It seems that a great many modern mathematicians are stuck back in the middle ages when it comes to the concept of infinity even though it has been around 60 years since many (if not all) of the issues were resolved with the mathematics of hyperreals.
Personally, I define a term for that particular number so as to make the math simpler to visualize;
infA ≡ [1+1+1+…+1]
and
smlA ≡ 1/infA
InfA represents a simple infinite series that does not converge and it is exactly defined and thus does not merely have the quality of being infinite, but represents an exact degree or scale of infinity.
InfA + 1 =/= infA
It can be said (and has been countless times) that anything added to infinity is still infinity. That has been both true and false, depending on how the term “infinity” is meant. It is true in that both sides of that equation have the quality of being infinite, but it is not true that they have the same value or scale of infiniteness.
By definition if something is added to or subtracted from something else, the result is not equal to the its former value, even if it still has many similar qualities. One can say that a large number “L” plus a small number “n”, is still a large number. But that does not mean that L+n = L.
If for example we multiply infA times 2, we get;
2 * infA = 2 * [1+1+1+…+1] = [2+2+2+…+2]
[1+1+1+…+1] =/= [2+2+2+…+2], even though they are both infinite in quality.
This can be seen by merely beginning a comparison of the series and see where it leads.
[1] =/= [2]
[1+1] =/= [2+2]
[1+1+1] =/= [2+2+2]
and since the degree of difference between them is always the same (times 2), it can be seen that;
[1+1+1+…+1] =/= [2+2+2+…+2]
Even though both sides are infinite, they are not equal in their scale or degree of infiniteness.
But I find it interesting that mathematicians seem to commonly operate on a series in an effort to compare them in a way similar to the following;
[1+1+1+…+1] = [(1+1) + (1+1) + (1+1)+…(1+1)] = [2+2+2+…2]
One cannot arbitrarily couple terms within a series and declare them to be equal to the sum and then presume to do so for an infinite length and still have a valid comparison. The reason for this is that the infinite series has been “stretched” to compensate for the inadvertently added terms. And in this case, has been stretched to twice the original scale.
I noticed in a Wiki article concerning the Harmonic Series, that a comparison was being made in the following manner;
Note that in the first series, the terms are getting smaller and smaller. But in the second series, they are not. The second series never approaches a zero term, whereas the first series does. In order for the first to be actual equal to the second, it would have to be exponentially stretching merely to keep up with the growing number of terms of the second series. Term by term comparisons show that the first series is becoming greatly smaller than the second as the number of terms approaches infinity.
But as you can see, the first series is getting stretched so as to make it seem as though it is equal in value to the second series. Those are not equal values because the number of terms determines the scale of the infinity involved.
[1] = [1]
[1+1/2] = [1+1/2]
[1+1/2 + 1/4] [size=150]<[/size] [1+1/2 + 1/2]
[1+1/2 + 1/4 + 1/4] [size=150]<[/size] [1+1/2 + 1/2 + 1/2]
Yet this proposed comparison (from the middle ages) is still used to prove that the Harmonic series sums to an infinite value. It is an invalid proof. One cannot stretch nor shrink infinite series’ when comparing them unless both series are altered equally.