Zermelo set theory


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Zermelo set theory

(mathematics)
A set theory with the following set of axioms:

Extensionality: two sets are equal if and only if they have the same elements.

Union: If U is a set, so is the union of all its elements.

Pair-set: If a and b are sets, so is

a, b.

Foundation: Every set contains a set disjoint from itself.

Comprehension (or Restriction): If P is a formula with one free variable and X a set then

x: x is in X and P.

is a set.

Infinity: There exists an infinite set.

Power-set: If X is a set, so is its power set.

Zermelo set theory avoids Russell's paradox by excluding sets of elements with arbitrary properties - the Comprehension axiom only allows a property to be used to select elements of an existing set.

Zermelo Fr?nkel set theory adds the Replacement axiom.

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With hindsight, we know that if the set-theoretic membership relation is logical and if the axioms of (say) Zermelo set theory are logically true, then (Terms) and at least (Basic Truths) hold, and logicism is established.
But why think that Zermelo set theory or this cumulative type theory passes Dummettian muster as logic?
Thus, the theory would resemble Zermelo set theory without the principle of infinity.