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The Potential Infinity Contradiction

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While having a conversation about space and time with my 5 year old little boy, he made a statement I found myself unable to disagree with, but I believe rejects a premise of Objectivism (and possibly of Aristotle's). My son has always been fascinated with oxymorons and likes to point them out whenever he notices one (He did this when I once told him that going to his room was his "only choice," so he'd better do it. "That's an oxymoron," he said "choice means more than one".)

So, this morning in our car ride to school, during a conversation about the infinite and finite nature of things, I made the statement to him that "Infinity suggests a potential but not an actual." When I asked him if that made sense he said it did, then he said, "Actually, no." He understands the word potential roughly as something that "could be" so he said something to the effect of, "[You said infinity can't ever be real (actual), so then how can it be a potential?] That's an oxymoron!". Basically, a potential to be what?

My struggled response was that infinity is an "idea" (concept) and that can be imagined but cannot ever be real. He says, "Like God?" (lol... whole different subject). In any event, we ended the conversation with him hopping out of the car for school and saying essentially, "[if it can't be then you can't say it is a potential]".

So I'm stuck trying to explain how infinity can never exist but can still be considered a "potential". Any help would be appreciated on that.

Two references:

An arithmetical sequence extends into infinity, without implying that infinity actually exists; such extension means only that whatever number of units does exist, it is to be included in the same sequence.

-Introduction to Objectivist Epistemology, 22.

There is a use of [the concept] “infinity” which is valid, as Aristotle observed, and that is the mathematical use. It is valid only when used to indicate a potentiality, never an actuality. Take the number series as an example. You can say it is infinite in the sense that, no matter how many numbers you count, there is always another number. You can always keep on counting; there’s no end. In that sense it is infinite—as a potential. But notice that, actually, however many numbers you count, wherever you stop, you only reached that point, you only got so far . . . . That’s Aristotle’s point that the actual is always finite. Infinity exists only in the form of the ability of certain series to be extended indefinitely; but however much they are extended, in actual fact, wherever you stop it is finite.

-Leonard Peikoff, “The Philosophy of Objectivism” | lecture series (1976), Lecture 3.

What Rand wrote in ITOE is perfectly logical and correct. I don't think it follows to call infinity a potential though.

_______

As a corollary issue, can someone help me out with a seeming contradiction in the Objectivist's definition of the Universe, versus the dictionary definition?

Peikoff says (in The Philosophy of Objectivism lecture):

"
The universe is the total of that which exists—not merely the earth or the stars or the galaxies, but everything."

This allows it to logically follow that there is not (and can never be) *anything* outside the universe. True, if that is the definition.

:

universe
- the totality of
known or supposed
objects and phenomena throughout space; the cosmos; macrocosm.

Webster's Definition
:

universe
- 1. the whole body of things and phenomena
observed or postulated

What the dictionary does (as Objectivism normally does) is leave out the arbitrary. The "unknown unknowns" are outside of, or not part of, the universe until the moment that they are postulated or known about (according to the dictionaries).

The issue with Peikoff's definition that I'm wondering about is that he is setting up one of two possible contradictions:

a.) He is considering the arbitrary and all potentials inside his definition of universe so as to be able to say that "Everything which exists is finite, including the universe."

~or~

b.) He is NOT considering the arbitrary, unknown, or yet-to-be-postulated as part of what would then be an "evolving" definition of an expanding, and then necessarily, potentially INFINITE universe. (Which also contradicts the statement that "Everything which exists is finite, including the universe.")

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So I'm stuck trying to explain how infinity can never exist but can still be considered a "potential". Any help would be appreciated on that.

I can see how, in this context, potential would be a bad word to describe infinity. Infinity is a concept of method, like the square root of -1. There is no such number (in the ordinary number system as I understand it, I don't want to get in a huge argument about some of the weird flavors of math out there, I GET it) that you can multiply it by itself and get -1. You just can't do it. But you can throw in a little placeholder, i, and still perform calculations as if there WERE such a number and get a useful answer provided you do something that gets rid of the i.

Infinity and i don't describe anything that can actually exist. They are used as conceptual placeholders.

The term "universe" is similar to this. When it is used to mean "everything" (which is the literal meaning of the term), what you are in effect saying is "I don't KNOW everything that exists, because my knowledge is finite, but whatever DOES exists, gets filed in this category". There is no conflict with the arbitrary or the "potentially" infinite, because the things that DO exist are NOT DEPENDENT UPON CONSCIOUSNESS OR KNOWLEDGE.

