitations. So an equation having a nonsensical infinite or zero value could  just as much signify a problem with the model as reflect reality. Often these ridiculous failures of our equations happens at boundaries or limits (absolute zero, speed of light, discontinuity in another parameter [phase change], etc.) But overall, we can do amazing things with infinities in models, giving real predictive power for experimental physics, chemistry and cosmology.

Most people have misconceptions about infinities, myself included--little is intuitive, only protracted analysis and mathematical rigor works here. One can do some really crazy things with them mathematically, or by logical induction (Cantor actually went crazy, but that's another story).

[PmbPhy]

If the universe was flat then Sean Carroll would not disagree with me.

[yor_on]

It's actually conceptually simple to think of any thought up outside as an inside too. If you do infinities becomes what we can't measure on, getting a linear answer. Doesn't matter if it is a actual experiment or a mathematical hypothesis. then what we are, to me looks more as a break than as 'what is'. We're an exception in my mind, and our ideas of what infinities should be defined as should then become a answer from what cultural and physical reality, we define as our seamlessly existing 'common' universe. Infinity loses its meaning.

[yor_on]

Eh, doesn't state that a Victorian description would be fitting. Just that if you define a inside and let the rest be a part, although not linearly measurable as we would prefer/expect a inside to be. Then our 'infinities' just may be what is, us being a slightly different version, from our linear definitions. It turns it upside down and force you to define what, and why, we are from a different base.
=
are, not 'exist'

[Bill S]

If the universe was flat then Sean Carroll would not disagree with me.

OK; but could you explain why, please.

[Bill S]

Does the fact that we cannot experimentally verify something make it untrue through?

I feel sure it doesn't.  However, such possible things can always be relegated to metaphysics or philosophy if they become irritating.  Otherwise, they can just be part of some theory. 

[PmbPhy]

If the universe was flat then Sean Carroll would not disagree with me.

OK; but could you explain why, please.

Absolutely. If the universe was flat and the cosmological principle (http://en.wikipedia.org/wiki/Cosmological_principle) is correct (both of which are widely beginning to be accepted as true) then the universe unbounded and not finite, i.e. the space is not bounded and goes on forever which means that the number of hadrons and hence the amount of matter is infinite. I can't imagine Sean disagreeing with me.

[jeffreyH (OP)]

If the universe was flat then Sean Carroll would not disagree with me.

OK; but could you explain why, please.

Absolutely. If the universe was flat and the cosmological principle (http://en.wikipedia.org/wiki/Cosmological_principle) is correct (both of which are widely beginning to be accepted as true) then the universe unbounded and not finite, i.e. the space is not bounded and goes on forever which means that the number of hadrons and hence the amount of matter is infinite. I can't imagine Sean disagreeing with me.

Surely infinity is an unbounded continuity whereas hadrons are discontinuous having gaps of varying magnitude between them. For hadrons to be infinite it would require there being only 1 hadron of infinite size. For multiple hadrons there would also have to be an infinity of empty space. If infinity is all inclusive how can you divide it in 2 or more portions? Mathematically this would not work.

[PmbPhy]

Surely infinity is an unbounded continuity whereas hadrons are discontinuous having gaps of varying magnitude between them. For hadrons to be infinite ...

No. It's the mass density that is uniform and as such when you multiply a constant mass density by the infinite volume of the universe you get an infinite mass. It's as simple as that. I used hadrons as an example so people can visualize it.

it would require there being only 1 hadron of infinite size.

Come on Jeff. Let's be real here. Hadrons are point particles having zero radius. A hadron can't be infinite in size. Let's try to understand the difference between the exact calculation and the example given to illustrate it.

For multiple hadrons there would also have to be an infinity of empty space.

That's what a flat universe implies Jeff. IK thought I explained that above. You didn't realize that?

If infinity is all inclusive ..

What in the world does that mean???

This is very simple, Jeff. Mass density = constant. Volume of universe = infinite.

If you need to see this from a university professors notes like a lot of people do to make them feel confident about the answer then see http://www.astronomy.ohio-state.edu/~ryden/ast162_9/notes40.html 

Precise measurements lead cosmologists to conclude that the universe is flat, and thus has infinite volume.

Do you understand now? Please stop confusing yourself with these bizarre and unphysical dreams of yours and stick with the physical universe. Okay?

