Infinity is nonsense ?

Great Debates
81

Well, it depends on what you mean by deciding the cardinality of the continuum. The cardinality of the continuum is exactly the cardinality of the continuum; of that there’s no doubt. It is equal to the power-cardinality of the integers, that much has been nailed down since people started looking at it [although in non-Boolean mathematics, this comes apart…]. This is larger than the cardinality of the integers, which is itself the smallest infinite cardinality. That’s all quite nailed down [again, depending on exact choice of formalization, but traditionally].

The thing that’s not nailed down is how many infinite cardinalities might be strictly between the cardinality of the integers and the cardinality of the continuum. The continuum hypothesis is the assertion that there are none.

And this is not nailed down, in the sense that it’s well understood how to take any model of the conventional set theory axioms, and produce from it a model which does or does not satisfy the continuum hypothesis, so that we know that the conventional set theory axioms do not fix the continuum hypothesis to go in one particular way. (Although this is perhaps a kind of nailing down in itself, given how well we understand how to make these models and what fine control we have over them (Easton’s theorem and the like)…)

Anyway, it’s not clear to me that any of these investigations has great relevance to ordinary calculus.

82

I think there are many simple applications of the simple concept of infinity, but I will agree with you that the landscape of transfinite cardinalities as traditionally presented (a linearly ordered scale whose purpose is to gauge “How large?/How infinite?”) is not one for which I know much use.

But, if one avoids so much thinking of cardinalities in those terms, and instead simply thinks of a cardinality as a description of a set up to bijection, the study of cardinalities is simply the study of sets up to bijection, which is to say, the study of datatypes up to isomorphism [which is to say, category theory]. And this naturally comes up in any context where a theory of datatypes would be useful; e.g., in the design and analysis of programming languages. A set theory* is a programming language (definable sets amount to definable datatypes; definable functions amount to definable programs; provable equalities between functions amount to definitional equivalences between programs), and the study of cardinalities is the study of abstract datatypes. (This is currently exploding with the development of “homotopy type theory”/“univalent foundations”, which takes the two ideas “A theory of the mathematical universe is a programming language” and “Isomorphic datatypes are as good as the same” and runs with them.)

Of course, programming languages are also just formal systems, pretty much by definition. But not purely so, in that the wonderful thing about the electronic computer is how it puts such abstract formalities to such concrete use… Application in programming is application in real life.

[*: Pedantic note: This account of a set theory doesn’t account for every datatype having a canonical way of understanding its elements as themselves datatypes. But that’s ok; looking at sets only up to bijection erases any such information anyway. It’s bizarre that “set theory” has come to be understood as meaning specifically the study of the cumulative hierarchy of sets of sets of sets…, and even more bizarre that cardinalities are considered the province of that particular niche study, rather than, as they actually are, a central concept of the field which precisely studies the idea of objects up to isomorphism: category theory]

83

There’s a useful (to mathematicians) notion of “transfinite induction” that allows you to do induction arguments or constructions using transfinite ordinals, not just the counting numbers. Iterating some construction countably many times may not get you where you need, but doing it a lot more times may. An important example of this is in abstract homotopy theory, where transfinite induction is used in what’s known as Quillen’s small object argument, a key to many applications of the theory (within mathematics).

Then there’s an interesting article I read recently, Real Analysis in Reverse (pdf), which appeared in the May 2013 issue of the American Mathematical Monthly. Essentially, the article looked at the question of what statements about an ordered field imply Dedekind completeness (every cut has a cut point), hence imply that the field is isomorphic to the usual real numbers. One thing shown in the article that surprised me is that Cauchy completeness (convergence of every Cauchy sequence) does not imply Dedekind completeness. The counterexamples involve ordered fields that are so large that the set of all integers is bounded (so the fields are not Archimedean) – in fact, every countable sequence is bounded. For a sequence in such a field to be unbounded it has to have transfinite cardinality. This also gets at something Indistinguishable has been saying in this thread and others: There are useful systems of “numbers” in which you have elements larger than all integers.

84

In a way, I think transfinite ordinals are a lot easier to present than transfinite cardinals, in that everyone is already familiar with them (and can easily see their transfinitude) in at least one guise: alphabetical ordering. If you consider finite strings of characters alphabetized by the usual rules (X comes before Y if at the first position where they disagree, either X has a character which comes before Y’s or X has ended and Y has not), then we get a transfinite ordering like so:

“a” comes before “aa”, which comes before “aaa”, which comes before “aaaa”, …, all of which comes before “b”, which comes before “ba”, which comes before “baa”, …, all of which comes before “bb”, and so on.

I’ve used some strings here that don’t correspond to any actual words, but you could get the same effect just as easily with suitable sentences, if you insist on strings which are meaningful English.

The order type this produces is actually not a “well-order”, in the jargon sense of that term [it is N * [1 + Q]], but it embeds many well-orderings of interest [and the study of order types more generally is of interest as well].

85

We tend to use the word imagine in a loose sort of way. I know there may be a score of entries for the one word in a dictionary, but that just reflects the vagaries of common usage.