So, really, your problem here boils down to a small failure to really understand the idea of the primacy of existence. The concepts we use to describe reality (and the various mental file-folders we slot things in) don't determine reality and mean that reality winds up with some sort of contrary nature where it does and doesn't include things that may only exist inside our heads. A is A. Our method of conceptualizing has nothing to do with it.

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Freestyle, good post. Your suspicion about infinity not being a potential is correct -- it isn't. Infinity is a pure, albeit extremely useful, abstraction. It has no physical form -- none. One has to be careful not to get hung up on a particular instance of a word. When you see a puzzling usage, you have to ask, "Wait, what is the context in which this is being used?" In Peikoff's quote, the "it" that he's saying is potential is not infinity as such, but merely the fact that there are always other numbers to be potentially counted that you haven't enumerated yet. In 1, 2, 3, 4... "5" and everything else can be thought of as a potential, a potential quantity in your list of numbers. But infinity is a different concept. It is "endlessness". Its quantity will never be named; it has none. The thing to remember is that the concept "potential" is only meaningful with respect to some thing that actually could exist. Clearly, a physical infinity is impossible. To be is to be something.

As to existence, that's another understandable miscomprehension. "Existence" is all that exists, whether man knows about particular far-off or subatomic existents or not. If it is, it's included in that biggest of all things, existence. It doesn't matter whether man has discovered it or not. Man is saying, "There are things out there. Things are. I don't know all the things that exist, and that's demonstrated by the fact that man regularly discovers new existents. I want a concept that stands for everything that is, known or unknown. There's no problem of appealing to the arbitrary, because I'm not saying anything about anything that is unknown to man. I'm simply saying that for anything that is unknown to man, if it exists, then it exists in existence. I'll talk about it itself when it gets discovered."

That could only be arbitrary if you extended your statement to saying something about the unknown, such as its attributes, its location, etc. Observe that you're not talking about a particular unknown as if it existed. That would be arbitrary. No, you're just saying that everything that is, is in existence, with the implicit recognition that man's knowledge doesn't encompass every existent; that claim is for a primacy of consciousness.

Lastly, the universe is infinite only with respect to its spatial boundaries. There is nothing outside or apart from the universe. But in the context of the universe being the sum of all physical objects (entities), that is most definitely finite.

You have one hell of a sharp five-year-old. Your conversations must be a blast!

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I can see how, in this context, potential would be a bad word to describe infinity. Infinity is a concept of method, like the square root of -1. There is no such number (in the ordinary number system as I understand it, I don't want to get in a huge argument about some of the weird flavors of math out there, I GET it) that you can multiply it by itself and get -1. You just can't do it. But you can throw in a little placeholder, i, and still perform calculations as if there WERE such a number and get a useful answer provided you do something that gets rid of the i.

Infinity and i don't describe anything that can actually exist. They are used as conceptual placeholders.

Every so-called real number other than integers or ratios of integers are also "place holders". A non-rational real number is the limit of a sequence of rational numbers. It does not exist except as a conceptual entity, namely the limit of an infinite (non-terminating) Cauchy sequence.

You are taking the position argued by Leopold Kronecker in the late 19th century when he argued that Cantor's theory of transfinite numbers was nonsense. Kronecker made the famous rhetorical declaration ---- God invented the integers, the rest is the work of Man ---. Kronecker would have no truck with irrational numbers. Fortunately for the future of physics, most mathematicians simply ignored what Kronecker had to say on this particular issue.

David Hilbert the greatest mathematician of the late 19th and early 20th century had a similar dispute with the Dutch mathematician L.E.J. Brouwer. Brouwer would not permit any kind of mathematical object that could not be finitely constructed from the integers. Hilbert's rhetorical rant to Brouwer's position was --- From the Paradise created for us by Cantor no one shall expel us. ----

Again, most mathematicians simply ignored Brouwer's restrictions and mathematics went on to bigger and better things.

Bottom line: Mathematicians went on, pretending that these non-existent things existed and produced the tools needed by our most advanced theories of physics.

Most mathematicians will deal with any system of objects that does not lead to logical paradoxes. They rarely pay attention to philosophical arguments. Why? They are too busy proving theorems.

