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Here's what I've tried so far.

Let $A$ be a set and suppose $|A| < |\mathbb{N}|$. By the definition of less than for cardinalities (I'm reading out of Hrbacek's Introduction to Set Theory), this means that there exists a one-to-one function $f$ from $A$ onto a subset of $\mathbb{N}$, but there does not exist a one-to-one function from $A$ onto $\mathbb{N}$ itself.

In the book, finite is defined as in bijection with a natural number $n \in \mathbb{N}$. The natural numbers are defined in a weird way:

$$0 = \varnothing$$

$$1 = \{0\} = \{\varnothing\}$$

$$2 = \{0,1\}$$

$$3 = \{0,1,2\}$$

... and so on ad infinitum.

My intuition is this: I want to show that $A$ is in one-to-one correspondence with a finite subset of $\mathbb{N}$, and once I've done this it's trivial to show that $A$ is finite itself (the composition of bijections is a bijection, so $A$ is in bijection with some $n \in \mathbb{N}$, and we're done). I just have no idea how to find the correspondence with a finite subset of $\mathbb{N}$.

If I show that any infinite (infinite is simply defined as not finite) subset of $\mathbb{N}$ is in bijection with $\mathbb{N}$ itself, this would work ...

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  • $\begingroup$ This has been asked before. What you want to show is that for any infinite set $A$, $\aleph_0\leqslant \# A$. This can be proven using the axiom of choice. $\endgroup$ – Pedro Tamaroff Dec 17 '14 at 2:33
  • $\begingroup$ @PedroTamaroff what is $\#A$? EDIT: Are you sure this requires the Axiom of Choice? $\endgroup$ – Vincent Luo Dec 17 '14 at 2:36
  • $\begingroup$ Consider the following argument: since $A\neq \varnothing$, there is $x_0\in A$. Since $A$ is infinite, there is $x_1\in A\smallsetminus x_0$. Since $A$ is infinite, there is $x_2\in A\smallsetminus \{x_0,x_1\}$. Since $A$ is infinite, $\ldots$. Then $x_i\longleftrightarrow i$ is a bijection, i.e. there is a subset $A'\subseteq A$ equipotent with $\Bbb N$. $\endgroup$ – Pedro Tamaroff Dec 17 '14 at 2:37
  • $\begingroup$ Then there is a bijection between $A$ and $\mathbb{N}$ via the function that takes $\{x_0, x_1 ... x_n\}$ to $n$? $\endgroup$ – Vincent Luo Dec 17 '14 at 2:39
  • $\begingroup$ @PedroTamaroff Okay I think this works, the recursion part seems a bit non-rigorous but it makes complete intuitive sense to me $\endgroup$ – Vincent Luo Dec 17 '14 at 2:40
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Given an infinite set of $A$ we can define an injection from $A$ to $\mathbb N$ by the inclusion mapping. define the injective map from $\mathbb N$ to $A$ where $1$ goes to the minimum element of $\mathbb A$, where $2$ goes to the minimum element of $A\setminus f(1)$ and in general map $n$ to the minimum element of $A\setminus \{f(i)|1\leq i\leq(n-1)\}$.

So if you have an infinite subset of $\mathbb N$ it has the same cardinality as $\mathbb N$

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  • $\begingroup$ Oh, I thought I could do that, sorry I haven't taken set theory yet. $\endgroup$ – Jorge Fernández Hidalgo Dec 17 '14 at 2:58
  • $\begingroup$ You can prove that $A$ can be well-ordered. Define the order $\prec$ over $A$ such that $x\prec y$ iff $i(x)<i(y)$, where $i$ is an injection from $A$ to $\Bbb{N}$. $\endgroup$ – Hanul Jeon Dec 17 '14 at 2:58
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    $\begingroup$ I was just using the order on $\mathbb N$ $\endgroup$ – Jorge Fernández Hidalgo Dec 17 '14 at 2:59
  • $\begingroup$ $A$ is well ordered using the induced order from $\mathbb{N}$. $\endgroup$ – copper.hat Dec 17 '14 at 3:00
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    $\begingroup$ You could even avoid Cantor-Bernstein by noting that the function mapping $f(n)$ to the minimum of $A\setminus\{f(i)\mid i<n\}$ is a bijection. $\endgroup$ – Asaf Karagila Dec 17 '14 at 5:54
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Let $f:A \to \mathbb{N}$ be an injection. Consider the following recursively defined function $g:D\subseteq \mathbb{N} \to A$:

$$ \DeclareMathOperator*{\argmin}{argmin} g(n) = \argmin_{a\in A\setminus\{g(k)|k<n\}}(f(a)) $$

Notice that $g$ is a bijection from its domain of definition $D$ to $A$, so $D$ cannot equal all of $\mathbb{N}$ (since $|A| < |\mathbb{N}|$). Therefore there exists some $n = \min(\mathbb{N}\setminus D)$. Observe that $D = \{0, \ldots, n-1\} = n$. So $g$ is a bijection from $n$ to $A$, so $A$ is finite.

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