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Binary addition on natural numbers is defined using simple statements:

$$n + 0 = n$$ $$a + succ(b) = succ(a + b)$$

Binary addition of real numbers may be again defined using simple statements (I won't list them here).

Could we come up with a list of similar simple statements that would define summation of infinite sequences of real numbers, in a way that does not, even intutitively, rely on the notion of limit or convergence?

These are examples of such "simple, not-limit-using" statements (though I'm not claiming they are necessarily good candidates):

  • Some base cases:
    $sum(0, 0, 0, ...) = 0$
    $sum(0.5^1, 0.5^2, 0.5^3, ...) = 1$
  • Respects binary sum:
    $sum(n_0, n_1, n_2, ...) = r$ implies $r = n_0 + sum(n_1, n_2, ...)$
    $sum(n_1, n_2, ...) = r$ implies $n_0 + r = sum(n_0, n_1, n_2, ...)$
  • Respects binary multiplication:
    $sum(n_0, n_1, n_2, ...) = r$ implies $sum(n_0 * c, n_1 * c, n_2 * c, ...) = r * c$
  • Zigzaging rule:
    $sum(A) = a$ and $sum(B) = b$ and $zigzag(A,B,C)$ implies $sum(C) = a + b$

Helper definitions:

For sequences of reals $A$, $B$ and $C$, $zigzag(A,B,C)$ holds iff there exists a sequence of naturals $I$ such that $A = (C_{I_0}, C_{I_1}, ...)$ and $B$ equals $C$ with the elements at indices $I$ removed.

I'm afraid that if this question does not have an affirmative answer, it would be very hard to give any, because I have not given precise enough description of what is simple and non-limit-using. So I'll be happy to also accept counter-examples that cannot be proven to equal to their limits using my example statements (or other simple statements of the answerer's choosing, if they wish to add some).

I'm also aware none of my examples can be used to prove a particular sum diverges. Perhaps we could define summation as a least fixed point? But feel free to restrict yourself to converging sequences, I guess.

This question was motivated by this one, where the asker has trouble accepting the conventional definition of infinite summation, which got me thinking about whether one may avoid using limits for defining it.

Jozef Mikušinec
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    Simply put, no. An "infinite sum" is not a sum. You can't define it as a sum because nothing is being added. It only exists as some sort of limit, like it or not. – MPW Dec 01 '23 at 14:36
  • You wrote "Binary addition of real numbers may be again defined using simple statements (I won't list them here)." Maybe you should try to list them, because you'd need to address what a binary representation even means with real numbers. Describing expansions in any base is going to use some limiting process even when you may not directly use the word "limit" (you'll have to address why some real numbers have more than one binary representation while others don't). – KCd Dec 01 '23 at 14:40
  • In your second base case the result is not equal to $1$ if you do not use a limit. It will always be less than $1$ – Vasili Dec 01 '23 at 14:42
  • @MPW I'm not claiming infinite sum is or is not a sum. I'm not asking for it to be defined as a sum. I'm asking whether there exists an equivalent definition of it that does not use limits – Jozef Mikušinec Dec 01 '23 at 14:49
  • @KCd I don't need to use expansions. Reals can be defined axiomatically and we need not care what a real number is. The axioms for real numbers seem simple to me, perhaps with the exception of the least-upper-bound property, but even that one I consider simpler than the definition of a limit. – Jozef Mikušinec Dec 01 '23 at 14:53
  • @KCd I disagree that we need to use any form of limits to define binary addition of real numbers. We define a semigroup on subsets of the integers which have a maximum element (mimicing the concept of binary addition, I won't write it here as it is a bit long). We can then quotient this semigroup by letting a subset $S$ which contains all values less than $n$ but does not contain $n$ to be equal to $S$ but with $n$ and without anything less than $n$. This new object is equivalent to binary addition on positive reals, we can then use a standard method to get the reals with addition from this. – Fishbane Dec 01 '23 at 14:53
  • @Vasili The base case forms a proposal for an alternative definition of $sum$. The base case then equals 1 as a matter of definition – Jozef Mikušinec Dec 01 '23 at 14:54
  • @JozefMikušinec starting with the description of real numbers that you gave, how do you make contact between that description and binary representations, or even show each (positive) real number has a binary representation? – KCd Dec 01 '23 at 15:30
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    @KCd I think you are misunderstanding the use of the term binary. As I understand it they mean the sum of $2$ numbers can be defined (as in a binary operation). It has nothing to do with binary representations. – Fishbane Dec 01 '23 at 15:34
  • Note: I just noticed my sandwich rule is very likely wrong, because it is the only rule that assumes $sum(X) = lim(X)$ instead of $sum(X) = lim_n^{\infty}(\sum_m^nX_m)$. I'll double check and fix it. – Jozef Mikušinec Dec 01 '23 at 15:46
  • I agree with the "no" answer of @MPW ... for series of real numbers (or complex numbers) we must use limits. There are other cases though. Definition for series of cardinal numbers does not use limits. Another case to investigate: surreal numbers; can we define sum of an infinite series by specifying some left and right options for it? – GEdgar Dec 01 '23 at 16:47
  • P.S. I have an old (1860s) "algebra" textbook. There is a chapter on series that does not mention limits. But (it turns out) the only series covered are geometric series. – GEdgar Dec 01 '23 at 16:50
  • @Fishbane indeed I was misunderstanding what "binary" was about. – KCd Dec 01 '23 at 17:39
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    I think the answer's going to be "no", but I like the attempt so I up voted. – JonathanZ Dec 01 '23 at 17:50

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In your comments, you said you consider the least-upper bound property to be "simpler than the definition of a limit".

These two concepts are closer to each other than perhaps you might realize.

To demonstrate this, let me rewrite the definition of an infinite sum which is expressed purely in terms of finite sums, of the least upper bound property, and of its (easily derived) consequence known as the greatest lower bound property.

Given a sequence of real numbers $(a_n)_{n \ge 1}$, with its associated sequence of partial sums $$S_N = \sum_{n=1}^N a_n $$ and given a number $L$, the infinite summation expression $$\sum_{n=1}^\infty a_n = L $$ means this:

the sequence of real numbers $D_N = \bigl| S_N - L \bigr|$ is bounded above and below ("below" is easy because $D_N \ge 0$); and applying the least upper bound property to define the sequence $$U_M = \text{l.u.b.} \{D_N \mid M \ge N\} $$ we have $$\text{g.l.b.}\{U_M \mid M \ge 1\} = 0 $$

In order to write this definition, all that I really did was to apply a well-known equivalent statement for the definition of a limit. Namely, given a sequence $S_N$ of real numbers and another number $L$, the limit expression $$\lim_{N \to \infty} S_N = L $$ means this:

[repeat what's written above]

In fact I agree with you that the least-upper bound property is "simpler than the definition of a limit", and that perhaps is why it was chosen to be one of the axioms of the real numbers, as opposed to trying to axiomatize the concept of a limit.

But the least-upper bound property is rather clumsy to apply in some situations. So from an axiomatic viewpoint, what one does is to realize that many, many, many clumsy applications of the least upper bound property may be reformulated as elegant applications of the concept of limits.

Perhaps one might think it hard to learn limits, but they are easier to apply once you practice enough.

Lee Mosher
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