Lambda the Ultimate

I agree with your conclusion, but not with your argument. ∞/∞ only leads to paradoxes when ∞ = ∞ + 1 = 2*∞ and so on, which the guy specifically presents as being distinct numbers. Also, he specifically says that only non-mathematicians would consider the Infinite Hotel manipulations to be paradoxical.

The problem with infinity not being a number is, however, quite a big issue. On page 9, he writes that the sets {1, 2, ...} and {1, 2, ..., â‘ } are the same, which leads him to ridiculous conclusions later on, like "The set â„• is not a monoid under addition because â‘  + 1 does not belong to â„•".

It's a shame that he discards ordinal numbers early on, because his base â‘  system is strikingly similar to the Cantor normal form (b_k*ω^k + ... + b₂*ω² + b₁*ω¹ + b₀*ω⁰) for ordinals. His use of fractional coefficient and negative exponents, however, seems like an interesting extension which I hope has also been explored by a non-crackpot mathematician. He also has fractional, infinite and infinitesimal exponents, which is weird, but not necessarily devoid of interest.

It's also a shame that he shuns axiomatic systems, regarding them as too restrictive. There seems to be a fascinating number system screaming to get out of his paper. Let's release it!

The gross numbers are defined by the following rules.

1. zero is gross.
2. if E is a finite set of (distinct) gross numbers, and c is a function associating each element of E to a non-zero real number, then the sum of terms c(i)*â‘ ⁱ for each i in E is gross.
3. only the numbers obtainable by a finite number of application of the first two rules are gross.

It is true that inf/inf=c is only a problem when inf is a divergent process rather than a number. But trying to form a system with distinct values of infinity and treating each as a distinct value causes other issues with closure.

I spotted the claim about N not being a monoid under addition after I'd posted. I'd love to hear his explanation for why a number system designed by shunning axioms is more useful than one that has algebraic properties.

I'm not sure that trying to liberate a number system from his paper is a good idea. There is a serious danger of being trapped inside it with the author. But putting that worry to one side, there seem to some problems with your definition of rule 2.

Firstly I would claim that (1) is not well-defined (and my html-fu is terrible, but that is an aside), and so I don't know how to evaluate the terms c(i)*(1)**i. Secondly I can't apply rule 2 in an inductive manner to build the set of gross numbers as the only gross number that I know when I start the rule is zero.

I can see where you are going with rule 3, but it would appear that we could only construct gross numbers a finite number of steps from an ordinal (assuming that is what the place-holder in rule 2 would be). There are many systems defined in that way. What do you the think that the "extra" would be that results from fractional coefficients (and actually the thread below reminds me that Conway handles these) or negative exponents?

Let me first reveal my html-fu trick for rendering advanced characters like â‘ : I use unicode! Just copy-paste the character from somewhere else (this post, for example) into the comment box. No HTML required.

Let me also clarify that in rule two, I did not mean that the expression c(i)*â‘ ⁱ should be evaluated in order to obtain a new gross number. As you correctly pointed out, that would require a definition for â‘ , which we don't have (nor want).

Rater, I meant to use â‘  syntactically. Compare with the following definition of natural numbers...

1. zero is natural.
2. if n is natural, then suc n is also natural.
3. only the numbers obtainable by a finite number of applications of the first two rules are natural.

Here, we don't fret about the fact that suc is undefined, since we simply use it to syntactically distinguish zero from suc zero, from suc (suc zero), and so on.

Now if we use â‘  syntactically, like suc, I hope it is more clear how to use rule two in an inductive manner to get more numbers. Starting with zero, the only non-empty set of distinct gross numbers I can construct is the singleton set {0}. But I can build 2^ℵ₀ different functions from a singleton set to ℝ, so the first stage of induction yields:

1*â‘ ⁰, 2*â‘ ⁰, 3*â‘ ⁰, ...
-1*â‘ ⁰, -2*â‘ ⁰, -3*â‘ ⁰, ...
1/2*â‘ ⁰, 1/3*â‘ ⁰, 1/4*â‘ ⁰, ...
ℯ*â‘ ⁰, Ï€*â‘ ⁰, √2*â‘ ⁰, ...

