Topological Forcing Semantics with Settling

1. Topological Forcing Semantics with Settling Robert S. Lubarsky Florida Atlantic University

2. background Classical 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.

3. background Classical forcing: • σ = {〈σi, pi〉 | σi a term, pi a condition, i ∊ I} • ground model embeds into the extension • p ⊩ φ defined inductively on formulas Topological 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

4. Classical forcing: • σ = {〈σi, pi〉 | i ∊ I}, ground model V embeds into the extension, p ⊩ φ defined inductively on formulas Topological semantics: • σ = {〈σi, Ji〉 | i ∊ I}, ground model V embeds into the extension, J ⊩ φ defined inductively on formulas Topological 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.

5. The settling-down functions σr (r ∊ T) is defined inductively on σ: σr = {〈σir, T〉 | 〈σi, Ji〉 ∊ σ and r ∊ Ji} ∪ {〈σhr, T〉 | 〈σh, r〉 ∊ σ }

6. The settling-down functions σr (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.

7. 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 = τi J ⊩ φ ∧ ψ iff J ⊩ φ and J ⊩ ψ J ⊩ 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)

8. Topological semantics with settling J ⊩ 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 ⊩ψr J ⊩ ∃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)

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

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

11. Application with intuition Example 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

12. Application with intuition Example 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)

13. Application with intuition Example 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: • f the elementary embedding from M to M’

14. Application with intuition Example 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: • f the elementary embedding from M to M’ • σ ↦ f(σ)s

15. Application with intuition Example 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: • f the elementary embedding from M to M’ • σ ↦ f(σ)s Truth Lemma r ⊨ φ iff Jr ⊩ φ for some Jr ∍ r.

16. Application with intuition Example Let T be ℝ (the reals). Two transition functions: • f the elementary embedding from M to M’ • σ ↦ f(σ)s Truth 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”.

17. What is valid under settling?

18. 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)

19. 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

20. 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

21. Does Separation really fail so badly? Definitions T is locally homogeneousaround 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.

22. Does Separation really fail so badly? Definitions T is locally homogeneousaround 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.

23. 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.

24. 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)”.

25. 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 …

26. 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)”.

27. 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.

28. Does Power Set really fail so badly?

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

30. 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.

31. 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