Topological Forcing Semantics with SettlingRobert S. LubarskyFlorida Atlantic University

backgroundClassical forcing: • A term σ is a set of the form {⟨σi, pi⟩ | σi a term, pi a forcing condition, i ∊ I, I an index set}. • The ground model embeds into the forcing extension, by always choosing pi to be ⊤. • p ⊩ φ is defined inductively on formulas.

backgroundClassical forcing: • σ = {⟨σi, pi⟩ | σi a term, pi a condition, i ∊ I} • ground model embeds into the extension• p ⊩ φ defined inductively on formulasTopological semantics:• σ = {⟨σi, Ji⟩ | σi a term, Ji an open set, i ∊ I}• ground model embeds into the extension, by always choosing Ji to be the whole space T• J ⊩ φ defined inductively on formulas

Classical forcing: • σ = {⟨σi, pi⟩ | i ∊ I}, ground model V embeds into the extension, p ⊩ φ defined inductively on formulasTopological semantics:• σ = {⟨σi, Ji⟩ | i ∊ I}, ground model V embeds into the extension, J ⊩ φ defined inductively on formulasTopological semantics with settling:• σ = {⟨σi, Ji⟩ | i ∊ I} ∪ {⟨σh, rh⟩ | rh ∊ T, h ∊ H} • The ground model V embeds into the extension, by choosing Ji to be T and H to be empty.• J ⊩ φ is defined inductively on formulas.

The settling-down functionsσr (r ∊ T) is defined inductively on σ:σr = {⟨σir, T⟩ | ⟨σi, Ji⟩ ∊ σ and r ∊ Ji} ∪ {⟨σhr, T⟩ | ⟨σh, r⟩ ∊ σ }

The settling-down functionsσr (r ∊ T) is defined inductively on σ:σr = {⟨σir, T⟩ | ⟨σi, Ji⟩ ∊ σ and r ∊ Ji} ∪ {⟨σhr, T⟩ | ⟨σh, r⟩ ∊ σ } Note:a) σr is a (term for a) ground model set.b) (σr)s = σr .Notation: φr is φ with each parameter σ replaced by σr.

Topological semantics ⊩J ⊩ σ = τ iff for all ⟨σi, Ji⟩ ∊ σ J∩Ji ⊩ σi ∊ τ, and vice versa, J ⊩ σ ∊ τ iff for all r ∊ J there are ⟨τi, Ji⟩ ∊ τ and Jr ⊆ Ji such that r ∊ Jr ⊩ σ = τiJ ⊩ φ ∧ ψ iff J ⊩ φ and J ⊩ ψJ ⊩ φ ∨ ψ iff for all r ∊ J there is a Jr ⊆ J such that r ∊ Jr ⊩ φ or r ∊ Jr ⊩ ψ J ⊩ φ → ψ iff for all J’ ⊆ J if J’ ⊩ φ then J’ ⊩ ψ J ⊩ ∃x φ(x) iff for all r ∊ J there are σr and Jr such that r ∊ Jr ⊩ φ(σ)J ⊩ ∀x φ(x) iff for all σ J ⊩ φ(σ)

Topological semantics with settlingJ ⊩ σ = τ iff for all ⟨σi, Ji⟩ ∊ σ J∩Ji ⊩ σi ∊ τ, and vice versa, and for all r ∊ J σr = τr J ⊩ σ ∊ τ iff …J ⊩ φ∧/∨ψ iff …J ⊩ φ → ψ iff for all J’ ⊆ J if J’⊩ φ then J’⊩ ψ, and for all r ∊ J there is a Jr ∍ r such that for all K ⊆ Jr if K ⊩ φr then K ⊩ψrJ ⊩ ∃x φ(x) iff …J ⊩ ∀x φ(x) iff for all σ J ⊩ φ(σ), and for all r ∊ J there is a Jr ∍ r such that for all σ Jr ⊩ φr(σ)

Application with intuitionExample Let T be ℝ (the reals).Equivalent description of the topological model as a Kripke model.

Application with intuitionExample Let T be ℝ (the reals).Equivalent description of the topological model as a Kripke model.Starting node r ∊ ℝ.

Application with intuitionExample Let T be ℝ (the reals).Equivalent description of the topological model as a Kripke model.Starting node r ∊ ℝ.r ⊨ σ ∊ (resp. =) τ iff for some Jr ∍ r Jr ⊩ σ ∊ (resp. =) τ

Application with intuitionExample Let T be ℝ (the reals).Equivalent description of the topological model as a Kripke model.Starting node r ∊ ℝ.r ⊨ σ ∊ (resp. =) τ iff for some Jr ∍ r Jr ⊩ σ ∊ (resp. =) τ The node s extends r if s is infinitesimally close to r. (set-up: r ∊ M ≺ M’ ∍ s)

Application with intuitionExample Let T be ℝ (the reals).r ⊨ σ ∊ / = τ iff for some Jr ∍ r Jr ⊩ σ ∊ / = τ The node s extends r if s is infinitesimally close to r. (set-up: r ∊ M ≺ M’ ∍ s)Two transition functions:1. f the elementary embedding from M to M’

Application with intuitionExample Let T be ℝ (the reals).r ⊨ σ ∊ / = τ iff for some Jr ∍ r Jr ⊩ σ ∊ / =) τ The node s extends r if s is infinitesimally close to r. (set-up: r ∊ M ≺ M’ ∍ s)Two transition functions:1. f the elementary embedding from M to M’2. σ ↦ f(σ)s

Application with intuitionExample Let T be ℝ (the reals).r ⊨ σ ∊ / = τ iff for some Jr ∍ r Jr ⊩ σ ∊ / =) τ s extends r if s is infinitesimally close to r.Two transition functions:1. f the elementary embedding from M to M’2. σ ↦ f(σ)sTruth Lemma r ⊨ φ iff Jr ⊩ φ for some Jr ∍ r.

