## Are there more numbers than numerals?

0

If the universe is finite and numbers are infinite, would that mean that there are more numbers than numerals to name them?

4What do you mean exactly by the existence of a number / a numeral? – None – 2015-08-14T12:58:54.700

1Each natural number n has a numeral n that expresses it. Just define n to be 0 preceded by n-many occurrences of the successor function S. Then n expresses n. So there is a bijection between the natural numbers and the numerals that name them. Obviously, though, we'll run into problems if we actually try to write down all of the numerals - once the numbers get big enough, there likely isn't enough physical stuff around to facilitate writing down their corresponding numeral. – possibleWorld – 2015-08-14T13:37:30.187

Can you compare infinite with infinite? Is there something bigger than infinite? – Mark Knol – 2015-08-18T23:06:54.303

@MarkKnol There are mathematical constructs which allow comparisons of infinities. In particular "countable infinity" is less than "the infinity of the continuum," meaning there are more real numbers than there are integers. The fact that such a statement about real numbers and integers is considered meaningful for study suggests just how far down the rabbit hole mathematics goes when it comes to infinity. – Cort Ammon – 2015-08-22T18:55:31.180

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Let's take numerals as the representations of numbers - as @possibleWord suggests. His/her description is indeed similar to the mathematical definition of natural numbers from set theory according to von Neumann:

• The empty set is the numeral representing the number 0,

• the set with single element the numeral representing 0 is the numeral epresenting the number 1,

• and in general, the union of (the numeral representing the number n) with (the set with single element the numeral representing n) is the numeral representing the number n+1.

By definition, the set of all numerals and the set of all natural numbers map bijectively to each other, hence they have the same cardinality.

Hence the answer to your question: No, there are not more numbers than numerals.

In my opinion your question does not presuppose to write down numerals. And the answer is independent whether the universe is finite or not. Both numerals and numbers are abstract concepts.

@jobermark I cancelled the reference. – Jo Wehler – 2015-08-17T17:24:36.680

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Wording is tricky in this case. Mathematics has a very exacting definition of things like numbers, finite, and infinity. By bringing in the finite universe, you have clearly stepped just outside the realm of pure abstract math. I'll try to answer what I perceive your question to be, and maybe after the fact you will agree (or have ways to correct my assumptions).

The most typical definition of numerals is the "natural numbers," which are integers counting from 0, 1, 2, 3... and so on. That counting progresses towards infinity, but we need to be very precise because mathematics actually has multiple infinities. This particular infinity is known as "countably infinite." Yes, there is a concept of counting to infinity. The process takes infinity long, which sounds circular, but when you look at the actual mathematical definitions, they aren't circular. The circular feeling comes from my three-sentence overview of the concept taking a lot of shortcuts.

Now, when we talk about the universe being finite, we usually are talking about there being a finite amount of some important resource, like space or energy. There is some limit to the universe. Because you mention the finiteness of the universe, I believe when you say "numerals to name them," you are referring to what would be more technically termed "enumeration." Your question I believe is technically worded "Can you enumerate the natural numbers within the resource limits of a finite universe." And for that question, we will need to turn to machines which can do enumeration.

The Turing machine is the easiest example to look to, because it defines how a computer works, and we're all pretty comfortable with them. It is trivial to see that no matter how many computers we gang up in parallel, they will never be able to enumerate all the natural numbers... or more technically precise: "A finite set of Turing machines will require a countably infinite quantity of time to enumerate the natural numbers." So this first pass at your question would be "no, we cannot name all the numbers." However, this rabbit hole goes deeper than that, and the answers get more profound as you go down.

The next step is to explore what are known as hyper-Turing machines. These, simply put, are machines that can do tasks that would take a Turing machine infinite time to do, but can do those tasks in finite time. Such a machine would clearly be capable of naming all the natural numbers, but does one exist? It turns out that one important class is theoretically possible, a "real computer."

Enter the real numbers. In between 0 and 1 is an infinite continuum of numbers, and it stretches outwards towards infinity. This sounds like just infinity squared, but it is much much bigger. Consider a thought experiment: make a countably infinite 2d grid of values - 2 dimensions of natural numbers.

