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The observations made above about the Mostowski function and their relation to transitive sets as well as their relation to order embeddings could be valid only for the shown examples, but not in general. Therefore, we want to summarize these observations in a theorem and prove that indeed the observations hold also in every case. This theorem was first proven by Mostowski (1913 – 1975).

## Theorem: Mostowski's Theorem

Let $U$ be a universal set, $(V,\prec)$ with a well-founded relation $”\prec”$.

Then the corresponding Mostowski function defined by $\pi(x):=\{\pi(y)\mid y\in V\wedge y\prec x\}$ exists and its Mostowski collapse $\pi[V]\subseteq U$ is a transitive set.

Moreover, if $”\prec”$ is in addition extensional, then the function $\pi$ is an order embedding, i.e. fulfills the property $x\prec y\Longleftrightarrow \pi(x)\in\pi(y).$

| | | | | created: 2019-03-03 11:46:44 | modified: 2019-03-07 15:01:13 | by: bookofproofs | references: [8055]