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Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has no nontrivial automorphisms.

Under the Axiom of Choice, every set is well-orderable, and since well-orders are rigid, it follows under AC that every set does have a rigid binary relation.

My questions are: does the converse hold? Does one need AC to produce such rigid structures? Is this a weak choice principle? Or can one simply prove it in ZF?

(This question spins off of Question A rigid type of structure that can be put on every set?A rigid type of structure that can be put on every set?.)

Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has no nontrivial automorphisms.

Under the Axiom of Choice, every set is well-orderable, and since well-orders are rigid, it follows under AC that every set does have a rigid binary relation.

My questions are: does the converse hold? Does one need AC to produce such rigid structures? Is this a weak choice principle? Or can one simply prove it in ZF?

(This question spins off of Question A rigid type of structure that can be put on every set?.)

Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has no nontrivial automorphisms.

Under the Axiom of Choice, every set is well-orderable, and since well-orders are rigid, it follows under AC that every set does have a rigid binary relation.

My questions are: does the converse hold? Does one need AC to produce such rigid structures? Is this a weak choice principle? Or can one simply prove it in ZF?

(This question spins off of Question A rigid type of structure that can be put on every set?.)

typo "only" should be "no"
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Joel David Hamkins
Joel David Hamkins
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Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has onlyno nontrivial automorphisms.

Under the Axiom of Choice, every set is well-orderable, and since well-orders are rigid, it follows under AC that every set does have a rigid binary relation.

My questions are: does the converse hold? Does one need AC to produce such rigid structures? Is this a weak choice principle? Or can one simply prove it in ZF?

(This question spins off of Question A rigid type of structure that can be put on every set?.)

Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has only nontrivial automorphisms.

Under the Axiom of Choice, every set is well-orderable, and since well-orders are rigid, it follows under AC that every set does have a rigid binary relation.

My questions are: does the converse hold? Does one need AC to produce such rigid structures? Is this a weak choice principle? Or can one simply prove it in ZF?

(This question spins off of Question A rigid type of structure that can be put on every set?.)

Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has no nontrivial automorphisms.

Under the Axiom of Choice, every set is well-orderable, and since well-orders are rigid, it follows under AC that every set does have a rigid binary relation.

My questions are: does the converse hold? Does one need AC to produce such rigid structures? Is this a weak choice principle? Or can one simply prove it in ZF?

(This question spins off of Question A rigid type of structure that can be put on every set?.)

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Joel David Hamkins
Joel David Hamkins
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Does every set admit a rigid binary relation? (and how is this related to the Axiom of Choice?)

Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has only nontrivial automorphisms.

Under the Axiom of Choice, every set is well-orderable, and since well-orders are rigid, it follows under AC that every set does have a rigid binary relation.

My questions are: does the converse hold? Does one need AC to produce such rigid structures? Is this a weak choice principle? Or can one simply prove it in ZF?

(This question spins off of Question A rigid type of structure that can be put on every set?.)