Concept
Axiom of empty set
Overview
In axiomatic set theory, the axiom of empty set, also called the axiom of null set and the axiom of existence, is a statement that asserts the existence of a set with no elements. Although Zermelo originally stipulated the existence of a set with no elements as an axiom , it can be treated as either an axiom or a derivable truth depending on the specific set-theoretic context.
It is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstrable truth in Zermelo–Fraenkel set theory, with or without the axiom of choice.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
1 As Empty Conception without object, _ens rationis_ 2 3 Empty object of Empty intuition a conception, without object, _nihil privativum_ _ens imaginarium_ 4 Empty object without conception, _nihil negativum_ We see that the ens rationis is distinguished from the nihil negativum or pure nothing by the consideration that the former must not be reckoned among possibilities, because it is a mere fiction—though not…
Immanuel Kant, The Critique of Pure Reason
Close in meaning to Axiom of empty set; the term itself is not named.
VII. That thing is called free which exists from the necessity of its own nature alone, and is determined to action by itself alone. That thing, on the other hand, is called necessary, or rather compelled, which by another is determined to existence and action in a fixed and prescribed manner. VIII. By eternity, I understand existence itself, so far as it is conceived necessarily to follow from the definition alone…
Benedictus de Spinoza, The Philosophy of Spinoza
States what Axiom of empty set is.
Others hold it to be a principle which, antecedently to any verification by experience, we are compelled by the constitution of our thinking faculty to assume as true. Having so recently, and at so much length, combated a similar doctrine as applied to the axioms of mathematics, by arguments which are in a great measure applicable to the present case, I shall defer the more particular discussion of this controverted…
John Stuart Mill, A System of Logic
Close in meaning to Axiom of empty set; the term itself is not named.
Related ideas
- Formal language
- First-order logic
- Zermelo–Fraenkel set theory
- Ernst Zermelo
- Axiom of infinity
- Set (mathematics)
- Set theory
- Axiom of extensionality
- Axiom schema of specification
- Axiom schema of replacement
- De Morgan's laws
- Axiom of choice
- Existential quantification
- Axiom
- Empty set
- Kripke–Platek set theory
Place in the tradition
Part of First-order logic.