Cantor's method of comparing set size uses one to one correspondence i.e. existence of a bijection. Now, set A = (0, 1) and set B = (0, 2). Using the function x → 2 x, every element of set A can be uniquely mapped to that of set B. This is irrefutable.

However, what also appears correct is that size of set B has to be greater than set A because every element of set A is of the form 0.xxxxxx... and every element can be transformed by replacing 0 with 1, this makes it a member of set (1, 2). Set B consists of this set and the initial set A. Clearly this is also intuitively correct. And therefore set B is larger than set A. What is the explanation of this counter intuitive result? Is it something about our understanding of infinite sets? Has any philosopher seriously considered the possibility that set theory might be just wrong or our interpretation might be skewed?


It is simple.

We have two approaches to "measuring" the size of two sets. The first one is based on counting and it is the usual one we are accustomed to with finite sets : if the number of elements of set A is lesser than the numbe rof elements of set B (and this amounts to saying that A is a proper subset of B) then we say that B is Greater than A.

When we move to infinite sets, the issue is that we have no easy way to extend the usual way of counting the elements of a sets.

Thus, Cantor's intuition is that we can generalize the usual way of counting the elements of a set - i.e. defining a bijection between the set and a natural number - and founding it on the concept of bijection between sets whatever :

two sets (finite or infinite) are equinumerous when there is a bijection between them.

This new definition does not add nothing in the case of finite sets : two finite sets are equinumerous exactly when they have the same number of elements (the number we get counting them).

Cantor's new way of counting the elements of a collection is based on the notion of one-to-one correspondence, which is prior to (and independent of) the concept of counting number.

The basic result is that the new definition is applicable also to infinite sets and give us a way to measure their size.

The "unpleasant" result is that two infinite sets can be equinumerous also if one is a proper subset of the other.

This means that - for infinite sets - the well-know principle expressed by Euclid with :

Common notion 5. The whole is greater than the part,

is not applicable.

The fact that the use of the said principle may lead to problems with infinite sets was already known since at least Middle Ages; see the so-called Galileo's paradox.

So yes, the issue is related to

our understanding of infinite sets.

With infinite sets modern mathematics has stretched our intuition that is clearly based on our experience with "finite" facts.

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  • Does that mean conventional meanings of the words 'equal in size' don't hold in case of infinite sets? All we can say is that we can show the existence of a bijection, and that's it. That it is a purely technical result, and any philosophical discussion (i.e. talking about it in natural language) only leads to wrong distorted ideas? – Ajax Jun 21 '19 at 15:20
  • Because, if taking bijection to literally imply equal number of elements, one needs to list them -to be able to 'say' that there are equal number of elements. Since this cannot be done, and we can always list more numbers in either set to satisfy bijection, they are sort of equal in 'some' way, but definitely not in a strictly countable sense? That ideas of countability don't work here. – Ajax Jun 21 '19 at 15:24
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    @Ajax - the conventional meanings of the words 'equal in size' implies - for finite sets - that neither of them is a proper subset of the other. This is not the case for infinite sets. – Mauro ALLEGRANZA Jun 21 '19 at 16:04

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