A favorite notion held dear by Platonists is the notion of an 'infinite decimal expansion'. For a fraction, the decimal representation of the fraction is simply a fraction with 10, 100, 1000, 10000, etc as the denominator (the denominator is the number under the line in a fraction), so that we can have, for example 9⁄10, 45⁄100, 230⁄1000, 123⁄10000, etc. But it would be a mistake is to think that there is such a thing as a fraction with a limitlessly large denominator.
Of course, when trying to calculate the value of the square root of 2 (the value that, when multiplied by itself gives the value of 2) we can continue to generate more and more fractions (or decimal places), viz.
This process of generating more and more decimal places can be continued indefinitely. There is no limit to the number of times you can add on an extra digit.
and that there are limitlessly many such fractions that continue to get smaller and smaller, all added together.
Unfortunately for the Platonist, this attempt to solve the conundrum doesn't work. Every one of the fractions has to have a denominator of a definite size, that is, the denominator must be a 1 followed by a definite number of zeros. The alternative is that there are fractions in this infinite sum that do not have a definite value, and that their denominator is 1 followed by an infinite number of zeros – which contradicts the original definition.
And if you have more fractions to choose from, you will have many more different finite sums of fractions. The above was a finite collection of fractions from which there can only be a finite number of different sums. But you can also have a concept of a limiting value for all the finite sums that can be got from a limitlessly large collection of fractions. In this case, there are a limitless amount of such sums – but the sums are still all finite. Since all these finite sums are derived from a concept of a collection of fractions that has limitlessly many fractions, then, in order to be able to be able to determine if there is a limit which none of these finite sums can exceed, there must first of all be a way of expressing that concept of a collection of limitlessly many fractions in a finite way, using a finite combination of symbols. Since the concept involves a limitless amount of such fractions, the only way to express that concept is by way of a definition. And similarly, once you have that finite definition, you may be able to use that finite definition to define a limiting value. And that limiting value will be a value that no finite sum of fractions, taken from the overall collection that is given by a definition, can exceed. None of this is anything new; most of the methods of calculating the limits for various series of fractions were known over a hundred and fifty years ago.
and so on, which are all numbers that are close to the number Pi, relative to the number 3. And a number has been calculated to billions of digits and this is also closer to the number Pi, relative to the number 3.14159265358979323846. But, in the same way as it is a mistake to think of a 'large' number as being close to infinity, it is a mistake to think that the number 3.141592653 is close in some absolute sense to the value of Pi. It isn't, since there are still infinitely many real numbers between 3.141592653 and Pi. It is only close in relative terms, in that it is closer to Pi than 3.1416.
For more demonstrations of contradictions arising from the Platonist beliefs in the 'existence' of 'actual' infinite sets, see The Courant & Robbins contradiction and Sums of infinitely many fractions: 1.