All naturals are T-finite, all finite sets are T-finite In Jech's Set Theory, there is defined T-finite, where a set $S$ is T-finite if every non-empty $X\subseteq\mathcal{P}(S)$ has $\subseteq$-maximal element.
[ie. there is $u\in X$ s.t. there is no $v\in X$ with $u\subsetneq v$]
The following exercises are being related to this.


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*Each $n\in \mathbb{N}$ is T-finite

*$\mathbb{N}$ is T-infinite (not T-finite)

*Every finite set is T-finite

*Every infinite set is T-infinite


I completed 2 and 4 (considering first $\mathbb{N}\subset \mathcal{P}(\mathbb{N})$ since the naturals are linearly ordered by $\subseteq$, and for $S$ infinite, $\{u\subseteq S\vert u \text{ finite}\}$ ). I am so far unable to solve the others. Many thanks for you kind help.

For (1) try to show by induction that each $n \in \mathbb{N}$ is T-finite.


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*Since $0 = \emptyset$, then $\mathcal{P} ( \emptyset ) = \{ \emptyset \}$, and we can analyse both subsets of this to show that they have $\subseteq$-maximal elements.

*Going from $n$ to $n+1$, note that if $X \subseteq \mathcal{P} ( n+1 ) = \mathcal{P} (\{ 0, \ldots , n \} )$ has no $\subseteq$-maximal element, then the family $Y = \{ a \in X : n \in a \}$ must be nonempty.  Reduce this down to a question about a subset of $\mathcal{P} ( n )$.
For (3), note that it follows easily from (1) once you show that if $X$ is T-finite and $f : X \to Y$ is a bijection, then $Y$ is T-finite.  ($f$ will induce a bijection $\hat{f} : \mathcal{P} ( X ) \to \mathcal{P} (Y)$.)
