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One, Two, Three...Infinity: Facts and Speculations of Science
What makes this book awesome?

A very interesting book. Daniel, have you read this? How do you define infinity. Is it possible to define?

Picking this up today from the library.

Not sure if that question was directed at me (there is/was another Daniel, I think.) But it is something I know a little about.

Simply, there are multiple ways to define infinity.

First comes the (possibly innate?) concept we all have: really, really, really, really... ... really big. That might not satisfy as a definition, but there it is.

Second, is the "real number extension" infinity of real analysis. This is the infinity you are always using in calculus (e.g. integrate 1/n-squared from 1 to infinity.) This can be defined thus: Having defined the real numbers (integers, rational numbers, irrational numbers) extend the set by the addition of two symbols: "�" and "-�" defined such that for any real number x, x < � and x > -�. (If you look at this carefully, you will see why you had math teachers and/or math nerds excitedly tell you things like, "� is not a number!" and, "� - � does not equal zero!!!")

Third, from set theories. Which I'm not really qualified to discuss. In any case, this is where the "different infinities" come from, the idea that some infinities (e.g. the number of real numbers between 0 and 1) are larger than others (the number of natural numbers, 0, 1, 2, ....) This last part, the set theoretic definitions, are what cause people to loose sleep, tear out hair, etc. though in **some** senses these are also the 'simplest' definitions.

And there are probably more that I am either not thinking of or simply don't know about.

It's ok to post here even if I hadn't meant you. This is a free group.
By the way, yes, I mean Daniel Cunningham :). Lol.
1. Any reason to think that one of those definitions is the correct one?
2. Are there really different types of infinity?
3. Do you have a different definition?
4. Is the universe infinite? Physics tells us that something that is infinitely big doesn't exist. And even though string theorists have come out with multiple 'lalaland' (other people's words, not mine) to prove the existence of one dimensional strings to replace the problematic concept of zero-dimensional elementary point-like particles, both infinitely small and infinitely big 'things' haven't been proven to physically exist. And here comes the bullet: if they don't exist, is there really any valid way to describe infinity?
We can read about string theory here:

Andreas wrote:
1. Any reason to think that one of those definitions is the correct one?


Well, no... I guess it depends on your question. As a part of daily speech I suppose the first is perfectly adequate. The second is used in real analysis/calculus (and in fact is something of a formalization of the first.)

The third is a distinctly different concept from the second, to my understanding/thinking. The set cardinalities � and � are different beasts.

2. Are there really different types of infinity?

So there are at least two types --again, I am not an expert on this-- as I mentioned above: � and �. But within the "aleph infinities" there are different size infinities: the cardinality of natural numbers, �-naught, is smaller than the cardinality of reals, �-one (there is some argument about this, though it is technical and doesn't matter unless you really, really care about precisely which axioms you build your math from.)

The fact is any range of the reals, say 0-1, 1-pi, 17-555,491,494,012,9333 has cardinality �-one, the same as all reals; and this means there are more reals between each integer than there are total integers.

What I consider one of the most simple, beautiful proofs in all of mathematics is involved with this:

3. Do you have a different definition?

No. Nope. Not my area, but one I've dipped my toes into enough to have a healthy respect for the deep complexities I don't really understand. I'll let people who really think about this stuff much more than I be the ones to put forward different definitions.

4. Is the universe infinite?

What do you mean by infinite? It is not infinite in time into the past, so we think. But it may exist infinitely far into future. Infinite in space? But how measured? If you set out in a direction at maximum speed (c) and never get to "the end", is that infinite?

Physics tells us that something that is infinitely big doesn't exist.

It does? I mean, I get what you're (probably) saying, but you can have an open universe that expands indefinitely... perhaps at an ever increasing rate. Again, if you imagine trying to get anywhere in this universe (which may be ours!) you find that, past a certain expansion rate, e.g. light can't even make it between neighboring galaxies... then, eventually even between neighboring stars (or their cooling remnants.)

...to replace the problematic concept of zero-dimensional elementary point-like particles...

That is already replaced: by QM. You have probability waves/densities, not point particles flitting about. An even with point particles, there are still classical radii based on force interactions.

