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The problem I'm dealing with is the following.

Show that every set $I$ which satisfies the conditions $$\emptyset\in I,\quad (\forall x)[x\in I\Rightarrow\{x\}\in I]$$ is Dedekind-infinite.

Where Dedekind-infinite is defined as follows.

A set is Dedekind-infinite if there exists an injection $$f:A\rightarrow B\subset A$$ from A into a proper subset of itself.

My attempt to solve this was to construct a function $$f(x): \begin{cases} x&\text{if } x\in R\subset I \\ \{x\}&\text{if } x\in I\backslash R \end{cases}$$ where my thought was that I would map $R\subset I$ onto itself, and $I\backslash R$ onto something else. This doesn't fulfil the criteria as I must map all of $I$ injectively into $R$. Now I'm stuck with the thought that if I map something onto $R$ then I can't map more into $R$. Of course $I$ is infinite in size, but I don't know how to exploit this in order to obtain a satisfactory injection.

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  • $\begingroup$ Actually the injection is very easy: simply consider $f(x)= \{ x\}$ (i.e. $R= \emptyset$). $\endgroup$ – Crostul Sep 3 '16 at 9:40
  • $\begingroup$ @Crostul Mustn't $R$ be any subset of $I$ according to the definition? Also: $f(x)$ doesn't map every element of $I$ injectively into $R$. In fact it doesn't map any element into $R$. Am I wrong? $\endgroup$ – Logarithmic Derivative Sep 3 '16 at 9:54
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You can use:

$$f(x)=\{x\}$$

which is an injection from $I$ to $I\backslash\{\emptyset\}$. The function is well defined since $\forall x\left[x \in I \implies \{x\} \in I\right]$. $I\backslash\{\emptyset\}$ is a proper subset of $I$ since $\emptyset\in I$

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  • $\begingroup$ I must be missing something obvious. Please observe my comment to Crostul! $\endgroup$ – Logarithmic Derivative Sep 3 '16 at 10:00
  • $\begingroup$ I now see that the answer was obvious. My apologies. Thank you anyway! $\endgroup$ – Logarithmic Derivative Sep 3 '16 at 10:05

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