There are a large number of mathematical puzzles associated with infinity, and because I don’t accept the notion of a completed infinity, I feel that most of them are not actually paradoxical. Many people (even mathematicians who do not study mathematical philosophy) are not clear about the difference between potential and actual (completed) infinity: here is the first paragraph on that difference from wikipedia (https://en.wikipedia.org/wiki/Actual_infinity):
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted with potential infinity, in which a non-terminating process (such as “add 1 to the previous number”) produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps.
It is important to note that both systems are happy to use the term infinity: if they did not believe in infinity, then they would hold a different mathematical position called finitism. They use the term in significantly different ways, however. Mathematicians who believe in completed infinities (like Aleph Null) treat infinity like a number. Mathematicians who do not accept completed infinity, and believe only in potential infinity, treat infinity like a limit process (which has a more complicated representation than a finite number).
The subject of this article is a paradoxical statement which is widely held to be true by mathematicians who believe in a completed infinity:
Most people are not intuitively comfortable with this statement, because it implies that two different numerical (decimal) representations correspond to a single number. The “proof” looks as follows:
3) Multiply both side by 10 to get :
4) Subtract (3) from (2) to get:
There are at least two ways out of this paradox. Most mathematician these days believe in completed infinity, so they are forced to accept (2) & (5) and therefore (1). If you hold an intuitionistic view of mathematics, however, you can get rid of the paradox by handling infinite representations as limit series.
Before we go farther, I should mention that whether or not my reasoning is sound, it is mathematically novel (at least according to some of my mathematician friends). If you happen to have conversations about intuitionism with narrow-minded mathematicians, they will not be sympathetic to your view (despite the fact that a number of genius mathematicians have advocated this stance, such as Arend Heyting, L. E. J. Brouwer, Stephen Kleene, and Henri Poincare).
To return to how an intuitionist might represent a number which has “infinite length”. Obviously, a representation like where the nines run “to infinity” will not be acceptable. However, we can turn it into a limit series as follows:
In other words, repeating decimal representations are written as a limit where the number of terms approaches infinity. The significant difference between the limit representation and the repeating decimal representation is that the former is more clearly seen to be a process, not a “number whose length has a cardinality of aleph null”, which again is disallowed (in intuitionism). Given this representation, let’s re-write (4):
6) Subtract (3) from (2) to get:
The result from (6) is not 9, it is (commonly written as) , and for an intuitionist, it is 8.1 after one term, approaching a limit of 9 only “at” infinity (which, on that account, we never reach, and therefore we do not find that (1)). Now, why did we arrive at a different result when we used a limit formula, instead of the subtraction formula? My “intuitive” answer is that we paid attention to the number of terms in the series expansion. In other words, when in (4) we perform the subtraction , we are subtracting term i in the second series expansion from term i+1 in the first series expansion (i.e. we are ignoring the first term). While this might be OK if you have a completed infinity of number of terms, an intuitionist account must yield a correct series at every (finite) step, and combine terms under the appropriate loop variable, i. That means that after three terms, we have:
In other words, , and as a result, intuitionists who do not believe in a completed infinity do not have to accept that .
OK, mathematicians who can either logically tear me apart or pat me on the back, I welcome your feedback.