 ...                            ...
(0, 3)  (1, 3)  (2, 3)  (3, 3)

(0, 2)  (1, 2)  (2, 2)  (3, 2)

(0, 1)  (1, 1)  (2, 1)  (3, 1)

(0, 0)  (1, 0)  (2, 0)  (3, 0) ...


Is the number of pairs in this grid countably infinite? A quick intuition may say "no, there's more than there were before," but this is where the math of infinities starts to diverge slightly from intuition. Consider this diagonal meandering path, which keeps going back and forth along diagonal paths, getting further out each time (start at (0,0), and follow the lines):

 ...                            ...
^^  \\      \\      \\
(0, 3)  (1, 3)  (2, 3)  (3, 3)
\\      \\      \\      \\
(0, 2)  (1, 2)  (2, 2)  (3, 2)
^^  \\      \\      \\      \\
(0, 1)  (1, 1)  (2, 1)  (3, 1)
\\      \\      \\      \\
(0, 0)->(1, 0)  (2, 0)->(3, 0) ...


Now I've put the numbers on a straight line, not 2 dimensions. It starts to look like I could count them, and I'd be right. This is the exotic world of cardinality. If I have a set of natural numbers, I can say "the cardinality of the set of natural numbers is countably infinite." I can also say "the cardinality of the set of points in the grid, above, is also countably infinite."

Why does this matter, because real numbers break the mold. If I were to ask for "the cardinality of the set of real numbers," the answer would not be countably infinite. The proofs are a bit more complicated, but this is a big enough deal that mathematics has another infinity: the infinity of the continuum. That infinity is defined as "the infinity of the continuum is the cardinality of the set of real numbers." It is bigger than countable infinity.

All of this does matter, because while the universe is believed to be finite, it is also believed to be well described by real numbers. This creates an interesting opportunity for computation: Turing machines only operate on 1 and 0.. what if a "Turing machine" could operate directly on real numbers? The answer is that it becomes a "real computer" that actually can enumerate all the natural numbers. Instead of consuming more power to do so, it relies on finer and finer grained details with less energy in each detail. This is theoretically possible, so your answer here is "yes, you can count the numbers." Some even theorize that the brain may actually be a real computer, though that is still pure speculation.

There is, of course, a limit: the more detailed you get, the more it gets messed up by small things like gamma rays, or even quantum disturbances. However, there is no theoretical reason why these issues cannot be overcome... we just haven't found out how to do so yet.

If we're not cool with relying on diminishing energies like that, there's even a third possibility. Dan Willard has been working on some interesting worlds which start by defining a set as having a cardinality of countable infinity, and working backwards, dividing to get towards the small, rather than adding and multiplying to get to the big numbers like we're used to. These worlds have the curious property of being too weak to admit the diagonalization technique shown above, so they have unusual properties. For instance, if you nest one of these worlds inside another world, a number can appear and behave as countably infinite on the inside, but appear and behave as finite on the outside. This can create interesting worlds where you can construct an infinity in finite time, just by being clever with the word choice in the problem!

Infinities are weird! If you will excuse me, I'm going to go play on the swing set and watch the sun rise, and do all sorts of normal human stuff now! You should too!

I like this answer and especially the distinction between the abstract and the physical, however I must take issue with your claim that maths includes an exact definition of the general concept of number. Set Theory, for example, includes an exact definition of certain types of numbers (ordinal and cardinal), but the general concept of number is one that has resisted definition. Frege spent a considerable amount of time on this question, but ultimately failed. – Nick – 2015-08-14T22:07:48.857

1@Nick R. You say that the general concept of numbers resisted definition. In my opinion, natural numbers can be defined according to von Neumann. And from natural numbers you can construct integers as pairs of naturals modulo an equivalence relation, rational numbers, real numbers as limits of sequences of rationals, complex numbers, quaternions, Hamiltonian octaves. What is missing? – Jo Wehler – 2015-08-14T23:28:41.110

1@jowehler I agree that von Neumann's approach is perfectly satisfactory in the context it is expressed, but I do not intuitively feel that a number is a set or that an integer is an ordered pair, etc.. I have a copy of Frege's "The Concept of Number", which I shall re-read and hopefully get back to you. I also have a copy of Benacerraf's "What Numbers Could Not Be", which I shall try to read soon. My ideas are not yet very sophisticated as I have only just completed my second year. – Nick – 2015-08-14T23:56:54.373

