Perhaps some consensus can be arrived at in regard to what infinity is. After that, perhaps its nature can then be discussed.

One approach to defining infinity is to first define what finite means and then say something is infinite if it is not finite. Rather than define infinity by what it isn't, let's try to define it by what it is.

This definition is intentionally vague, in part because it has precise definitions in math. The sort of thing I want to discuss is what isn't covered in math. A definition of infinity that isn't in terms of what infinity isn't will be motivated by what we think its nature is, will dictate what its nature is, or both.

So let's give it a go...

Infinity is a quality or quantity for which it is possible to be reduced in a way that the reduction is, in some sense, equivalent to the original.

What a reduction is and what it means to be equivalent is, of course, crucial. A particular example of infinity would be an infinite set which is infinite if reduction means removing a single element of the set and two sets are equivalent if there is a one-to-one correspondence between them (i.e., there is a bijection between them).

What it means to be finite could then be a quality of quantity that is not infinite. For example, if reduction means subtraction and equivalence is taken to be equality, no counting number has the quality of infinity since, when reduced, no counting number is equal to the original.

Then perhaps we can answer some basic questions such as is there anything in the universe (or is the universe itself) infinite?

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If anything is possible, then the statement, "anything is possible," is possibly false.

Great thread and thoughts M.T.

What can be observed is limited and therefore finit. Does the universe really exist, if it's not observed? Or do we create it the minute we look at it/for it?

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What is infinite is, by definition, limitless. How can you discuss something that is limitless? How can you differentiate the whole and make it comprehensible?

I am limited; I cannot imagine something limitless. Not being able to imagine it, how can I discuss it in any intelligible way.



Only partially true. You can, sometimes, define things by defining the opposite; Dark, for example, is defined as the absence of light, and cold as the absence of heat. But this is pretty meaningless when applied to infinity.

What is finite has limits, therefore what is infinite has no limits. And once again, we are left with something that cannot be reasonably discussed.

Inifinity is a cipher.



Which seems to suggest that you define infinity as something that, if you subtracted infinity from it, it would still be infinite. That sounds, to me, like a simply undefined term being reiterated. I do not find it illuminating...

Even iumagining the finitudes of the cosmos is daunting. I use to have a scale model of the Earth and moon hanging from the ceiling in my class room. The distance between them was proportionally correct.

When students noticed it, I would explain it. Sooner or later, one would ask me where the Sun would be, in this model. I would take them to the window and point to a building about half a kilometer away. That's where the sun would be.

On the same scale, the nearest star would be 268,000 times more distant, almost half way to the Moon. (I have some idea of how far away the Moon is; I once owned a car for 10 years, and put about 370,000 miles on it -- roughly equivalent to a trip to the Moon, and half way back (I drive a lot, but that still took a long time!)).

But even that distance is almost incomprehensible, yet scarecly moves us off our side of the street, astronomcially speaking. We are still a long way from coming to terms with even galactic scales.

Andf the universe contains perhaps a couple of billion galaxies, sepearted by areas of void so vast as to beggar description.

No. I cannot imagine inifinity. I doubt that anyone can, in any meaningful (or at least communicable) way.

"Be seeing you..."

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Grassaf, Eolas

It's not a circular definition. It would be circular if it was, "infinity is a quality or quantity for which it is possible to be reduced in a way that if the reduction is done infinitely many times, the reduction is, in some sense, equivalent to the original." One way to interpret my definition is that the reduction is possibly very small and done only once. If after the reduction the reduced version is, in some sense, equivalent to the original, then the original is infinite. If "reduction" is subtraction and "equivalent" means numerically equal, then no mathematically-finite integer is infinite since after subtracting one (doing a reduction) it is not equal (what equivalent is in this case) to the original. That's a good sign since whatever the broad definition of infinite is, no mathematically finite thing should be infinite.

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If anything is possible, then the statement, "anything is possible," is possibly false.

I said it was unillumintaing. And so it is, to me, anyway.

I understood what your words implied, I simply don't think it tells us much. This is why mathematicians say that operations involving infinity are impossible (or "undefined" to use the precise term).

Whether the reduction is small, or large, without being able to quanitfy (in any meaningful way), what the original term is, the exercise is pointless. Certainly, if I subtract 3 from inifnity I suppose it might well still be infinity...but since I did not know what infinity was to begin with, how am I more informed?

We cannot quantify infinity because it is not a limited quantity; and a limitless quantity is so far beyond human experience as to be meaningless.

Similarly, we cannot limit the universe because it is boundless by definition....and, as I tried to suggest, even our best attempts at understanding realtively small distances (such as the "real" distance to the nearest star.. a celestial hop, skip and a jump) founder miserably. Saying it is 4.2 Light Years is meanignless, since I have never travelled at the speed of light, neither have you; nor are either of us terribly likely to do so.


"Be seeing you..."

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Grassaf, Eolas

You didn't say it was circular but your rephrasing of the definition was circular.

Do you find the definition "infinity is that which is not finite*" illuminating?



*bounded, limited,...

