0 / 0, transfinite numbers, & related.


Rational Skeptic
Valued Senior Member
This Thread was started due to some questions by Esoteric. I have tried to answer his questions without using esoteric mathematical notation and language.

I am aware that thiese topics have probbly beeen covered in other threads, and apologize if any are offended by redundant threads.

Note that divison is defined as the inverse of multiplication.
  • If Divisor * Quotient = Dividend, then Dividend / Divisor = Quotient

    More simply: If A * B = C, then C / B = A and C / A = B
Consider 0 / 0 in the context of the above.
  • 50 * 0 = 0

    1000 * 0 = 0

    Hence 0 / 0 could be 50 or it could be 1000.

    In general, X * 0 = 0, for any finite value of X

    Hence, it could be claimed that 0 / 0 = X, where X is any finite value which pleases you.
The above surely suggests that 0 / 0 is ambiguous (in ordinary English) or undefined (in mathematical jargon).

The above provides some insight into the problem of division by zero, although it does not help much in dealing with infinity.

Now consider 1 / 0 = X, which implies that X * 0 = 1
  • X cannot be any finite number.
Saying that X is infinity is somewhat ambiguous since infinity is not a very well defined number.
  • It can be argued that the number of integers is a transfinite number (jargon for an infinite number).

    It can also be argued that the number of points on a line segment is a transfinite number, which is larger than the number of integers.

    There are more transfinite numbers than the above two.

    If 1 / 0 = Infinity, which transfinite number is it?
Once again, there is ambiguity in the absence of some special context and/or some special definitions, causing mathematicians to state that 1 / 0 is undefined or perhaps an unallowable operation.

Back on the subject of 0 / 0

As suggested above, 0 / 0 must be considered ambiguous or undefined without some context.

For context, consider sin(X) / X, with X measured in degrees.
  • For X = 60, sin(X) = 69.282 032

    For X = 10, Sin(X) / X = 57.587 705

    For X = 1, sin(X) / X = 57.298 688

    For X = 0.1, sin(X) / X = 57.295 809

    For X = 0.01, sin(X) / X = 57.295 780

    As X gets smaller and smaller, sin(X) / X gets closer and closer to 180 / Pi,
    which is approximately 57.295 779 513

    More formally: As X approaches zero, sin(X) / X approaches 180 / Pi
In the above context, 0 / 0 is said to be approximately 57.295 779 513

On the subject of arithmetic with infinity, which mathematicians usually call transfinite arithmetic.

Since there are a lot of different transfinite numbers, expressions involving infinity are considered ambiguous or improper or unallowable.

The subject of transfinite numbers is a bit esoteric, and I would have difficulty doing it justice in a thread at this forum. It is usually handled using what is called “Set Theory.” A man named Cantor worked out most of the principles prior to 1900.

In general, I do not think that transfinite arithmetic is handled like ordinary arithmetic. For example, I think that addition of transfinite numbers is viewed as the combining of two sets rather than as addition in the ordinary sense of the word.

While I remember a lot of the concepts developed by Cantor and others, I do not remember much about doing arithmetic with transfinite numbers.

The following are some basic ideas developed by Cantor, expressed informally.
  • Two sets have the same number of members if the members can be matched up with none left over.

    A set has a transfinite number of members if it has the same number of members as a subset of itself.

    The first transfinite number is called Aleph<sub>0</sub> and is the number of members of the set of all integers. Note that you can match 1 with 2, 2 with 4, 3 with 6, n with 2n, et cetera. This matches (or pairs) all the integers with just the even integers, leaving no member of either set unmatched. The even integers are a subset of all the integers, hence Aleph<sub>0</sub> is a transfinite number.

    A set can not be matched with the set of all subsets of itself. Hence, the set of all subsets has more members than the original set.

    The set of all subsets of Aleph<sub>0</sub> has more members than Aleph<sub>0</sub>, and is called Aleph<sub>1</sub>. This can be kept up, defining more and more transfinite numbers.

    There is a tricky method of matching all the rational numbers with all of the integers, showing that the set of all rational numbers has the same number of members as the set of all integers (both have Aleph<sub>0</sub> members).

    There is a cute proof that you cannot match all the real numbers with all the integers, showing that the set of all real numbers has more members than the set of all integers.
I hope the above gives some clues about the subject of transfinite numbers.
I was thinking Cantor was referring to infinity as a number, which is where the confusion stemed from, which is why my equations were bad. Thanks for clearing that up. I was thinking of transfinite numbers as a number in laymens terms, but it is a way of measuring/numbering infinite sets by determining the size of the set. He was talking about infinite sets (a collection of numbers or objects) is not a number, I was mixing up the two. A set or a group of sets can be numbered. From that deduction, infinite sets can be numbered, thus the transinfinite number(s). Infinity itself is not a number. So sets of infinite numbers come in different sizes (inf+1(set2) > inf(set1). Good.
Quick comment:

It is more usual to consider the expression sin(x)/x with x in radians rather than degrees. In that case, the limit of sin(x)/x as x goes to zero is 1.
x/0 is not infinity at all, it is impossible.

x/0+ (where 0+ is a number as close as we want it to be to zero) grows to infinity as 0+ gets smaller.
A Ht: I am a bit rusty on this topic, but I think it is more proper to say that
  • For finite n, n / x grows without bound as x approaches zero.
Saying that n / x grows to infinity implies some definition of infinity, while grows without bound expresses the concept without requiring a definition of the term infinity.

Frankly, I am not sure what transfinite number (if any) can be assigned as the limit of the above. If forced to give an opinion, I would say the limit is the transfinite number associated with the set of all real numbers.
Dinosaur said:
Saying that n / x grows to infinity implies some definition of infinity, while grows without bound expresses the concept without requiring a definition of the term infinity.

"Grows to infinity" is acceptable terminology and does not require you to define "infinity" as any kind of number at all. It has meaning in the sense of limits, the "grows without bound" that you mentioned. Just like the symbols "lim f(x)=infinity" are given meaning in calculus texts without defining the symbol "infinity" on it's own. lim=infinity is just a convenient way of writing that the limit diverges in a specific way, it's not an equation in the usual sense.