There is a philosophical objection to declaring theoretical 'artifacts' to be questionable, too. (This was a big deal to various philosophers throughout a lot of the 20th century; you probably are aware of this.) Suppose you have a theory that makes predictions that are confirmed out to 10 or 12 decimal places, the limit of your experimental precision. Suppose that theory rests upon e.g. point particles. The theory clearly works; it works incredibly well. On what basis, then, are you rejecting the existence of the point particle? Discomfort? Worse still, if you do reject it you now have two problems, not one: you still don't know what the fundamental 'stuff' is AND you now to have explain why the theory you don't believe is so incredibly accurate.

Personally, I have difficulties with the second and third concepts:
1. What does 'real' mean in 'real numbers'. Why are 'some' (how many are 'some') numbers called rational numbers and the others are called irrational numbers? Is the number system completely arbritary? Is it even valid? In other words, numbers and the decimal point in a number are an effort to make something exist out of nonexistent, and this system of making something exist out of the nonexistent will never be true (true is a philosophical term). So numbers are never true.
2. Please correct me if I'm totally inaccurate. The real number system can be considered a topological space. If we have two sets of real numbers, these sets must be homemorphic. If we have infinite sets of real numbers, all these sets must be homeomorphic. So why do we say that two sets are two sets? It is completely arbritary to say that there are different ways to establish the homeomorphism between two, three, four, infinite sets of real numbers as much as it is arbritary to say that there are different types of infinity. We made that up, didn't we?

So far, all my study conclude that existence doesn't exist and it is only consciousness that exists. I really want to be sure.

Ok. Math first, physics later. One by one :). I wonder what the consensus is, as of last year, QM predicts that there was no beginning of time:

Andreas wrote:
"1. What does 'real' mean in 'real numbers'. Why are 'some' (how many are 'some') numbers called rational numbers and the others ..."

Don't get tied up in names :) They are real because they aren't complex or imaginary numbers. They could just as well be called 'common numbers.'

Rational numbers are numbers that can be written as a ratio: 1/2, 7/8735, 3/3, and so on. They are 'ratio'nal numbers.

Irrational numbers are simply numbers that can't be written as ratios, pi and e being the most famous examples.

There are other divisions/classifications as well: algebraic and transcendental numbers are an example (algebraic being numbers that are root of a polynomial, transcendental numbers are not the root of any polynomial.)

There are infinitely more irrational numbers than there are rational numbers: in fact rational numbers have the same cardinality as natural numbers!

Andreas wrote:
"...numbers and the decimal point in a number are an effort to make something exist out of nonexistent, and this system of making something exist out of the nonexistent will never be true (true is a philosophical term). So numbers are never true."

To be honest, I don't know what to make of this statement. Does "3" exist? I own three pair of glasses. I have 3 pen's sitting on my desk right now. My last name starts with the third letter of the English alphabet. But none of those things is "3". Neither is "3", "three", "tres", or "..." But does "3" exist? I don't know. I probably don't care.

As far as a number being true... you've really lost me. Can "3" be true? Can a "car" be true? Can "yellow" be true? Whether or not I have 3 pens on my desk, a car in my bathtub, or yellow hair... these are all propositions with truth values. But not "3", "car", or "yellow". Or "numbers."

Andreas wrote:
"2. Please correct me if I'm totally inaccurate. The real number system can be considered a topological space."


I've not studied topology; however, I do actually start a course in topology in a couple of weeks, so I can get back to you a few months from now :>)


If we have two sets of real numbers, these sets must be homemorphic.


No. Not all sets. {1,2} is not homeomorphic to R. Homeomorphisms are built on top of isomorphisms (which come from algebra, a subject I have studied.) Both {1,2} and R are sets of real numbers (with R just being the entire set), but clearly these are not isomorphic.

Now the set (0,1) is isomorphic to R.


If we have infinite sets of real numbers, all these sets must be homeomorphic. So why do we say that two sets are two sets?


Yes and no. No in that I can have a set of real numbers consisting on of integer-valued numbers; this is still infinite, but no longer isomorphic with the reals.

Yes! There are an infinite number of subranges (sets) of the real numbers that are isomorphic to the whole set R. And to each other.