With the addition of the zero we already have from rule one (and which, come to think of it, we could obtain from rule 2 using an empty set of gross numbers), we now have all real numbers. So you can already see that I'm neither aiming for ordinals nor for Sergeyev's system here.

With another induction stage, we can use real number exponents:

x*â‘ ² + y*â‘ ¹ + z*â‘ ⁰,
x*â‘ ^2.5 + y*â‘ ^Ï€ + z*â‘ ^-1, ...

I have written real number exponents as x*â‘ ², but I really should have used the full syntax x*â‘ ^2*â‘ 0. This is because, of course, our next step of induction can use non-real numbers as exponents:

1*â‘ ^1*â‘ 1 for â‘ ^â‘ ,
x*â‘ ^3*â‘ 2+1, and many other numbers I can't think of any use for.

I don't know what good the fractional, negative and gross coefficients yields, and I'm not really curious enough to find out. If Conway did something even remotely similar, then that's good enough for me.

There is a good intro to Conway's surreal numbers here.

Let |S| be the cardinality of S. We have these properties:

1. |S|=1 if S is finite
   
2. |A union B| = |A|+|B| if A and B are disjoint
   
3. |A x B| = |A||B|
   
4. |S|=|T| if there is a bijection from S to T.
   

He's given a new definition for |.| that dumps 4 because bijections between infinite sets need an infinite number of steps (whatever that means). Except, he allows some isomorphisms because he's unable to recognise then as such. (Eg. he doesn't recognise that by his own rules, the bijection from the positive integers to the negative integers is a bijection that requires "an infinite number of steps".)

If you're trying to define a new way of counting, and you can't recognise bijections, I think that qualifies you for being a crackpot.

All of his sets are effectively finite. His (1) is just a large natural number with certain divisibility properties.

When he writes { 1, 2, 3, ...} he really means { 1, 2, 3, ..., (1) }. Since he chooses (1) to be divisible by all small numbers, he has results like: |{ 2, 4, ..., (1) }| = |{ 1, 3, ..., (1)-1}|. This is just ordinary cardinality of finite sets.

As the union of two finite set means that 1 = 1 + 1 with your rules 1 and 2..

|S| = 1 if S={x} for some x. That his idea of cardinality agrees with the usual one for finite sets should follow immediately.

The discussion of Piraha did nothing for the article, besides showing how the author was willing to discuss topics he hadn't mastered.

Didn't Knuth do something like that?

I picked up that book once, and didn't get all the way through it. A work of fiction in which "a young couple turned on to pure mathematics and found total happiness".

Both are defined like Dedekind cuts in that they consist of pairs of sets (the left and the right) of values. Think of them as games where the left set is left's moves and the right set is right's moves. The surreal numbers have the condition that every element on the left is less than every element on the right. Games in general have no such restriction. Symmetric games like Nim, ie. games where the legal moves are identical no matter whose turn is next, are typically not surreal numbers.

I'd be surprised if there isn't any computational content via game semantics.

I'd be surprised if there isn't any computational content via game semantics.

Indeed, this paper by Paul-Andre Mellies explains some of the history behind this idea, some of the problems, and some solutions.

Even if I've only understood parts of this paper (by Mellies) at a first reading, it was already mind-blowing. Thanks for the reference!

Conway numbers are a generalization of the Dedekind reals, and there's a question of whether even those (as opposed to the Cauchy reals) are a useful basis for implementations of real arithmetic. That they are useful is what Andrej Bauer and Paul Taylor argue in the work described here.

Going from memory, it takes an infinite number of steps before the infinitesimals emerge from the system of numbers that he constructs. In fact I seem to remember that it take an infinite number of steps to construct any non-dyadic fraction. The problem is that the number of steps relates to the size of the description of that particular value; essentially anything that is not a binary string with a floating point would require a description of infinite size.

It's been a while since I read it, perhaps somebody else could correct me?

The problem is that the number of steps relates to the size of the description of that particular value; essentially anything that is not a binary string with a floating point would require a description of infinite size.

Yes, but the same goes for any representation of the real numbers, yet "exact real arithmetic" exists. The point is that to run a computation, you have to specify a desired degree of precision -- and you can specify any degree you want. But for a given degree of precision, the computation will only depend on a finite portion of the input (by continuity).

