What's wrong with this idea implying a contradiction in the concept of infinity?

Question

Let be a set of something with infinite cardinality. Let ℕ be the set of natural numbers ℕ. There is a bijection f: ℕ → . They both have the same cardinality, denoted as |ℕ| = ||. Bijection means that every element of ℕ can be mapped to every element of . "Every" means that there are no further elements in ℕ that can be mapped to anything else. Now consider a new set ₂, which contains all the elements of , with the addition of element x, which is not an element of : ₂ = ∪ {x} (x ∉ ). Yet, it is a basic principle that || = | ∪ {x}|, in which case there is a bijection between ℕ and ₂. So there are no further elements in ℕ after mapping each of its elements to , and at the same time there is an extra element in ℕ which can be mapped to ₂. That is a contradiction.

To clarify:
Proposition (1): Each of the infinite elements of ℕ is used in the bijection between ℕ and . No elements of ℕ remain unused in this bijection.
Proposition (2): There is an element of ℕ remaining unused in this bijection, and it is this element that is required to create a new bijection between ℕ and ₂.
Proposition (1) contradicts Proposition (2).

The contradiction cannot be resolved by stating that we can create separate bijections for and ₂, since they are both bijections from the same set ℕ. Although ℕ has infinite cardinality, all its elements are employed in the bijection between it and (which also has infinite cardinality, and so requires the use of all elements of ℕ).
Such an attempted resolution amounts to saying that part of an infinite set is as large as the whole infinite set, which is clearly a logical contradiction.

Answer 1

There is no contradiction. Your usual intuition of cardinality with finite sets breaks down when you have infinite sets.

To generalize your problem a bit more, let's say there are infinitely many elements in $E_2$ that are not in $E$. For instance, take $E$ to be the set of positive even integers and $E_2=\mathbb{N}$. Obviously, $|E_2|=|\mathbb{N}|$ (choose the identity map for a bijection). However, we can still form a bijection between $\mathbb{N}$ and $E$ (different than one between $\mathbb{N}$ and $E_2$), namely $f:\mathbb{N}\to E$, $f(n)=2n.$ Hence, $E$ and $E_2$ have the same cardinality (both are countable).

Hilbert's hotel is a good illustration of how our intuitions of finite sets break down with infinite sets.

Answer 2

If $f:\mathbb N \to E$ is your bijection, and $x$ is your new element that's not in $E$, then $ g:\mathbb N \to E \cup \{x\}$ given by $$ \begin{align} g(1) &= x\\ g(n) &= f(n-1) \textrm{ for }n \ne 1\\ \end{align} $$

is also a bijection.

When you say

So there are no further elements in ℕ after mapping each of its elements to , and at the same time there is an extra element in ℕ which can be mapped to ₂.

the 'map' in "... after mapping each ..." refers to the map $f$, and the 'map' in "... which can be mapped ..." refers to the map $g$.

Additionally, when you say in your comment that

If every element of N is required to create a bijection between N and E,

you're mistaken. Not every element of $\mathbb N$ is required to create a bijection with $E$ - you could create one with just $\{15, 16, 17 ....\}$. That's the fundamentally weird thing about infinite sets: they can be put in 1-1 correspondence with a proper subset of themselves.

Answer 3

To judge whether one thing exists, an example of other things alone is irrelevant.

To judge whether there's a lion outside your village, catching a wolf and then claiming "this animal is not a lion" is irrelevant.

To judge whether there's a bijection from $\mathbb N$ to $\mathbb E_2$, proposing a function that is not surjective (hence not a bijection) is irrelevant. If one can propose an actual bijection, then there's a bijection.

Answer 4

"The Case Against Infinity", which I have now found (https://philarchive.org/rec/SEWTCA), by Kip K Sewell, argues much better than I can why, indeed, the contradiction I identify, and many more, mean that infinity is not a coherent or logical concept, even if it can be used operationally in mathematics. Well worth a read, and I think is a more detailed answer to my question than any of the previous answers given. Specifically, with regards to an infinite set of marbles used as an example, he writes:

by virtue of being complete, the infinite set of marbles can be divided up leaving two infinite sets with one for you just as in the equation ℵ0 - ℵ0 = ℵ0. However, by virtue of being limitless, the infinite set of marbles cannot be divided up, and so one person is left without any marbles just as the equation ℵ0 - ℵ0 = 0 states. The mathematical contradiction in the quantity left over is a result of the logical contradiction of an infinite set being both “divisible” and yet “not divisible.” Conversely, the mathematical contradictions in dividing or subtracting infinite sets also imply logical contradictions in “removing” infinite subsets from infinite sets of objects, like marbles, in the real world. Subtracting or “removing” the odd numbered subset of marbles from the total set of marbles (ℵ0 - ℵ0) cannot leave a remainder that is both infinite (ℵ0) and not infinite (0) simultaneously, which is both a mathematical and a logical contradiction. Since logical contradictions cannot manifest in reality, there can be no infinite set of marbles or an infinite set of anything else in the real world.

This relates directly to my original question, and I consider it the correct answer.