[Bill S]

This is very simple, Jeff. Mass density = constant. Volume of universe = infinite

Pete, what you say is undoubtedly right, but, as has already been mentioned, mathematical reality and physical reality are not necessarily the same thing.  E.g. with the “book stacking problem” it is possible, in theory, to reach an infinite overhang with an infinite number of books, but would you claim that is physically possible?

[PmbPhy]

Pete, what you say is undoubtedly right, but, as has already been mentioned, mathematical reality and physical reality are not necessarily the same thing.

You're wrong, Bill. What math does is to allow us to describe physical reality. Start with a universe having a uniform uniform mass density which as the value at time t of rho. Let the volume of the universe be V. Then the total mass in a universe at time t is M = rho*V. For larger and larger universes we have more and more mass given the same mass density. If that volume increases without limit it means that the mass increases without limit. That's what it means to be infinite.

E.g. with the “book stacking problem” it is possible, in theory, to reach an infinite overhang ..

What do you mean by an infinite over hang? Do you mean that the center of mass of the next book to be put on the stack is not right above the previous one and thus there is an increasing over hang? If so then that's not possible because it'd collapse before that.

with an infinite number of books, but would you claim that is physically possible?

Of course not but those two problems are like comparing apples and oranges and as such it's not the same thing as in this or with all other physics problems. Try using your argument on every single exercise problem in any physics textbook and you'll see that it's the wrong answer.

This is a very simple idea. If people are having a difficult time with it then you're missing something, i.e. there is a hole in your education somewhere. Let's find out where that hole is.

[Bill S]

What do you mean by an infinite over hang? Do you mean that the center of mass of the next book to be put on the stack is not right above the previous one and thus there is an increasing over hang? If so then that's not possible because it'd collapse before that.

You must be familiar with the book-stacking problem.  The books have to be stacked so that the centre of gravity of the stack never lies beyond the edge of the table.  Once it does they will start to topple.  If each book has size 1 then the maximum possible overhang of N books is just one half of the sum of the harmonic series up to N terms:

    Maximum Overhang Distance = 1/2X {1+1/2+1/3+1/4+1/5…+1/N}

John Barrow points out that the overhang distance can be made as large as you like by making N big enough. 

"To make the overhang 10 times the size of a single book would need a stack of 272,400,600 books.  In an ideal world without friction and imperfect surfaces and smallest particles of matter, the overhang could be infinite.” 

Barrow concedes that: “This is possible in principle, not in practice.”

I mentioned this only as an example of something that might work in principle but would not work in practice.

To suggest that this particular example could be applied to every problem in physics would be absurd, and is certainly not something I said. 

[Bill S]

If that volume increases without limit it means that the mass increases without limit. That's what it means to be infinite.

I am neither qualified, nor would I wish, to suggest that this is not a mathematical reality.  However, if you are talking of something "increasing", that something is not infinite.  If it is not infinite it is finite and therefore it can never become infinite.   

[PmbPhy]

I am neither qualified, nor would I wish, to suggest that this is not a mathematical reality.  However, if you are talking of something "increasing", that something is not infinite.  If it is not infinite it is finite and therefore it can never become infinite.

Hi Bill,

May I ask you a personal question? Have you ever studied calculus or real analysis? If so then surely you learned about what infinity is. If not then I see where the problem lies. If that is indeed the case then let me explain it to you if you, if you don't mind that is? By definition, a quantity is defined as infinite when it increases without bound. It's said not to have a limit. That's what it means to be infinite. For example; what is the value of the function f(x) = 1/x as x approaches 0? This is called a limit. In this case f(x) increases without bound and we say that f(x) becomes infinite as x approaches 0.

A limit is the value that a function or sequence "approaches" as the input or index approaches some value. If there is no such number and the sequence increases without bound then

[PmbPhy]

To suggest that this particular example could be applied to every problem in physics would be absurd, and is certainly not something I said.

I wasn't referring to the example. I was referring to your comment "mathematical reality and physical reality are not necessarily the same thing". In what I've been talking about I've only been using math to describe why there's an infinite amount of matter in the inverse. I don't see how your example applies to this problem. In fact it doesn't.

However, suppose I was asked to prove that E = mc² and I posted the math to prove it's true. Could someone logically respond to my derivation by saying "mathematical reality and physical reality are not necessarily the same thing" and have a valid point? If not, why?