@NickR I agree there are some questions on definitions, because it is rare that the meaning of a word has perfect agreement. The point of was that mathematics does have very specific definitions, and thus those definitions seem particularly useful when answering a question about infinities. One does not have to accept these definitions as the only valid definitions, but this at least provides a foundation to explore other potential meanings without spending a considerable amount of time on the question and failing. If this is an idle curiosity for Aryeh, this may be enough... – Cort Ammon – 2015-08-15T01:09:50.400

... if this is a lifelong pursuit question, hopefully Aryeh can come back and continue the conversation for definitions of number, infinity, and finite which accurately suit the desired conversation. – Cort Ammon – 2015-08-15T01:10:32.590

@CortAmmon Yes. Set theory, and von Neumann's articulation in particular, is very elegant and creates very beautiful mathematics and (even surprising) truths. Who could fail to be inspired to want to know more. – Nick – 2015-08-15T02:21:38.643

@NickR Can you give me something searchable on what you are mentioning. I'm very comfortable with numbers within set theory, but von Neumann's articulation is a new phrasing to me, and got very few hits. Is that just leading to class theory, or is there something interesting he did within set theory? – Cort Ammon – 2015-08-15T03:15:40.200

@CortAmmon von Neumann's formulation is one which is familiar to us today - sometimes called "The Cumulative Hierarchy" or "The von Neumann Universe" V. Have a look at https://en.wikipedia.org/wiki/Von_Neumann_universe as a good starting point. You probably know this formulation, but maybe not by name.

– Nick – 2015-08-15T03:31:05.680

@Cort Ammon A simple introduction to von Neumann's idea exemplified by the natural numbers is the section "Constructions based on set theory" in https://en.wikipedia.org/wiki/Natural_number

– Jo Wehler – 2015-08-15T04:39:21.183

@jowehler Ahh thank you. Nick was right, I have seen that before. That's the set construction for numbers I use most often when I need one! – Cort Ammon – 2015-08-15T16:12:11.653

@Cort Ammon Could you please reference some paper on hyper-Turing-machine. And explain why a hyper-Turing-machine is able to count the infinite set of natural numbers in a finite time. - Any convergent series, e.g. the geometric series sum(i=0,.., infinity) (1/2)^i = 2 gives a theoretical recipe: You must not use more than (1/2)^i seconds to count number i. But for sufficiently big numbers i this time falls below any physical duration. Hence the recipe cannot be implemented in a physical system. – Jo Wehler – 2015-08-16T04:27:18.103

@jowehler There's several hyper turning machines, https://en.wikipedia.org/wiki/Hypercomputation is actually a nice list. The one you mention feels a lot like the first one of the list. The real number machine I mention takes a different tact. If you have real numbers, you can handle infinite bits of information per operation. Thermal noise is still an issue, but there may be some very "pure" real number operations available to reality. We'll find out as our culture progresses.

– Cort Ammon – 2015-08-16T04:40:23.190

Being able to visit all of the real numbers with some kind of infinite bandwidth Turing machine still doesn't count as naming them in any sense that most folks would allow for. This is 'naming' only in Madeline L'Engle's sense of 'considering their existence relevant'. You still run out of names that might be input by a human before you run out of potential answers. If I cannot say, write, remember or otherwise process the vast majority of the names, how are they names? – None – 2015-08-17T16:54:37.527

@jobermark That is a fun line of reasoning that pulls away from the mathematical side of the question quite nicely. What does it mean "to name" something anyway? I elected to translate that as the ability to enumerate the numbers. A counter argument could be that, if I were to define a constructive approach which, given a number, can provide a string to identify it (a name), can I claim to have named all of the numbers? If the names need physical representations, such as slips of paper, do they need to all exist in the same temporal slice, or can they be spread across eternity? – Cort Ammon – 2015-08-17T19:16:36.917

