Transcript Topological Semantics with Settling

Topological Forcing
Semantics with Settling
Robert S. Lubarsky
Florida Atlantic University
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.
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
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.
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 }
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.
Topological semantics ⊩
J⊩σ=τ
iff
J⊩σ∊τ
iff
J⊩φ∧ψ
J⊩φ∨ψ
iff
iff
J ⊩ φ → ψ iff
J ⊩ ∃x φ(x) iff
J ⊩ ∀x φ(x) iff
for all 〈σi, Ji〉 ∊ σ J∩Ji ⊩ σi ∊ τ, and
vice versa,
for all r ∊ J there are 〈τi, Ji〉 ∊ τ and
Jr ⊆ Ji such that r ∊ Jr ⊩ σ = τi
J ⊩ φ and J ⊩ ψ
for all r ∊ J there is a Jr ⊆ J such
that r ∊ Jr ⊩ φ or r ∊ Jr ⊩ ψ
for all J’ ⊆ J if J’ ⊩ φ then J’ ⊩ ψ
for all r ∊ J there are σr and Jr such
that r ∊ Jr ⊩ φ(σ)
for all σ J ⊩ φ(σ)
Topological semantics with settling
J⊩σ=τ
iff
J⊩σ∊τ
iff
J ⊩ φ∧/∨ψ iff
J ⊩ φ → ψ iff
J ⊩ ∃x φ(x) iff
J ⊩ ∀x φ(x) iff
for all 〈σi, Ji〉 ∊ σ J∩Ji ⊩ σi ∊ τ, and
vice versa, and for all r ∊ J σr = τr
…
…
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
…
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 intuition
Example Let T be ℝ (the reals).
Equivalent description of the topological
model as a Kripke model.
Application with intuition
Example Let T be ℝ (the reals).
Equivalent description of the topological
model as a Kripke model.
Starting node r ∊ ℝ.
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
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)
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:
1. f the elementary embedding from M to M’
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:
1. f the elementary embedding from M to M’
2. σ ↦ f(σ)s
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:
1. f the elementary embedding from M to M’
2. σ ↦ f(σ)s
Truth Lemma r ⊨ φ iff Jr ⊩ φ for some Jr ∍ r.
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 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