Bob Kolker

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"Infinity suggests a potential but not an actual."

That's not quite what Peikoff is saying. He is saying that infinite is a valid concept, in the context of mathematics, when it is used to indicate a potentiality. Is it not true that no matter how high we count, we can always count higher? Yes, it is. Well then, we need a word that designates a series as "extendable indefinitely". That word is infinite. However, an actual collection of numbers is not infinite, only the series (which does not exist in reality, there's no such thing as 1, 2, 3, 4,...,n,..., infinite. apples (where the 1st group contains an apple, and the n-th group contains n apples, etc.), for instance. There is such a thing as 1, 2, 3, ..., n. apples.

The same is true if instead of apples you use existents, and each element in the series is the total number of existents in an expanding Universe, at a given time. At each time you look at the Universe, you will have n number of existents, and you'll be able to end your increasing series, which denotes something that exists in actuality, with that n. You won't have n+1 existents until the next instance, when you can do a new count, and you'll never have infinite number of existents. There will always be a number: in fact that's what the word infinite means, when applied to the set of natural numbers. So, by definition, there will always be a number large enough to end your series denoting the number of existents with, no matter when it is that you look at your expanding Universe.

What the dictionary does (as Objectivism normally does) is leave out the arbitrary.

No, the dictionary leaves out the unknown. Objectivism never does that, we use concepts to refer to all existents in a category. If we say birds, for instance, we mean all birds, including all the ones no one's ever seen. When we mean Universe, we mean everything, including all the things no one knows about. The simple acknowledgment that some of the things that exist are unknown is not an arbitrary statement. It is backed up by the fact that new things are discovered every day.

Besides, Webster Online does have other meanings to the word, including something to the effect of "the entirety of the celestial cosmos", so they're not necessarily in disagreement.

potentially INFINITE universe

A common misconception is that infinite is a number, in math (a very large number). It is not. It's just something we can say about a set defined in a certain way (like a series). Of course, there are different kinds of "infinite". (for instance real numbers aren't infinite the way natural numbers are, they are "more" infinite)

Size, on the other hand (such as the size of a planet, or a galaxy, or this thing that's supposedly expanding after the Big Bang, and about we can't say whether it is everything there is or not, since we can't even see its edge), is measured in numbers. So you definitely aren't saying anything when using "infinite" as an adjective ahead of something other than a set, in the context of mathematics. It's not a valid concept outside of math. Obviously, various infinite sets do have a relationship with reality, but that is the only link the concept "infinite" has to the Universe. The Universe is not, and cannot become, infinite any more than it can become reptilian. The concept does not apply.

You should try to explain infinite to your son precisely by looking at natural numbers, and a few more types of infinite series (such as the series 2, 4, 6, ..., and 1, 5, 10, 15,...). He won't fully understand until he understand concept formation first, but at least he won't think infinite is something it is not. (such as a number) But, above all else, you should explain infinity to your son in the context of math, and never use it (or allow him to use it) in any other context, except math. You can't say "infinity indicates a potential", and leave it at that. Infinity indicates a potential in the context of math, when applied to something specific.

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Infinity is a concept of method, like the square root of -1. There is no such number (in the ordinary number system as I understand it, I don't want to get in a huge argument about some of the weird flavors of math out there, I GET it) that you can multiply it by itself and get -1. You just can't do it. But you can throw in a little placeholder, i, and still perform calculations as if there WERE such a number and get a useful answer provided you do something that gets rid of the i.
That explanation, at best, ignores the context of 20th century mathematics.

It is correct, as you alluded, that in the real number system, there is no number x such that x*x = -1.

However, in the predominant modern mathematical context, the number i, of the complex number system, is not a "placeholder" that is gotten "rid" of. In the most common method of "constructing" complex numbers from real numbers, we take complex numbers to be ordered pairs of real numbers, then we define various operations (including a multiplication operation) on ordered pairs of real numbers. It turns out that in complex multiplication <0 1> times <0 1> is <-1 0> (which we may call '-1' since the real number -1 is embedded in the complex numbers as <-1 0>).

Reference, e.g., Rudin's 'Principles Of Mathematical Analysis'.

As to potential vs. actual infinity, in context of the mention of infinite sequences:

In classical mathematics (the by far most common and popular approach), a sequence is a function and every function has a domain, so the domain of an infinite sequence is an (actual) infinite set, and all as follows from the set theoretic axioms.