OK, but we know a name need not be written down to be a name. I can mentally note a name. Given infinite time, I can mentally note the kinds of things that math identifies as enumerations. If I am restricted to finite time, I still can't use your names. So this is all evasion. People have considered the words we use in math for centuries, they make sense. Why discard them? – None – 2015-08-17T20:02:26.223

@jobermark The question is phrased in terms of the physical world, as you have noticed, so the words used in pure math will not be quite as useful as words designed for dealing with math in the physical world - and infinity is quite a tricky word to work with. I've had too many times where a closed definition of wording paints one in a corner. The corner is an illusion, of course, an artifact of choosing to hold to word choices that bind us. I chose to push on those words a little to point in directions which may offer new opportunities, instead of dead answers. – Cort Ammon – 2015-08-17T21:16:41.383

Personally I like to remember that, even if we cannot expand outwards forever, we can become more detailed and intricate, expanding inward. – Cort Ammon – 2015-08-17T21:16:44.663

@jowehler Last night, I had a quick look at Benaceraff's "What Number Could Not Be". In it he describes an interesting thought experiment where children are (successfully) taught elements of Set Theory without any reference to number before being told that "this is what other people mean by number". The argument is convincing (on a quick read), so I guess my naive intuition needs some refinement. – Nick – 2015-08-22T18:22:09.307

@NickR Do you like to explain a bit: Did the children agree that set theory supports our intuition about numbers? – Jo Wehler – 2015-08-22T18:26:39.877

@jowehler The impression I got was that the children's intuition was accommodative toward the teaching. I'll need to read it more carefully to give you a more informed view. (I have mild dyslexia so I often need to read things a few times. This is why I often delete answers. I post an answer and the re-read the question only to discover that I have misunderstood it in the first place.) – Nick – 2015-08-22T18:36:36.923

1

Assuming by "number" you mean nonnegative integer and "numeral" you mean some standardized form of decimal notation....

Numerals are mathematical objects too, and there are exactly as many numerals as there are numbers. The point you are asking about, I think, is that there may not be enough physical material in the universe to create physical representations (e.g. ink on paper) of all of the numerals.

(similar things can be said with various other common meanings of "number" and "numeral")

0

We cannot name all the numbers. But it does not have so much to do with the expanse of the universe, but with the infinite divisibility of space. As @possibleWorld's comment above shows, we can cover an infinite expanse with arbitrarily long sequences. Each of these would still be finite.

We know, however that we cannot name all of the Real numbers unambiguously. Since most numbers are transcendentally irrational, the names themselves would mostly need to be unending, non-repeating expansions, truly infinite in length, and not simply arbitrarily long. It is hard to consider something of truly infinite length a 'name', or a 'numeral'.

(it should probably be clarified that by "name", you mean to write in the standard way via decimal notation, assuming that is what you mean) (also, most rational numbers cannot be "named" in this fashion either; e.g. 1/3 has a truly infinitely long decimal expansion, even though it repeats) – None – 2015-08-18T05:46:30.750

I didn't really have decimals in mind, but bits. And that is not the only kind of name this applies to. The relevant aspect is need. Whatever convention you use, most of these numbers cannot be represented finitely and genuinely need an infinite form. I can change irrational to transcendental if it will make you less picky. – None – 2015-08-18T14:12:28.323

Well, we can always resort to using definitions as names, and it's actually consistent with ZFC that every real number (in fact, every set) has such a name. Although if true, we can't always tell which alleged names really are names, or whether two names both name the same number. – None – 2015-08-18T14:34:27.320

No, those definitions would be infinitely long, or we would fail to address each number with its own name. So whether we get less specific, or run out of sequences there are more numbers than names. – None – 2015-08-18T14:52:43.037

This is all obvious, and you are just being needlessly overprecise. It is clear what I said, and this is not the Principia... – None – 2015-08-18T14:57:01.253

It's obvious, well known... and wrong. See this question on math.se.

– None – 2015-08-18T16:28:05.997

There is a model where it is false, by Lowenheim-Skolem, but only in the first order. In the second order, the reals are uncountable and the names are countable. If you want to extend the result, you need infinite sentences. A model of ZF is not necessarily the preferred model of ZF that people actually use. – None – 2015-08-18T16:51:27.007