potentialist reflection

Potentialist Logic Modal Set Theory Reflection Revisited

Potentialist Reflection

James StuddPostdoctoral Researcher

Inexpressibility and Reflection in the Formal Sciences

19th March 2012

[email protected]



The Iterative Conception

Sets are formed in a well-ordered series of stages:

Plenitude Any (zero or more) sets formed at any stage will forma set at every later stage.

Priority Any set ever formed is formed from some sets thatwere formed at an earlier stage.

[email protected]



LanguageI L∈. First-order language of set theory (with ∈ and =)

I L^∈ . adds two primitive modal operators:

�>ψ ‘it will be the case at every later stage that ψ.’

�<ψ ‘it was the case at every earlier stage that ψ.’

Defined modal operators: �≥, �≤, �.

NB: Tense not to be taken literally.

Logic Governed by a tenselike bimodal logic:

I Negative free logic (Ev =df v = v).

I Earlier-later relation is a serial well-order

I Stages are non-decreasing

I Atomic predicates are stable

I � satisfies the S5 axioms; �≥ and �≤ satisfy S4.3

[email protected]



Comprehension vs CollapseNon modal Modal

P-Comp !∃xx∀z(z ≺ xx ↔ φ(z)) % ^∃xx�∀z(^z ≺ xx ↔ ψ(z))

Collapse % ∀xx∃y∀z(z ∈ y ↔ z ≺ xx) ! �∀xx^∃y�∀z(^z ∈ y ↔ ^z ≺ xx)

Standard response

I Deny Collapse.

I What does it take for some sets to be able to form a set?

Potentialist response

I Retain Collapse^

I Explain why the pathological instances fail.

[email protected]



MetalogicHow does set theory in L^∈ relate to ordinary set theory?

I Modalisation [·]^ : L∈ → L^∈

Φ 7→ ^Φ (for atomic subformula Φ)

∀ 7→ �∀

∃ 7→ ^∃

e.g. [Foundation]^ = ^∃z(z ∈^x)→ ^(∃z0 ∈^x)�(∀t ∈^x)(¬t ∈^z0)

Invariance Theorem[φ]^ is invariant (i.e. �[φ]^ ∨ �¬[φ]^)

Mirroring TheoremΓ `FOL φ only if Γ^ `LST φ

^.

[email protected]



PlenitudeAny sets formed at any stage will form a set at every later stage.�∀xx�>∃y�∀z(z ∈^y ↔ z ≺^ xx)

How can this be approximated as a (modal) first-order schema?

First try. �>∃y�∀z(z ∈^y ↔ [φ(z)]^)

Second try. Restrict this to ‘inextensible’ conditions.

I Informally: ψ is inextensible at a stage if:

I Some sets formed then comprise every ψ ever formed.I Every ψ ever formed is formed then.

I Formally: ¬extz [ψ] ≡ �>∀z(ψ(z)→ ^<Ez)

Plenitude. �(¬extz [ψ] ∧ inv[ψ]→ �>∃y�∀z(z ∈^y ↔ ψ(z)))

[email protected]



Priority

Any set ever formed is formed from some sets that were formed atan earlier stage.�∀x^<∃xx�∀z(z ∈^x ↔ x ≺^ xx)

This admits of a non-schematic first-order formulation.

I ¬extz[ψ]: ‘every ψ ever formed has been formed’I ¬extz[z ∈^x]: ‘every element of x has been formed’.

Priority. �∀x^<¬extz[z ∈^x]

[email protected]



Modal Stage Theory

MST = Extensionality^ + Priority + Plenitude

S = Extensionality + Regularity + Emptyset + Separation+ Pairing + Union + Powerset

ZF = S + Infinity + Replacement

ρ ∀x∃U∃α(Rα(U) ∧ x ⊆ U) where Rα(U) formalizes U = Vα.

TheoremMST `LST [φ]^ iff S + ρ `FOL φ

Questions

(1) What about ZF?

(2) What about formulas ψ not of the canonical form [φ]^?

[email protected]



On Question (1)

Response to question (1): Add reflection

Refl0 [φ]^ → ^φ

I Linnebo: ‘The principle says that we should recognise anysituation that is realized in the potential hierarchy as genuinelypossible.’

Refl1 �^>∀x1, . . . , xk ([φ]^ ↔ φ)

I No matter how far we go, no condition can distinguish theentire hierarchy from some later stage.

Refl2 �∀x^∃y[φ(x, y)]^ → �^>∀x∃y[φ(x, y)]^

I Parsons: Potentially infinite sequences are actualised.

[email protected]



On Question (2)Every modalised formula is (equivalent to) an invariant formula? Is everyinvariant formula equivalent to a modalised formula?