However, there are certain constructivist (the less common, though much informative, approach) understandings by which the infinitude of a sequence may be understood without reference to an actually infinite domain, but rather in a scenario in which an ("ideal") mathematician (call him an 'agent') may at any given finite point choose a next term in the sequence.

How any of this does or does not line up with Objectivist notions of 'actual', 'potential' and 'concept of method' is another matter.

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an actual collection of numbers is not infinite, only the series
In ordinary mathematics (even as informal as Calculus 1), a series is a certain kind of sequence. So, since for the purposes of this discussion we don't need to complicate by referrring specifically to series when we can as easily refer to sequences in general, my question for you is what do you take to be the domain of an infinite sequence? Or, similarly, what do you take to be the domain of a function on real numbers such as the function f where, for any real number x, we have f(x)=x^2? I would think the answers would be: the set of natural numbers and the set of real numbers, respectively. But both those sets are infinite. That is just Calculus 1 (even high school algebra, for that matter). If not, then what is your own rigorous account of such functions? Edited by Schmarksvillian
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In ordinary mathematics (even as informal as Calculus 1), a series is a certain kind of sequence. So, since for the purposes of this discussion we don't need to complicate by referrring specifically to series when we can as easily refer to sequences in general, my question for you is what do you take to be the domain of an infinite sequence? Or, similarly, what do you take to be the domain of a function on real numbers such as the function f where, for any real number x, we have f(x)=x^2? I would think the answers would be: the set of natural numbers and the set of real numbers, respectively. But both those sets are infinite.

Yes, they are infinite, that is the definition, that is how the term is used, in math. They just happen to not be an actual collection of numbers. Both the set of natural numbers and the set of real numbers are theoretical devices, only used in the context of mathematics. The word actual refers to reality, not math theory. An actual collection of natural numbers would always have a number of elements. It would be nonsensical to call it infinite, in actuality, as opposed to in the context of math theory.

the domain of an infinite sequence is an (actual) infinite set, and all as follows from the set theoretic axioms.

You can call something actual all you want, because it follows from mathematical axioms. You'll just be ignoring the context of our discussion, in favor of a speech on irrelevant aspects of mathematics. I think there is a math subsection on this forum, but this isn't it. This is definitely the philosophy subsection, and this thread is a discussion on the actual actual, not the actual math, that's actually in a book.

Also, there's absolutely no reason for JMeganSnow to consider i in the context of 20th century mathematics, when picking an example of a mathematical concept, to explain her point. That concept does exist in some context, and that makes her right enough. If she was asked what is i, then it would've been appropriate to call her out on her answer being incomplete.

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Yes, they are infinite, that is the definition, that is how the term is used, in math. They just happen to not be an actual collection of numbers. Both the set of natural numbers and the set of real numbers are theoretical devices, only used in the context of mathematics.
I think it's fair enough that you may wish to distinguish between what we may call 'mathematical existence' and 'real existence'. That raises the question though of whether you consider there to be any mathematical objects (or concepts, or what have you) - such as sets, numbers, etc. - that have real as well as mathematical existence. At least our conversation is made easier though by our agreement that in mathematics (classical mathematics; I'll take as tacit that as the mathematical context hereon) the set of natural numbers is infinite.

The word actual refers to reality, not math theory. An actual collection of natural numbers would always have a number of elements. It would be nonsensical to call it infinite, in actuality, as opposed to in the context of math theory.
I have no special attachment to the word 'actual' in mathematics; I could just as well not use it in context of mathematics. However, when people do raise a distinction between actual and potential in a mathematical context, then I am interested as part of the inquiry to pursue how such a disctinction is explicated IN MATHEMATICS.

And as to the very point of the CONTEXT, in fact the MATHEMATICAL context was introduced in the FIRST post with a quote of Peikoff that starts with:

"There is a use of [the concept] “infinity” which is valid, as Aristotle observed, and that is the MATHEMATICAL use." [emphasis mine].

Granted, that suggestion did not itself require that the context be that of 20th century mathematics (rather the contrary, since indeed Peikoff was mentioning Aristotle), but my point is that if people are going to address the mathematical context then it is productive not to ignore also the context of 20th century (actually, even earlier, since the notion of complex numbers as ordered pairs of reals goes as far back as Hamilton and Gauss) mathematics.