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If anything is possible, then the statement, "anything is possible," is possibly false.

Not at all.

Let's discuss meanignful, as it applies to limits.

What is 100?

That's not too hard, we have all counted to 100, probably many times. Some people can look at a group of things and estimate 100 pretty easily -- it's a skill that cops and reporters use quite a bit. It's easy to grasp and think about 100 of anything.


What is 1,000?

Well, you may not have counted to 1000 -- most people get bored or lose track somewhere around 600 or so -- but you have certainly added things up to 1000, and likely more. You can conceptualize it; an array that is 31 by 32 will (almost) add up to 1000. I can see that im my mind's eye and grasp it easily enough. If you have ever been to a large sporting event or concert, you have see thousands of people, and subdividing that mob into smaller units of a thousand isn't hard.

What is 1,000,000?

That's a bit harder. You may well do math that results in a million, but I know no one who has counted to a million, even by installments. My son would dearly love me to count to a million for him, but doesn't clearly understand that I do not have the time. I can't really form a picture of a million in my mind; I can't conceptualize it except with extreme difficulty. I can look at some things and say "Oh, there must be a million there," but I have no idea if that is correct or not. I have never seen a crowd of one million, and even though I live in a city of more than 2.7 million, that pretty much doesn't "mean" anything beyond bare numerical reference. An array 1000 units by 100 units will give you a million, but I would have to trust you that there were a thousand rows and columns, due to the diffiuclty of counting that many and not losing track.


What is 1,000,000,000?

A Billion (or Milliard if you prefer) is much harder to grasp even than a million. Some people may have counted to a million, but I rest assured that no one has ever counted to a billion, and no one likely ever will. I cannot even remotely conceptulaize a billion; even looking at sand on a beach, or piles of salt or stars doesn't help. The concept of "individual things" gives way and the focus of the mind falters; I can see the beach, perfectly well..but not a billion grains of sand. An array 31,622 by 31623 will give you almost a billion, but I can't really imagine creating such a thing. Where would you put it? How would you find the time? My son is very dissapointed when I tell him that, even if he tried to count to a billion, he would never be able to do it. (He couldn't, either. Even if he was to count one number a second, 12 hours a day, 365 days a year, it would take 63 years to count that far...and since he is 7 now, he would be 70 when he finished...and that is pretty much leaving out doing anything elese in life, such as having an education, or a job, or a family.)

A billion is, essentially, an incomprehensible number. That there are people, like Bill Gates, with multi billions of dollars in wealth, or countries with a billion or more populations, doesn't help. We have left the realm of things graspable to the human mind here. Any larger numbers, a trillion, a sextillion, a google...all are incomprehensible. They are "empty" words, for all theat they may have mathematical meaning.

So, to make a long story short, even fairly finite things are totally impossible to grasp in any real way.

Size is also something the human mind boggles over. My son is still struggling with the fact that there are larger units than cities. He has travelled by car from Ontario to Florida twice, and from Ontario to Newfoundland once, but, even so, the fact that we have left his native place eludes him.

He isn't alone in that. If you converse with someone from Europe, you will often be asked, "Oh you live in Canada; you must know Bill who lives in ______." (Fill in the name of any distant city you please.) Or, should someone be visiting, they may well say something like "Well, while we are visiting Chicago, can we pop over and visit the Grand Canyon for the day."

It isn't just uneducated people who say things like that, by the way. People tend to judge everything by what they are familiar with, and the distances of North America are pretty much menaingless to people familiar with much smaller scale geographic units. But even if you are used to large spaces and long distances, you may still not really grasp such things; after all, people routinely underestimate the time it will take them to do things like drive across country.

And from one side of Canada to the other is 7228 km, or about 4336 miles. Thats about 18% of the distance around the equator.... Can I conceive that? Not really. I have certainly driven a distance equivalent to 25 times around the equator, but that is over 30 years of driving expereince. And we are still not at a million miles.... its hard to really get a grasp of that.

As I said earlier, longer distances becopme increasingly hard to imagine, even though they are quite finite.

So, no, your "definition" is, at a very fundamental level, unilluminating.

"Be seeing you..."

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Grassaf, Eolas

btw, I have trouble with even very small quantities like the square root of 2. I can't really wrap my brain around much over 100. One time I printed out 6 million numbers or so on about a dozen pages to give me a sense of how much 6 million is.

Saying infinity is nothing like any of those finite numbers is illuminating, imo, because it's saying that it's somehow fundamentally harder to imagine than every finite number...but not very illuminating, fundamentally because it's a definition in terms of what it's NOT. So in that sense, I think my definition is better than that since it's in terms of what's possible rather than in terms of what infinity isn't.

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If anything is possible, then the statement, "anything is possible," is possibly false.

BTW, we cannot quantify MOST finite numbers either so the ability to quantify it is irrelevant.

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peace for all
love for all
bliss for all

...may your journey be graceful...

If anything is possible, then the statement, "anything is possible," is possibly false.

I think a better way to understand the natur of infinity is to think of it as intangible. If you can define it, it ain't it.

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