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 {〈σ i, p i 〉 | σ i a term, p i a forcing condition, i ∊ I, I an index set}. The ground model embeds into the forcing extension, by always choosing p i to be ⊤. p ⊩ φ is defined inductively on formulas.

3. background Classical forcing: σ = {〈σ i, p i 〉 | σ i a term, p i a condition, i ∊ I} ground model embeds into the extension p ⊩ φ defined inductively on formulas Topological semantics: σ = {〈σ i, J i 〉 | σ i a term, J i an open set, i ∊ I} ground model embeds into the extension, by always choosing J i to be the whole space T J ⊩ φ defined inductively on formulas

4. Classical forcing: σ = {〈σ i, p i 〉 | i ∊ I}, ground model V embeds into the extension, p ⊩ φ defined inductively on formulas Topological semantics: σ = {〈σ i, J i 〉 | i ∊ I}, ground model V embeds into the extension, J ⊩ φ defined inductively on formulas Topological semantics with settling: σ = {〈σ i, J i 〉 | i ∊ I} ∪ {〈σ h, r h 〉 | r h ∊ T, h ∊ H} The ground model V embeds into the extension, by choosing J i to be T and H to be empty. J ⊩ φ is defined inductively on formulas.

5. The settling-down functions σ r (r ∊ T) is defined inductively on σ: σ r = {〈σ i r, T〉 | 〈σ i, J i 〉 ∊ σ and r ∊ J i } ∪ {〈σ h r, T〉 | 〈σ h, r〉 ∊ σ }

6. The settling-down functions σ r (r ∊ T) is defined inductively on σ: σ r = {〈σ i r, T〉 | 〈σ i, J i 〉 ∊ σ and r ∊ J i } ∪ {〈σ h r, T〉 | 〈σ h, r〉 ∊ σ } Note: a) σ r is a (term for a) ground model set. b) (σ r ) s = σ r. Notation: φ r is φ with each parameter σ replaced by σ r.

7. Topological semantics ⊩ J ⊩ σ = τ iff for all 〈σ i, J i 〉 ∊ σ J∩J i ⊩ σ i ∊ τ, and vice versa, J ⊩ σ ∊ τ iff for all r ∊ J there are 〈τ i, J i 〉 ∊ τ and J r ⊆ J i such that r ∊ J r ⊩ σ = τ i J ⊩ φ ∧ ψ iff J ⊩ φ and J ⊩ ψ J ⊩ φ ∨ ψ iff for all r ∊ J there is a J r ⊆ J such that r ∊ J r ⊩ φ or r ∊ J r ⊩ ψ J ⊩ φ → ψ iff for all J’ ⊆ J if J’ ⊩ φ then J’ ⊩ ψ J ⊩ ∃x φ(x)ifffor all r ∊ J there are σ r and J r such that r ∊ J r ⊩ φ(σ) J ⊩ ∀x φ(x)ifffor all σ J ⊩ φ(σ)

8. Topological semantics with settling J ⊩ σ = τ iff for all 〈σ i, J i 〉 ∊ σ J∩J i ⊩ σ i ∊ τ, and vice versa, and for all r ∊ J σ r = τ r 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 J r ∍ r such that for all K ⊆ J r if K ⊩ φ r then K ⊩ψ r J ⊩ ∃x φ(x)iff… J ⊩ ∀x φ(x)ifffor all σ J ⊩ φ(σ), and for all r ∊ J there is a J r ∍ r such that for all σ J r ⊩ φ r (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 J r ∍ r J r ⊩ 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 J r ∍ r J r ⊩ 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 J r ∍ r J r ⊩ 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’

14. Application with intuition Example Let T be ℝ (the reals). r ⊨ σ ∊ / = τ iff for some J r ∍ r J r ⊩ 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

15. Application with intuition Example Let T be ℝ (the reals). r ⊨ σ ∊ / = τ iff for some J r ∍ r J r ⊩ 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) s Truth Lemma r ⊨ φ iff J r ⊩ φ for some J r ∍ r.

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

17. What is valid under settling?

18. 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. Δ 0 ) 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 Δ 0 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 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.

22. 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.

23. Does Separation really fail so badly? Theorem If T is locally homogeneous then T ⊩ Full Separation. Counter-example Let T n be the topological space for collapsing ℵ n to be countable. Let T be ⋃T n ∪ {∞}. A neighborhood of ∞ contains cofinitely many T n s. T falsifies Replacement for a Boolean combination of Σ 1 and Π 1 formulas.

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

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

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

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

28. Does Power Set really fail so badly?

29. 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