You can call something actual all you want, because it follows from mathematical axioms. You'll just be ignoring the context of our discussion, in favor of a speech on irrelevant aspects of mathematics.
I'm not ignoring the context of discussion. Mathematics was raised as part of that context. My remarks addressed that mathematical context. I have not thereby ignored that there may be other parts of the context. Moreover, it would be my point that it would be unwise for people to ignore the actual content of the ordinary mathematical context of the subject when a mathematical context has been introduced into arguments and explanations.

Also, there's absolutely no reason for JMeganSnow to consider i in the context of 20th century mathematics, when picking an example of a mathematical concept, to explain her point. That concept does exist in some context, and that makes her right enough. If she was asked what is i, then it would've been appropriate to call her out on her answer being incomplete.
As I mentioned, since you are attempting to driving a wedge as to what is approriate context, the MATHEMATICAL context was suggested in the first post

In sum, rather than quibble over (and attempt to control) what is the proper context of this discussion, I think it is more productive to (along with whatever notions one may wish to present) consider the knowledge we have available from 20th century mathematics and especially to consider such personal explanations as Ms. Snow's in a context also of the rigorous public mathematics that is available to us.

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One useful concept to bring into the conversation is "unbounded". That means "can increase with no limit" but does not in any way imply that infinity is possible. An unbounded variable always has a particular, finite value at any given moment.
In consideration of the previous post, I take it that the context here at least includes mathematics. That said, the quoted comment is fine except that it does not preclude that in the ordinary mathematics for the sciences (i.e., calculus) there are infinite sets. The notion of unbounded is, of course, important in ordinary mathematics. But the notion of unbounded does not eradicate the theorem (and usefulness of the notion) of infinite sets, nor, for that matter, theorems of non-standard analysis (which is within classical mathematics) that there are numbers beyond finite bounds; moreover that, even in ordinary ordinary calculus, for certain purposes we extend the ordering of the reals with two points that are each beyond finite bound. Edited by Schmarksvillian
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P.S.

Since Mr. Ellison has raised a point as to some proper context of this discussion, I point out that Mr. Ellison finished his earlier post with:

Infinity indicates a potential in the context of math, when applied to something specific.
[bold in original]

I think it quite appropriate that I (or anybody) include mathematical context and also remind where certain explanations of mathematical notions such as (infinity and complex numbers) are impoverished for lack of taking into consideration even basic mathematics such as available to us in even basic mathematical studies.

Edited by Schmarksvillian
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Remember, I'd like to be able to convey this all to my 5 year old (Nicholas). :) I have a bit of a reprieve since he was wondering about some other things today during our ride to school. But these replies have been very illuminating.

The term "universe" is similar to this. When it is used to mean "everything" (which is the literal meaning of the term), what you are in effect saying is "I don't KNOW everything that exists, because my knowledge is finite, but whatever DOES exists, gets filed in this category".

I definitely accept that. It makes sense. I agree and take "Universe" to mean that as well.

There is no conflict with the arbitrary or the "potentially" infinite, because the things that DO exist are NOT DEPENDENT UPON CONSCIOUSNESS OR KNOWLEDGE.
Well, this is the crux of my problem. Since this Universe contains everything that is but we don't know everything that is, it means that it does hold within it what we would surely consider arbitrary now (2010). That may or may not cause a contradiction. If a being were to arrive on earth and say, "Look at this primary color you've never seen before, and then showed us something real in that color, then we'd integrate that perception and try to figure out what was wrong with our understanding of the color spectrum. Twenty years ago, if someone had told me I could be walking from my office to my car and finish downloading an audio podcast from across the country (equivalent in file size to something that would take 40 5¼-inch floppy diskettes) to a mobile device in my hand --- It would have been difficult to contemplate that reality. If that was said to someone 200 years ago it would sound like pure witchcraft (capturing voices, playing them back, sending them through the air, etc... No way. I cannot begin to imagine what Aristotle would think if you made that claim and didn't include any of the science of how it works). I realize that these examples of arbitrary claims can all play out within the Universe and do not contradict or change the real nature of the Universe.

Here's a try at an arbitrary claim that might cause a problem: There is no finite amount of ping pong balls that would completely fill up the universe to the point where you couldn't add another. You could "infinitely" add ping pong balls into the universe and never run out of space. How can you do that inside a "finite" set?

