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YCor
YCor
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The$\DeclareMathOperator\Th{Th}$The random poset is the Fraisse limit of the class of finite posets, just like the random graph is the Fraisse limit of the class of finite graphs? That is, the random poset is the unique (upto isomorphism) countable poset $P$ such that every finite poset embeds into $P$ and every order isomorphism between finite subsets of $P$ extends to an order automorphism of $P$.

Now it has been proven that $Th(P)$$\Th(P)$ is countably categorical, with $P$ being its unique model upto isomorphism. But my question is, do we know exactly what $Th(P)$$\Th(P)$ is, or at least the complexity of the axioms of $Th(P)$$\Th(P)$? I’m hoping for a result similar to the $\Pi^0_2$ axiomatization of the theory of the random graph.

The random poset is the Fraisse limit of the class of finite posets, just like the random graph is the Fraisse limit of the class of finite graphs? That is, the random poset is the unique (upto isomorphism) countable poset $P$ such that every finite poset embeds into $P$ and every order isomorphism between finite subsets of $P$ extends to an order automorphism of $P$.

Now it has been proven that $Th(P)$ is countably categorical, with $P$ being its unique model upto isomorphism. But my question is, do we know exactly what $Th(P)$ is, or at least the complexity of the axioms of $Th(P)$? I’m hoping for a result similar to the $\Pi^0_2$ axiomatization of the theory of the random graph.

$\DeclareMathOperator\Th{Th}$The random poset is the Fraisse limit of the class of finite posets, just like the random graph is the Fraisse limit of the class of finite graphs? That is, the random poset is the unique (upto isomorphism) countable poset $P$ such that every finite poset embeds into $P$ and every order isomorphism between finite subsets of $P$ extends to an order automorphism of $P$.

Now it has been proven that $\Th(P)$ is countably categorical, with $P$ being its unique model upto isomorphism. But my question is, do we know exactly what $\Th(P)$ is, or at least the complexity of the axioms of $\Th(P)$? I’m hoping for a result similar to the $\Pi^0_2$ axiomatization of the theory of the random graph.

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Keshav Srinivasan
Keshav Srinivasan
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What is the theory of the random poset?

The random poset is the Fraisse limit of the class of finite posets, just like the random graph is the Fraisse limit of the class of finite graphs? That is, the random poset is the unique (upto isomorphism) countable poset $P$ such that every finite poset embeds into $P$ and every order isomorphism between finite subsets of $P$ extends to an order automorphism of $P$.

Now it has been proven that $Th(P)$ is countably categorical, with $P$ being its unique model upto isomorphism. But my question is, do we know exactly what $Th(P)$ is, or at least the complexity of the axioms of $Th(P)$? I’m hoping for a result similar to the $\Pi^0_2$ axiomatization of the theory of the random graph.