The Iterative Conception of Set Element Priority. At every stage, every set is formed from things that were formed at an earlier stage Maximality. At every stage, any things formed at an earlier stages will form a set

The Iterative Conception of Set A Modal Reading

∅ {∅} {∅ {∅}} {∅ {∅}, {∅ {∅}}} {∅ {∅}, {∅ {∅}}, {∅ {∅}, {∅ {∅}}}}, . . . {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}, {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}, . . .}, . . .

∅ {∅} {∅ {∅}} {∅ {∅}, {∅ {∅}}} {∅ {∅}, {∅ {∅}}, {∅ {∅}, {∅ {∅}}}}, . . .

.. .

James Studd DPhil Candidate, Oxford University [email protected]

.. . ∅ {∅} {{∅}} {∅ {∅}}

∅ {∅}

∅

The Maximally Liberal Attitude to Set Formation

I

Usual to axiomatise ‘Stage Theory’ in a non-modal language.

I

This suffices to recover (most of) Z.

I

Non-modal Stage Theory precludes our taking the maximally liberal attitude to set formation. Naive Comprehension^ Absolutely any things can form a set ∀xx ^∃y (xx forms y ) Naive Comprehension Any things do form a set ∀xx ∃y (xx forms y )

Four Ways to Understand φ Tense Operators: Sets are literally created(!) - φ: ‘No matter what sets are created it will always be the case that φ’ Context Operators: Forming sets consists in relaxing contextual restrictions on the background domain. - φ: ‘No matter how the context is admissibly relaxed it will always be the case that φ’ Interpretation Operators: Forming sets consists in liberalising the interpretation of ∀. - φ: ‘No matter how the interpretation is admissibly liberalised it will always be the case that φ’ Ontology Operators: Forming sets consists in adjusting the ‘third parameter’(!) - φ: ‘No matter what objects are admissibly postulated it will always be the case that φ’

Languages

Kripke Models I

L∈ : a (non-modal) plural first-order language. - ¬, →, ∀ - Logical predicates: =, ≺. - Non-logical Predicates: Set, ∈.

A frame is a triple F = hS ,