Welcome to transfinite math. It'll give you a headache.


It is completely arbritary to say that there are different ways to establish the homeomorphism...


So, in math, you will here lots of statements like " up to ", including "A is equivalent to B up to isomorphism", or, more likely "A and B are isomorphic". All this means is that A can be completely mapped to B and vice versa. If I have a set of 10 colored stickers and I'm moving, I can color-code the rooms of my house and label the boxes with those colors. Then I have created an isomorphism between sticker colors and rooms. What this means is that, as long as I have that mapping in hand, I can talk about the colors rather than the rooms. More directly, I can sort boxes into piles by color, and I have sorted boxes by rooms... because they are isomorphic. So then for classes of operations like e.g. sorting, under the isomorphism, the colors and rooms become interchangeable... they are, in this sense, now the same.

Now, of course, no one is confused that the color is the room. Nor was I required to use any specific color coding (though some were disallowed; I couldn't just label everything green.)

In the same way, I can map numbers to numbers. (I could have mapped colors to colors, or --perhaps with some slightly greater difficulty-- rooms to rooms.) Now: if I have a set of stickers and a set a rooms, I can say one of three things: I have more stickers, the same number of stickers, or fewer stickers than I do rooms.

THAT is what isomorphisms allow in the case of comparing infinities. I don't need to be able to put a number (whatever those are), I just create a special kind of mapping called an isomorphism and then see which set I run out of first. There might be many isomorphisms, but it doesn't matter (who cares if orange maps to kitchen or bathroom; as long as orange maps to one and no other room, and vice versa. Don't like orange? Pick green. No matter.) The mapping IS arbitrary, but the *properties* of the mapping (one-to-one and onto... aka, it being a bijection) are not arbitrary.

In this since it is not arbitrary to say there are different infinities; in fact, you are lead inexorably to the conclusion that transfinite "numbers" exist that are of different size, all by these simple counting exercises.

If you are really interested in this stuff, there are perfectly good introductory textbooks that cover the basics without getting into advanced topics too quickly. I don't know that they are the greatest books ever, but I have used:

Elementary Analysis: The Theory of Calculus (To understand the symbol �)
A First Course in Abstract Algebra (Homomorphisms, Isomorphisms... and to crush your will to live :>)
and
(Set theory/Different Infinities)
(Set theory/Different Infinities)

That really helps.
Thanks, Dan.
I will look for the books tomorrow.

But:
Of course {1,2} is not homeomorphic to R. {1,2} is a definite set which can be written because {1,2} is {1,2}. In mathematics, a set is a collection of distinct objects. Now R is not distinct, a collection of R is infinity and infinity can't be defined to any numbers, we can't separate R into sets of rational numbers and irrational numbers and then separate the rational numbers into integers, whole numbers, and natural numbers and then define what infinity means in each of these sets, because infinity is R itself and R doesn't have any value thus doesn't exist. What happens is: we separate R into sets of rational and irrational numbers and so on and define infinity in each sets, this is why we come up with different concepts of infinity. Or can we really define infinity using this approach? Is this approach valid -- or as I said earlier -- only something we made up? (I need guidance)

The implication: do we really think that there was definite amount of time in the past (i.e. big bang), or is this also something we made up because infinity doesn't exist (therefore existence doesn't exist) and we can't deal with the concept of infinity? We come up with the concepts: big bang theory, multiverse collisions, and the existence of strings and point particles. What law can confirm the existence of these concepts? We've never seen these concepts, they don't exist, we only know it, and it IS because we are conscious.

On the other hand, Boltzmann's brains don't exist floating in space (that we know of) so the fact that we are gathered on Earth is a big question. Evolution also happens as a process that is governed by nature (pretty random). So if consciousness doesn't govern the universe, and the universe doesn't exist if consciousness doesn't exist. W.H.Y.?

Andreas wrote: "In mathematics, a set is a collection of distinct objects. Now R is not distinct, a collection of R is infinity and infinity can't be defined to any numbers, we can't separate R into sets of rational numbers and irrational numbers and then separate the rational numbers into integers, whole numbers, and natural numbers and then..."