I Reverse translation 〈·〉U : L^∈ → L∈

〈x = y〉U =df x = y ∧ x ∈ U ∧ y ∈ U

〈x ∈ y〉U =df x ∈ y ∧ x ∈ U ∧ y ∈ U

〈¬ψ〉U =df ¬〈ψ〉U

〈ψ1 → ψ2〉U =df 〈ψ1〉

U → 〈ψ2〉U

〈∀xψ〉U =df (∀x ∈ U)〈ψ〉U

〈�>ψ〉U =df (∀U1 3 U)〈ψ〉U1

〈�<ψ〉U =df (∀U1 ∈ U)〈ψ〉U1

I U etc. reserved for ranks: ∀Uψ abbreviates ∀U(∃αRα(U)→ ψ)

I Define: 〈ψ〉? =df 〈ψ〉∅

[email protected]



MST vs S + ρMirroring Theorem(a) If S + ρ `FOL φ, then MST `LST [φ]^

(b) If MST `LST ψ, then S + ρ `FOL 〈ψ〉?

Normal Form Theorem(a) S + ρ `FOL 〈[φ]^〉? ↔ φ

(b) MST, inv[ψ] `LST [〈ψ〉?]^ ↔ ψ

Corollary Assume MST ` inv[ψ] and MST ` inv[δ], δ ∈ ∆1 ∪∆2.

(a) MST ` ψ iff S + ρ ` 〈ψ〉?

(b) MST + ∆1 = MST + ∆2 iff S + ρ + ∆?1 = S + ρ + ∆?

2

(c) MST + ∆1 ‘corresponds to’ S + ρ + ∆?1

i.e. MST + ∆1 ` ψ iff S + ρ + ∆?1 ` φ for some φ s.t. MST + ∆1 ` ψ↔ [φ]^

[email protected]



Partial ReflectionRefl0 [φ]^ → ^φ

Refl′0 [φ]^ → ^(Ex1 ∧ . . . ∧ Exk ∧ φ)

PR φ→ ∃U(x1 . . . xk ∈ U ∧ φU)

Proposition(a) S + ρ + 〈Refl′0〉

? = S + ρ + PR (= S + PR)

(b) MST + Refl0 = MST + Refl′0 = MST + [PR]^

(c) MST + Refl0 corresponds to S + PR

Theorem (Levy and Vaught 1961)S + PR is much weaker than ZF

I ZF ` φ→ Con(S + PR + φ)

[email protected]



Complete Reflection

Refl1 �^>∀x1, . . . , xk ([φ]^ ↔ φ)

CR ∀U(∃U1 3 U)(∀x1 . . . xk ∈ U1)(φ↔ φU1 )

Proposition(a) S + ρ + 〈Refl1〉

? = S + ρ + CR (= S + CR)

(b) MST + Refl1 = MST + [CR]^

(c) MST + Refl1 corresponds to S + CR

Theorem (Montague, Levy)ZF = S + CR

[email protected]



The Limit Principle

Refl2 �∀x^∃y[φ(x, y)]^ → �^>∀x∃y[φ(x, y)]^

LP ∀x∃yφ(x, y)→ ∀U(∃U1 3 U)((∀x ∈ U1)(∃y ∈ U1)φ(x, y)

)Proposition(a) S + ρ + 〈Refl2〉

? = S + ρ + LP = ZF = S + ρ + 〈Refl1〉?

(b) MST + Refl2 = MST + LP^ = MST + Refl1

(c) MST + Refl2 and MST + Refl1 both correspond to ZF

[email protected]



Nice features of potentialist reflection

I No extraneous content.The strength of PR over weak underlying theories suggeststhat it outstrips its standard motivation.

S0 = Extensionality + Foundation + Separation

I S0 0 Empty Set (formulated in Lβ,∈).I S0 + PR = S + PRI S^0 + Refl0 0 Empty Set

I Topic neutrality.Potentialist reflection can be applied outside set theory.I PR, CR and LP all require ∈.I Refl0, Refl1 and Refl2 are logical schemas.

[email protected]



Prolegomenon to strong potentialist reflection principles.

First thought: leave first-order setting.

I Plural formulas in Refl0 quickly leads to trouble.

I [∀xx∃y∀z(z ∈ y ↔ z ≺ xx)]^ → ^∀xx∃y∀z(z ∈ y ↔ z ≺ xx)

Two options

1. Simply modalise strong first-order formulations of reflection R^

2. Distinguish plural from monadic second-order quantification.

Plural Second-Order

P-Comp^ %[∃xx∀z(z ≺ xx ↔ φ(z))]^ ![∃F∀z(Fz ↔ φ(z))]^

Collapse^ ![∀xx∃y∀z(z ∈ y ↔ z ≺ xx)]^ % [∀F∃y∀z(z ∈ y ↔ Fz)]^

[email protected]


potentialist reflection

Documents