I take it literally, that to imagine means to employ mental imagery, which I claim means that we actually cannot imagine the infinite.

Take away the attempted mental imagery and all we are left with is the linguistic label.

So I take it to mean that at some point in time someone decided to apply a label to something which they could not imagine. And that point seem to be a mystery.

Language seems to have this property, I think, of being grammatically correct nonsense.

For example, “Did you count the sides of the five sided triangle ?”

Impossible to visualise, logically garbage, but very easy to say.

and-

How many five sided triangles have you got ?
What happens when you subtract a side from a five sided triangle ?
Can you arrange the five sided triangle in sets according to size ?

Hence, it seems to me, with infinity.

So here is my next question.

Does this mean there is a finite set of numbers ?

I mean for real - not according to some theoretical and impossible to imagine number line that goes on ‘for ever’ ?

That is, as humanity gets older and the universe keeps using numbers - will there come a time when every number which can be used, has been used ?

And I have this further question, which is - will the set of all the numbers that will ever be used have a pattern to it ? Maybe a fractal pattern resulting from the fractal nature of our brains ?

86

That’s it - but how do we know it is infinite ?

What did it look like ?
Did you measure it ?
How come it came to and end, if it was infinite ?
Is it still there ?
If I ask you to do it again, will it be the same infinite line or a new one ?

Can a finite amount of brain tissue, and a finite amount of brain fuel, produce an infinite mental image ?

Maybe it can, maybe I’ve got it wrong and I’m being simplistic about the need for an infinite brain fuel to produce an infinite image ?

But if that’s so, then how ? What would the brain scan look like for a person imagining an infinite line ?

Is anything else at work here?

87

I’m afraid you’re going to hate me, but I think the disagreement may come down to the usage and meaning of “grasp”. That’s probably my fault since I started talking about grasping numbers without stating what I mean by it. With regard to math concepts one usually says “grasp” or “understand” in allusion to the ability to solve a problem or work out a proof, but the ability to produce those results rests on layers of abstraction, obviously more so for solving multivariate differential equations than for counting three mastodons. While the fact remains that counting to three is also a process of abstraction, most of us are are sufficiently familiar and comfortable with such basic mathematical processes that it is difficult for us to recognize that there is abstraction going on at all. The concept of three just is, and it doesn’t seem possible that we would need to prove that adding one to two produces three. This is what I meant by “grasp”–that is, to be so utterly comfortable in the use of an abstraction that we forget it is an abstraction. In this sense, we can grasp infinity in the sense that it is always at least one more thing or some more “stuff” than we have currently accepted or verified to exist. But we can’t grasp infinity per se. At least, I don’t believe we can.

88

“If we up the ante to, say, a matrix of 253 or one of 17x39x8, visualization becomes more difficult if not impossible for everyone. Just so with an infinitude of like objects.”

So you’re saying that the lack of ability to be able to visualise a finite number of things, and the lack of ability to imagine an infinity of things, makes these two equivalent in a way ?

But OK, let’s say we trained you to imagine a large number of things in a matrix, as you say.
Let’s say on week one you could imagine a matrix of 300 things, week two you could imagine a matrix of 400 things.

How long would your training take to be able to visualise an infinite number of things ?

Would there come a sudden break break in your gradual development -a quantum leap after which you could imagine an infinite number of things ?

Has this been demonstrated anywhere ?

89

Spectre, out of curiosity, is there any empirical test you could do to determine if someone grasped X per se?

90

Well put.

91

infinity[sup]2[/sup], of course. :wink:

92

For those who don’t know, the very basics of how to deal with infinity in arithmetic (and the very basics are literally all I know) involve making a distinction that is not often made for non-infinite numbers–the distinction between ordinality and cardinality.

Roughly, the ordinality of a number is its position in the ordering of numbers. Meanwhile, the cardinality of a number is, very roughly, how “big” it is. For non-infinite numbers, on the standard ordering, the ordinality and cardinality are for all practical purposes the same–meaning a number further to the right in ordinality always has a higher cardinality and vice versa. But for infinite numbers (yes, “numbers”) the two can differ. Define “infinity” as “the first number greater than all counting numbers.” Infinity plus one is a different number (called “infinity plus one” as far as I know). It’s cardinality (how big it is) is exactly the same as infinity, but its ordinality is different from infinity. Infinity plus one is “further to the right on the number line” than infinity. Well that can’t be exactly right since nothing can be to the right of infinity. But the idea is that infinity plus one is the successor to infinity. It’s the next number. Don’t visualize it (you can’t) just understand the definitions and their implications.

As long as you’re adding or multiplying infinites together by pairs, you never get an infinite number with higher cardinality, but you do get a whole set of infinite numbers with different ordinalities. You could represent them on a “past-infinity” numberline like so: Infinity plus one, infinity plus two, infinity plus three, …, infinity times two, infinity times two plus one, infinitiy times two plus three… infinity squared, infinity squared plus one, infinity squared plus two… and so on…

There are, of course, an infinite number of these numbers with the same cardinality as infinity.