Putting aside that his approach is to put aside most mathematical results obtained this millennium, I don't think this is a case of a guy rediscovering a polynomial ordering and attributing undo significance. He's pretty much a crank.

On page 8:

The new numeral x allows one to write down the set, N, of natural numbers in the form

N = {1, 2, 3,...x − 3, x − 2, x − 1, x }

where the numerals indicate infinite natural numbers. It is important to emphasize that in the new approach the set (5) is the same set of natural numbers

N = {1, 2, 3, . . . }

we are used to deal with and infinite numbers (6) also take part of N.

I agree it was important to emphasize that he considers his set, which has "grossone" as its largest element, to be the same as the regular natural numbers. That saves readers some time.

His theory of "infinite sums" is also interesting. He apparently is just "adding up" infinities of terms without taking limits.

He doesn't discard or deny any well-established results. His own results seem reasonable.

I find he overstates the significance of what he is doing, and, more troubling, he doesn't make enough effort to tie in what he is doing to what has already been done, but this doesn't add up to being a crank.

Did you read the excerpt I provided above where the guy doesn't understand that the Naturals contain no upper bound? Did you see the part where he attempts to add "grossone" numbers without explaining what that even means?

Granted, I didn't study this paper much, but my crank-o-meter was going crazy for the part I did read. [Though, if you respond that I just misunderstood what he had written in these parts, I'll revisit it]

I second Charles' feelings about this paper, and add that, as presented, it is at best an amusement, since he really doesn't offer any motivation for his system, except to make infinities behave more like we expect numbers to.

Given my inclination is to feel that we tend to treat infinity too much like a number already, I have never stayed up nights wishing for this. ;-)

I'm curious what those who like this paper, e.g. Jacques, see in it? I'm guessing they are just warped in different way than I am. ;-)

Jacques response was not directed at this (the topic) paper.

I've edited my comment to make it clear that the paper I liked was the one by Mellies, thanking Noam for the reference. The topic paper rates highly on my crank-meter. I even believe that there are unexplored, serious approaches to dealing infinity still to be discovered - this particular one just doesn't strike me as a candidate.

I've given the paper that Noam and Jacques were referring to its own story.

That makes more sense. ;-)

I asked a question a week or so ago on MathOverflow:

Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by:

1. The leaves 1 and x, for x drawn from a class of variables; and
2. Closed under the binary functions of addition, multiplication, exponentiation, and division.

...

I'm interested in a decision procedure for [which of two functions in one variable is faster growing]. Do we have either the existence of canonical forms, or decidability for this problem?

As it happens, the Grossone numbers considered as polynomials seem to be a generalisation of these REXes, allowing subtraction in addition to the operations I described above. If that's right, then Sergeyev claims to have such a decision procedure. The MathOverflow turned up a nonconstructive result, but no constructive result.

I plan to look more closely at Sergeyev's other papers, but I thought that in the meantime, maybe this is of interest.

I revisited the paper last night and his viewpoint makes sense to me now, and I agree with many of the comments you made in your mini-review. I believe you're right that his "grossone" is a stand-in for a large natural that is divisible by every number up to some sufficiently large (smaller) natural. In fact, (and I fear I risk violating Postulate 2 by just talking about this) I think a model for his intended numbers would be arbitrary functions from admissible "grossone" values (some subset of the naturals) to the rationals. It's not just polynomials because he says things like this:

Numbers p_i in (13) called grosspowers can be finite, infinite, and infinitesimal (the introduction of infinitesimal numbers will be given soon), they are sorted in the decreasing order

Also he seems to think that his "record" defining e in (27) on p20 is a valid number, despite that it doesn't fit into his normal form. So if he's honest, it's not really a normal form. Luckily, no justification was given for his choice of normal form other than that it's a generalization of normal normal forms in decimal.

Of course, there's not going to be a total order on arbitrary functions, which he simply assumes is the case, so my model above doesn't exactly match what he has in mind.

So is the guy a crank? I guess it depends what exactly you mean, but the paper is certainly deeply flawed, and I don't really expect anything interesting to come of it. You can already do everything he can do in this system by working with a "sufficiently large integer N."