As you can see by my examples, I'm not great at this, but I'll *try* to articulate where I'm having an issue with the affirmative proclimation that the universe IS finite. If you get my meaning and have an easier way to express what I'm getting at, by all means skip to the meat of the issue.

If we accept the meaning of Universe to mean "the total of that which exists -- everything", then it is being described as a set, a finite set. As you said, whatever "everything" is, it is in that set. Is a set real, or is it a concept? If the Universe is understood as, and describes the entire "set" of everything, then can you really say that the Universe exists as an actual thing? Doesn't it exist the same way as "the bushel" in a bushel of grapes does?

So, really, your problem here boils down to a small failure to really understand the idea of the primacy of existence. The concepts we use to describe reality (and the various mental file-folders we slot things in) don't determine reality and mean that reality winds up with some sort of contrary nature where it does and doesn't include things that may only exist inside our heads. A is A. Our method of conceptualizing has nothing to do with it.

So does the Universe exist or exist only in our heads?

Lastly, the universe is infinite only with respect to its spatial boundaries. There is nothing outside or apart from the universe. But in the context of the universe being the sum of all physical objects (entities), that is most definitely finite.

This is the thing that I don't know how to explain to a 5 year old. What you said here is a clear contradiction on the surface.

Context 1 (spatial):

U = Infinite

Context 2 (entities):

U ≠ Infinite

That makes me think: U(s) ≠ U(e)

But U is U

Dog = Brown

Dog = Not Brown

(This can be true logically, but we know that we're talking about two different dogs)

When it comes to the universe, we're just talking about one universe, right?

Last month we touched on space and time a little bit (me and Nicholas). He wanted a synonym for circle, and the thesaurus said "sphere" in there. Explaining the differences got me to mentioning 2 dimensions vs 3 dimensions... After getting that, he wanted to know what the 4th dimension is.

My explanations included the thought experiment about the ant on the globe that could walk in one direction forever.

When he asked about the 5th dimension, I punted. I honestly can't explain it to myself.

You have one hell of a sharp five-year-old. Your conversations must be a blast!

lol, you have no idea. I feel a more urgent need to increase my intelligence.

You should try to explain infinite to your son precisely by looking at natural numbers, and a few more types of infinite series (such as the series 2, 4, 6, ..., and 1, 5, 10, 15,...). He won't fully understand until he understand concept formation first, but at least he won't think infinite is something it is not. (such as a number) But, above all else, you should explain infinity to your son in the context of math, and never use it (or allow him to use it) in any other context, except math. You can't say "infinity indicates a potential", and leave it at that. Infinity indicates a potential in the context of math, when applied to something specific.

He definitely knows that infinity is not a number. I wish I understood math better myself so I could use some creative ways to express these things without 1.) misleading him or 2.) needing to introduce details that he hasn't any conception of yet. What usually happens is #2 takes us into discussing those other things, which is fine but it doesn't answer his initial question always.

When he asked me what infinity minus 1 is, I just told him infinity isn't a number. It would be like asking, "What is Frank minus one?" (I feel guilty about that answer, btw)

One useful concept to bring into the conversation is "unbounded". That means "can increase with no limit" but does not in any way imply that infinity is possible. An unbounded variable always has a particular, finite value at any given moment.

Is the only way to conceptualize this in terms of math? When you say something can increase with no limit BUT doesn't imply infinity, the logical question that follows is: Doesn't that contradict? (or as my kid would put it... "isn't that an oxymoron?")

If it can only be explained through math then I definitely will have my work cut out for me.

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Here's a try at an arbitrary claim that might cause a problem: There is no finite amount of ping pong balls that would completely fill up the universe to the point where you couldn't add another. You could "infinitely" add ping pong balls into the universe and never run out of space. How can you do that inside a "finite" set?

There is a sub field of mathematics that can address that very question. It is called Point Set Topology. I should warn you, however, that if you wish to study it, you have (as you said) your work cut out for you. It is possible for metrically finite spaces to be unbounded in the sense they have no boundary points. Roughly a point p is a boundary point of set S if every neighborhood (this term has a technical meaning in topology) contains points that are in S and points that are not in S. Think of the edge of a disk on the plane. The boundary points of a disk is a circle.

