To me, infinity (in mathematics) means the sum total (either by counting or by the sum product.) of all numbers and all sets and subsets of numbers, regardless if they are “countable” or “uncountable”. Comparing individual sets of numbers may yield a difference in count, but the count of any set can never exceed the total count of all the numbers and sets and its value can never exceed the total value of infinity.
I agree, because mathematics is a calculus. If an algorithm of counting comes out with conflicting results, then either the algorithm of counting is wrong or we have miscounted.
Let us consider the odd numbers of (O) as oranges, the even numbers of (E) as apples and the natural numbers of (N) as orange and apples.
For finite sets, using one-to-one correspondence to determine whether there are more oranges, apples or oranges and apples, it is obvious since there are no apples in (O) and no oranges in (E) while (N) has both oranges and apples, therefore (N) must be bigger than either (O) or (E).
However, for infinite sets, the elements of (N), (O) and (E) are inexhaustible, by substituting apples for oranges and vice versa when we compare the size of (N), (O) and (E) (using one-to-one correspondence), the result is (N)=(O)=(E), which is how Cantor defined infinite sets.
Why is there a contradiction in the two results? Cantor wrote that as we “ascent into the realm of the infinite, we must accept this paradoxical result”. Are we asked to accept this absurd result which only satisfy Hume’s Principle, but contradicts Euclid’s principle: The whole is greater than the part? Has he any evidence to justify this claim or is it just an arbitrary rule so that he can define an infinite set?
So, where is the catch here?
1. We must compare apples with apples or oranges with oranges, not apples with oranges and vice versa. This rule must be applied rigorously to determine the size of sets with distinct identical elements.
2. Infinity is not a number, like 99999 and therefore cannot be treated as such.
3. An infinite set is a recursive rule to generate elements, it is not a list like the elements of a finite set. One should not compare a recursive rule to a list. They are different things.
4. Infinity is not defined precisely and cannot be defined precisely, otherwise it is not infinite. Therefore, one cannot proceed to (mathematically) treat infinity as though it is a known mathematical entity.
5. Infinity + infinity is meaningless (does not make any sense).
Cantor’s definition of an infinite set as (N)=(O)=(E) is wrong. The whole edifice of infinite mathematics is based on the wrong assumption of only using Hume’s Principle to compare the size of sets with distinct identical elements, (O), (E) and (N).
My appreciation for this essay on Wittgenstein’s Philosophy of Mathematics for clarifying some of the issues:
In sum, because a mathematical extension is necessarily a finite sequence of symbols, an infinite mathematical extension is a contradiction-in-terms.
Since a mathematical set is a finite extension, we cannot meaningfully quantify over an infinite mathematical domain, simply because there is no such thing as an infinite mathematical domain (i.e., totality, set), and, derivatively, no such things as infinite conjunctions or disjunctions.