A set is a collection of distinct objects, true. And we most easily think of this (sticking to math) as the set of integers; or even more simply, the natural numbers: 1, 2, 3, 4...

But you've already run into (an) infinity here: Just start counting, and let me know when you've named all the numbers :) (And, actually, here we again come to isomorphisms, since I've just asked you to create a one-to-one mapping between "numbers" and "names we call numbers.")

So the set of natural numbers, innocent, distinct, every-day counting numbers, to apply your phrase, "is infinity." Now I can take this set and divide it into evens and odds. Again, nothing tricky here; not continuum hypothesis, no density of the rationals here, etc. Just good old even and odd numbers. Count those off for me :) Again, for *both sets*, you're run right into infinity. To use your phrase again, these sets "are infinity".

I see at least a problem in phrasing in your statement: "...a collection of R is infinity and infinity can't be defined to any numbers..." No one is trying to define a **number** called infinity. And no one is saying the the **cardinality** of the reals is a number. (This is why separate terminology, hard as it is to maintain in casual conversation, really is important.)

Think about the examples above with the natural numbers and see if it makes the discussion of the reals any more clear. Reals are more dense than integers; the cardinality of the reals really is bigger than that of the integers. But the issue of dividing sets, infinities being inside other infinities, etc. happens long before you get to reals.

Andreas wrote: "...because infinity is R itself and R doesn't have any value thus doesn't exist."

Infinity does not equal R. First off, you're comparing apples to oranges. R is a set. Infinity is a measure of something's size (we should be talking about aleph's here, but we'll keep it simple for the moment: infinity.) Saying "infinity is R" is like saying "10 is fingers." You can parse a kind of sense out of it, but the statement is really just wrong. I do have 10 fingers, but "10 is fingers" is a nonsensical statement.

Andreas wrote: "do we really think that there was definite amount of time in the past"

Yes. This was the battle fought with the steady-state theorists throughout the 1900's. Hubble and his descendants won. Everything we now know points toward a beginning of the universe, roughly 13.5 billion years ago. Infinity never even came into that discussion (except with the steady state folks, who, out of some sense of unity/preference/etc. wanted the universe to be infinite in age.)

Andreas wrote: "infinity doesn't exist (therefore existence doesn't exist)"

???

Andreas wrote: "We come up with the concepts: big bang theory, multiverse collisions, and the existence of strings and point particles. What law can confirm the existence of these concepts?"

We don't have laws that confirm theories. We have to devise the theories (which then, at times, get elevated to become 'laws') and then... well, then you wait. If it survives, awesome; but you still have to wait, because something might disprove it. That's why science is rough on the psyche: you don't ever get to know, and, boy, people really love knowing.

Andreas wrote: "universe doesn't exist if consciousness doesn't exist"

You've utterly lost me.

I replied a pm and my browser closed, taking away all I've written (sigh).
Yes, it was my mistake, you're right, infinity does not equal R. But I just have to phrase it more accurately: R is infinite in size.
R, which means all numbers, is infinite in size, so if we can't prove that infinity exists, R, and hence all numbers, doesn't exist. What exist is mathematical ways to define infinity, to define R. Some examples:
In order to define the infinite size of R we can create certain mathematical ways: the Thomson's Lamp thought experiment where infinity can be defined as either the lamp being on, off, or both; an infinite series of odd numbers where the sum of this series can be defined to (�+1)square; the concept of potential infinity in a Turing machine; etc.
None of these is about the existence of the real infinity, thus the existence of R. Infinity doesn't exist, what exists are mathematical ways to define infinity.

First question: can we, with this premise 'infinity doesn't exist, what exists are mathematical ways to define infinity', conclude that infinity doesn't exist?
Second question: Is the universe infinite?
If the universe is infinite, and something that is infinite doesn't exist, existence doesn't exist, it is only we/life (and hence the mathematical ways to define existence) that exists. In other words: there is no 'truth', only mathematical ways to define truth; and there is no beginning of time, there is no everything.
If the universe is infinite: everything becomes not true without life.

"R, which means all numbers..."
No. It means real numbers. It does not contain e.g. imaginary numbers or complex numbers.

