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I've seen some posts on this website that consist in providing and comparing different proofs of a theorem (e.g. for Taylor's Theorem or trigonometric identities). Currently I'm reading Holz's Introduction to Cardinal Arithmetic and I find his derivation (a sketch of which is found at the end of the post) of the following theorem rather complicated:

Theorem/Definition: there is exactly one $\in$-isomorphism $\text{ON}\to\text{ICN}$, which we define as $\aleph$.

So I'm wondering if there are other ways of showing the existence of, or defining, $\aleph$.

As to why I find Holz's derivation complicated: the proof is a straightforward result of the following two corollaries:

Corollary 1.5.7 The class $\text{CN}$ of cardinal numbers is a proper class. The same holds for $\text{CN \setminus ω}$, the class of infinite cardinal numbers.

Corollary 1.3.11 a) Any extensional class $B$ is $∈$-isomorphic to a transitive class $A$.

The latter corollary is shown using

Theorem 1.3.10 (Mostowski’s Collapsing Lemma) If the relation $R$ is well-founded and extensional, then there is a unique transitive class $A$ and a unique isomorphism $π : \text{fld}(R) \to A$ with respect to $R$ and $∈$.

which proof makes use of the following

Lemma 1.3.9 If $A_1$ and $A_2$ are transitive classes and $π : A_1 \to A_2$ is an $∈$-isomorphism, then $A_1 = A_2$ and $π = \text{id}_{A_1} = \{(x, x) : x ∈ A_1\}$.

Theorem 1.3.5 (Transfinite Recursion Theorem Schema) If $R$, $G$ and $F$ are classes, then let $\text{rec}(F, R, G)$ be an abbreviation for the formula $$\text{Func}(F) ∧ \text{dom}(F) = \text{fld}(R) ∧ ∀x ∈ \text{fld}(R)(F(x) = G(F |`x)).$$ If $R$, $F'$ and $G$ are classes, then we can construct a class $F$, such that the following formula is provable in $\text{ZF}$: $$(\text{Rel}(R)∧\text{wf}(R)∧G : V \to V) \implies \text{rec}(F, R, G)∧(\text{rec}(F', R, G) \implies F = F').$$

With the exception of the last theorem, these results have not, up to this point in the book, been used. It seems they are here crucially to prove the existence of $\aleph$.

Sam
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  • If you are looking for a formal definition see here: https://us.metamath.org/mpeuni/df-aleph.html –  Jan 20 '24 at 09:00
  • As i understood you notations, I would try to prove the required fact by transfinite recursion, but it is hard for me to tell which preliminary results are needed for this. – Alex Ravsky Jan 22 '24 at 14:16

1 Answers1

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The $\aleph$-theorem is a corollary of the Hartog-Sierpinski Theorem on the existence of successor cardinals and also the Recursion Theorem, see Section 44 in the book Classical Set Theory: Theory of Set and Classes.

Taras Banakh
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