Now consider the surface of a sphere and restricting our attention to just the surface (as a topological space) define a neighborhood of a point p on the surface of the sphere as the set of all points q on the surface of the sphere which have a geodesic (great circle) distance from p that is less than epsilon (a small positive real number). It turns out every neighborhood of a point has only points that are on the surface of the sphere. So no point is a boundary point, hence the surface of a sphere is (in the technical sense) unbounded. Yet the surface of a sphere is finite. The surface is generated by a solid ball of finite radius R and the maximum great circle distance between any two points on the sphere is pi*R. Think of the earth with a circumference of 25,000 miles. The maximum distance between two points is 12,500 miles and the points are antipodal (poles apart).

Now extend this to four dimensions. The "surface" of a 4-sphere (or 4-ball, actually) of finite radius is a three dimensional space which is limited metrically but contains no boundary points, so there is no "edge" to this three-space.

Got it? That is about as simple as I can make it for a non-mathematician.

If you want to pursue this there is an elementary book on topology by Henle. Look it up on Amazon. I think there is a Dover edition out now that is pretty inexpensive.

A Combinatorial Introduction to Topology by Michael Henle (Paperback - Mar 14, 1994)

Bob Kolker

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If it can only be explained through math then I definitely will have my work cut out for me.

"Unbounded" is not a merely mathematical concept.

Discussing the size of the Universe with your 5 year old may not be much easier than math, though. If he has had any contact with astronomy, you might use the example of two galaxies moving away from each other - the distance between them is unbounded - always finite but ever increasing. It will never be infinite, but it will always be greater than it was just before.

It is hard to get good examples within our day to day context.

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One useful concept to bring into the conversation is "unbounded". That means "can increase with no limit" but does not in any way imply that infinity is possible. An unbounded variable always has a particular, finite value at any given moment.

In a purely abstract construction of mathematical objects, time does not enter into the conception. Pure abstract math is timeless. The only sequence you will find is in linear or partial ordering and in (the metamathematical sense) logical precedence. In logic premises come before conclusions, but this need not be comprehended as a temporal sequence.

Moments are in the real physical world, not the abstract idea world.

Bob Kolker

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One useful concept to bring into the conversation is "unbounded". That means "can increase with no limit" but does not in any way imply that infinity is possible. An unbounded variable always has a particular, finite value at any given moment.
P.S. to my previous response: Unboundedness doesn't exhaust the notion of infinity, as we see that there are sets bounded both above and below but still infinite. For example, the set of rational numbers (or also, the set of real numbers) between 1 and 2. Bounded above and below, yet infinite.

By the way, 'infinite' may be defined simply: x is infinite if and only if x is not in one-to-one correspondence with any finite set (of course, 'finite' will have been previously defined). Also, we may find the definition of 'infinite' as: x is infinite if and only if x is in one-to-one correspondence with a proper subset of itself. For example, the set of natural numbers is in one-to-one correspondence with the set of even numbers (also with the set of odd numbers), but by the pigeonhole principle, no finite set is in one-to-one correspondence with a proper subset of itself. The two definitions are equivalent in ordinary set theory if we allow the axiom of choice.

Still, there is another sense of 'infinite', which is of a member p of the field of a given ordering such that p is not within a finite number of steps (per the ordering) from some given "origin" such as 0. For example, to the reals we may add two "points" called 'infinity' (greater than any real) and 'negative infinity' (less than any real). The Archimedian principle that holds for the reals does not hold for this system with infinity and negative infinity added. That is, there is no real and no natural number such that their product is greater than or equal to infinity, and no real and no natural number such that their product is less than or equal to negative infinity. Also, the sense of infinite "points" is found in non-standard models of first order PA - those points being elements of a domain that satisfy the axioms of PA but are not themselves natural numbers (since every natural number is a finite number of steps from 0, but these "non-standard" points are not a finite number of steps from 0).

Edited by Schmarksvillian
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I'm not discussing math, I'm discussing the topic (how to address the issue of infinity versus the real world to a very smart 5 year old).

Your child and you are looking down straight railroad tracks as they stretch off into the distance.

"Daddy, where do the tracks meet?"

(I have four children -- now grown -- and five grandchildren, growing up). Been there. Done that.

Bob Kolker

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I'm not discussing math, I'm discussing the topic (how to address the issue of infinity versus the real world to a very smart 5 year old).
Fine, but also the subject of mathematics, and certain mathematical concepts have been raised along the way, as well as the original post also references the study of mathematics. So it makes fine sense for people also to share information about the mathematics of infinity. Edited by Schmarksvillian
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