"is infinite in size"
Has cardinality aleph-1.

"so if we can't prove that infinity exists"
Prove 3 exists.

"R, and hence all numbers, doesn't exist"
Nonsense. Pure nonsense. If I can't prove e.g. a fact about the class "horses", horses continue to exist. Moreover, why choose this fact about the cardinality to pin the existence of reals on? There are lots of statements about the reals that can't (or haven't) been proven, yet 1, 2, pi and 7/sqrt(5) continue to "exist."

"What exist is mathematical ways to define infinity, to define R."
Yes... but this is like saying there are musical ways to define scales. You seem to be after some Platonic ideal infinity or something similar; I can't help you there.

"Thomson's Lamp thought experiment"
Had to look this one up. Philosophers... sheesh. This isn't a paradox, and the thought experiment isn't even well framed. I invite you to learn why (the Elementary analysis book I suggested above will be helpful.)

These other things also don't define infinity; they each *use* a concept that can be called infinity.

"First question: can we, with this premise 'infinity doesn't exist, what exists are mathematical ways to define infinity', conclude that infinity doesn't exist?"
You've stated a tautology.

"If the universe is infinite, and something that is infinite doesn't exist, existence doesn't exist, it is only we/life (and hence the mathematical ways to define existence) that exists. In other words: there is no 'truth', only mathematical ways to define truth; and there is no beginning of time, there is no everything.
If the universe is infinite: everything becomes not true without life."
This is pure woo. I really don't know what to do with this.

Even if you had shown that there are not an infinite number of reals (which you haven't) how does this prove that, in general, infinities don't exist? And what would this have to do with the universe existing? (At a minimum, you've made a logical blunder here: "If infinities don't exist, and the universe is infinite, then the universe doesn't exist." ...but your initial assertion is falsified by the infinite universe existing. Your argument defeats itself.)

If you are interested in understanding infinities, you are going to have to do some work; I recommended some books, though there may be slightly shorter paths. A lot of thought during the last 100 years especially, the last 300+ more generally, has gone into infinities/transfinites/infinitesimals. You should devote some solid time to understanding that work --by some of the smartest people of the last several centuries-- before heading off into uninformed philosophical territory.

Now, I have a crying baby to take care of and I should be studying for the PGRE anyway!

I hope everything is ok, Dan.

First, you didn't answer my questions.

Second:
Daniel wrote: ""R, which means all numbers..."
No. It means real numbers. It does not contain e.g. imaginary numbers or complex numbers "
I totally forgot about the imaginary numbers and complex numbers. This just makes things worse �.

Daniel wrote: "Nonsense. Pure nonsense."
No, it is not. 1, 2, pi and 7/sqrt(5) exist because they are not infinity, they are some of those mathematical ways to define infinity, and do mathematicians love to say 'I found infinity' just as religious people say 'I found the black cat in the black room' without using any flashlight. Oh this is a good illustration: Have we tried to find infinity using a flashlight? Did we find it?

Daniel wrote: "but this is like saying there are musical ways to define scales"
No. Scales are not infinity, scales are mathematical ways to define infinity. So 'there are musical ways to define scales' means 'there are musical ways to define mathematical ways to define infinity.' Btw, I love musics, they are absolutely amazing. I teach music since college.

Daniel wrote: "This is pure woo. I really don't know what to do with this."
No, it is not Dan :). Quantum physics and the general theory of relativity both have to deal with infinity. They are forced to deal with it and they CAN'T deal with it. In quantum fields theory, we have vacuum energy and these renormalizations, they had to come up with string theories, we can't call string theories woo, can we? It is infinity. Even though we will never see whether the universe is infinitely 'large', we are forced to deal with the infinitely small in experiments, and so far there is no correct way to explain it, none, zero. And Einstein's general theory of relativity also predicts infinity, where the theory goes wrong.

I refuse religious explanations as much as I refuse the mathematical axiom that infinity must exist. Mathematicians have this axiom in their mind that infinity must exist. I'm sorry, this axiom is so not scientific that it's not even wrong.

Infinity doesn't exist. Now, the suspense is waiting for someone to be able to make us see that the universe is finite.