But it turns out something happens when you exponentiate by infinity. Infinity to the power of infinity doesn’t just have a different ordinality than infinity. It has a different cardinality! It’s literally an infinite quantity that’s bigger than infinity as defined above. It’s a new order of infinity. This cardinality we call aleph-one. (The first cardinality we called aleph-null).

Once again you can build a whole line of ordinalities within this cardinality. Then, exponentiate this cardinality by aleph-null again, and you get yet another cardinality.

How the cardinalities are standardly ordered I forget.

But anyway, that’s the basics. There are proofs for all this. While the proofs rest on conventions that one could negate to create alternative systems, the fact is that for what I’ve said so far, the proofs do pretty much the best job of using conventions that integrate the infinite numbers pretty neatly into the system we already have for dealing with counting numbers. Arithmetic turns out quite intuitive, and intuitions about which quantities should be “greater than” others check out quite nicely. So though this isn’t the only way you could do infinity, it is the way mathematicians standardly do do infinity.

93

Usually called ω (omega).

Why do you say that? (Haven’t you just given a nice way to see that there can be such things?)

I believe this is wrong, and that the cardinality of ω[sup]ω[/sup] is the same as ω.

ETA- As long as we are getting into explaining this sort of math, one good resource to look for the cardinality side is Suber’s A Crash Course in the Mathematics Of Infinite Sets

94

I should have called it omega of course. At the time I was thinking “don’t introduce more terminology than you need to” but, on second thought, “infinity” is ambiguous between omega and higher cardinalities so I really should have gone ahead and defined omega.

But anyway, if two to the omega has a higher cardinality than omega, surely omega to the omega does as well! (I may be just blatantly misremembering something, but this feels pretty certain to me… Omega has the cardinality of the entire set of counting numbers, and then if you raise two to that power, then you have a new cardinality.)

95

Okay some googling is bringing it back to me. I was confusing omega with aleph-null. Raising omega to the omega power gives you the same cardinality of omega. But that cardinality is aleph-null. And if you raise aleph-null to the aleph-null power, you get a higher cardinality (aleph-one).

Can you remind me where aleph-one fits on the number line? Is it the first element after all the omega ordinals?

96

omega-1 is the union of countable ordinals (where there are a lot of), and has cardinality aleph-1. I think it has cardinality aleph-1 by definition, but I’m not sure.

Ordinality of the infinite, to my mind, isn’t really the place to start with the infinite. To note that cardinality and ordinality break apart is helpful, but I think ordinality is probably too complicated to begin ones study with and that it is better to start with cardinality.

97

That’s fine. The basic idea is, take the size of the entire infinite number line, and raise that to its own power, and you end up with a size that’s even bigger. So infinite arithmetic doesn’t just give you the same quantity every time.

98

Because, as I pointed out in a bit you did not quote, "“Three is how many *s are in ***” only defines the number three as it pertains to *'s. By mapping objects to *'s, you can then quite easily define the set of all sets where you have exactly one * for each object in the target set. Then you can expand the definition of “three” to pretty much any type of object.

This is useful.

Defining 1 axiomatically and then defining some operation called “+” (we won’t get into all the mental gymnastics that this entails), so that we can say 1+1=2, and 2+1 = 3; well, this is less useful. We have no reason (from the definitions) to believe that these definitions have anything to do with objects outside of pure mathematics.

In order to bring our definition into line with our intuition, we can use the set theory type of definition. This makes it more powerful.

If you are looking to make some sort of philisophical statement that one type of definition is a priori better than another definition, then you are going to be disappointed. Basically any definition that is useful is a good definition, and any that is not useful is a bad definition. Of course, we now have to define what useful means; but as someone else has already pointed out, at some point any system becomes self-referential, and I’m not going to lose any sleep over it.

Incidentally, using sets to define numbers moves us a great deal further in showing us how to handle “infinity”: yet another sign of its usefulness that should increase your willingness to use it as a definition.

99

This is equivalence of cardinality. It is useful if you want to figure out when two sets have the same cardinality.

100

These are some ways to deal with arithmetics involving infinity. But they’re not the only way; for example, the projectively extended numbers noted before have nothing to do with ordinality or cardinality. [Well, “nothing” is a strong word, but they have about as much to do with ordinality or cardinality as -1/sqrt(2) does…]. Similarly for the affinely extended numbers (with just two infinities, one positive and one negative) or systems of hypernumbers (with many infinities, but satisfying all the usual arithmetic rules [so no solutions to x + 1 = x or things like that]).

As for the confusion about omega^omega = aleph_0^aleph_0 = N^N = whatever you want to call it, it turns out that exponentiation as standardly defined for cardinals (|A|^|B| = |the set of functions from B to A|), does not correspond to exponentiation as standardly defined for ordinals (A^B = the least upper bound of A * A^x over x < B for positive A and B; 0^B = 0 for positive B; A^0 = 1) when the inputs are infinite. So it’s a choice of what you want to call exponentiation.