Application with intuitionExample Let T be ℝ (the reals).Two transition functions:1. f the elementary embedding from M to M’2. σ ↦ f(σ)sTruth Lemma r ⊨ φ iff Jr ⊩ φ for some Jr ∍ r.Application This structure models IZFExp (and therefore “the Cauchy reals are a set”) + “the Dedekind reals do not form a set”.

What is valid under settling?

What is valid under settling?Theorem T ⊩ IZF with the following changes:• Eventual Power Set instead of Power Set: every set X has a collection of subsets C such that every subset of X cannot be different from everything in C, i.e.∀X ∃C (∀Y∊C Y⊆X) ∧ (∀Y⊆X ¬∀Z ∊C Y≠Z)

What is valid under settling?Theorem T ⊩ IZF with the following changes:• Eventual Power Set instead of Power Set: ∀X ∃C (∀Y∊C Y⊆X) ∧ (∀Y⊆X ¬∀Z ∊C Y≠Z)• Bounded (i.e. Δ0) Separation instead of Full Separation

What is valid under settling?Theorem T ⊩ IZF with the following changes:• Eventual Power Set instead of Power Set • Δ0 Separation instead of Full Separation• Collection instead of Strong Collection: every total relation from a set to V has a bounding set, but the bounding set may contain elements not in the range of the relations

Does Separation really fail so badly?Definitions T is locally homogeneous around r, s ∊ T if there is a homeomorphism between neighborhoods of r and s interchanging r and s.U is homogeneous if U is locally homogeneous around each r, s ∊ U.T is locally homogeneous if every r ∊ T has a homogeneous neighborhood.

Does Separation really fail so badly?Definitions T is locally homogeneous around r, s ∊ T if there is a local homeomorphism between neighborhoods of r and s interchanging r and s.U is homogeneous if U is locally homogeneous around each r, s ∊ U.T is locally homogeneous if every r ∊ T has a homogeneous neighborhood.Theorem If T is locally homogeneous then T ⊩ Full Separation.

Does Separation really fail so badly?Theorem If T is locally homogeneous then T ⊩ Full Separation.Counter-example Let Tn be the topological space for collapsing ℵn to be countable. Let T be ⋃Tn ∪ {∞}. A neighborhood of ∞ contains cofinitely many Tns. T falsifies Replacement for a Boolean combination of Σ1 and Π1 formulas.

Does Separation really fail so badly?Counter-example Tn ⊩ “ℵn is countable.” T is ⋃Tn ∪ {∞}. A neighborhood of ∞ contains ⋃n>I Tns. Let ω∞ be {⟨n, ∞⟩ | n ∊ ω}.Then T ⊩ “∀n∊ω∞ ∃!y (y=0 ∧ ℵn is uncountable) ∨ (y=1 ∧ ¬ℵn is uncountable)”.

Does Separation really fail so badly?Counter-example Tn ⊩ “ℵn is countable.” Then T ⊩ “∀n∊ω∞ ∃!y (y=0 ∧ ℵn is uncountable) ∨ (y=1 ∧ ¬ℵn is uncountable)”.Suppose ∞ ∊ J ⊩ “∀n∊ω∞(f(n)=0 ∧ ℵn is uncountable) ∨ (f(n)=1 ∧ ¬ℵn is uncountable)”. Then …

Does Separation really fail so badly?Counter-example Tn ⊩ “ℵn is countable.” Suppose ∞ ∊ J ⊩ “∀n∊ω∞(f(n)=0 ∧ ℵn is uncountable) ∨ (f(n)=1 ∧ ¬ℵn is uncountable)”. Then ∞ ∊ K ⊩ “∀n∊ω∞(f∞(n)=0 ∧ ℵn is uncountable) ∨ (f∞(n)=1 ∧ ¬ℵn is uncountable)”.

Does Separation really fail so badly?Counter-example Tn ⊩ “ℵn is countable.” Then ∞ ∊ K ⊩ “∀n∊ω∞(f∞(n)=0 ∧ ℵn is uncountable) ∨ (f∞(n)=1 ∧ ¬ℵn is uncountable)”. But K determines f∞(n) for each n, yet K does not determine whether ℵn is uncountable for each n – contradiction.

Does Power Set really fail so badly?

Does Power Set really fail so badly?Theorem If T is locally connected then T ⊩ Exponentiation.

Does Power Set really fail so badly?Theorem If T is locally connected then T ⊩ Exponentiation.Counter-example Let T be Cantor space. The generic is a 0-1 sequence, i.e. a function from ℕ to {0, 1}. So that function space does not exist as a set.

Does Power Set really fail so badly?Theorem If T is locally connected then T ⊩ Exponentiation.Counter-example Let T be Cantor space. The generic is a 0-1 sequence, i.e. a function from ℕ to {0, 1}. So that function space does not exist as a set. THE END