on 1-fixed point statements in kplogicatorino.altervista.org/steila/slides/munich2016.pdf · resume...
TRANSCRIPT
![Page 1: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/1.jpg)
On Σ1-Fixed Point Statements in KP
Silvia Steilaon-going work with Gerhard Jager
Universitat Bern
Arbeitstagung Bern–Munchen
MunchenDecember 8th, 2016
![Page 2: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/2.jpg)
Let us start from a binary relation r
![Page 3: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/3.jpg)
Well-founded part of a binary relation
The well-founded part of any binary relation r ⊆ a× a is the set of allx ∈ a such that there are no infinite decreasing sequences from x .
So, which is the well-founded part of our relation?
...on the blackboard.
![Page 4: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/4.jpg)
Well-founded part of a binary relation
It is known that we can obtain the well-founded part of any relationr ⊆ a× a has the least fixed point of the function
F (u) = {x ∈ a : ∀z ∈ a(z r x =⇒ z ∈ u)} .
This function is monotone, i.e.
u ⊆ v =⇒ F (u) ⊆ F (v).
![Page 5: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/5.jpg)
Well-founded part of a binary relation
If there exists a least fixed point for this function, then it is thewell-founded part of our relation.
Is leastness important?
...on the blackboard.
Does the least fixed point exist?
![Page 6: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/6.jpg)
Kripke Platek Set Theory
We work in extensions of Kripke Platek Set Theory (KP). We brieflyresume the axioms of KP.
I extensionality, pair, union, foundation, infinity,
I ∆0-Separation: i.e, for every ∆0 formula ϕ in which x is not freeand any set a,
∃x(x = {y ∈ a : ϕ[y ]})I ∆0-Collection: i.e, for every ∆0 formula ϕ and any set a,
∀x ∈ a∃yϕ[x , y ] =⇒ ∃b∀x ∈ a∃y ∈ bϕ[x , y ]
![Page 7: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/7.jpg)
A first question
Given a set a and any monotone function F : P(a)→ P(a), does thereexist a set which is the least fixed point of F?
We extend the standard language with Σ1-function symbols. F is aΣ1-function symbol if there exists a Σ1 formula ϕ such that:
I ∀x∃!y(ϕ[x , y ]) (i.e, functional);
I ∀x , y(F (x) = y ⇐⇒ ϕ[x , y ]).
![Page 8: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/8.jpg)
Σ1-least fixed point
Σ1-LFP
Given any set a and any Σ1-function symbol F such that
1. ∀x(F (x) ⊆ a) (i.e, bounded),
2. ∀x , x ′(x ⊆ x ′ =⇒ F (x) ⊆ F (x ′)) (i.e,monotone),
there exists z such that
I F (z) = z (i.e, fixed point),
I ∀x(F (x) = x =⇒ x ⊇ z) (i.e, leastness).
![Page 9: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/9.jpg)
Σ1-fixed point
Σ1-FP
Given any set a and any Σ1-function symbol F such that
1. ∀x(F (x) ⊆ a) (i.e, bounded),
2. ∀x , x ′(x ⊆ x ′ =⇒ F (x) ⊆ F (x ′)) (i.e,monotone),
there exists z such that
I F (z) = z (i.e, fixed point).
![Page 10: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/10.jpg)
Σ1-separation
Σ1-separation
For every Σ1 formula ϕ in which x is not free and anyset a,
∃x(x = {y ∈ a : ϕ[y ]}).
![Page 11: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/11.jpg)
Σ1-separation implies Σ1-LFP
I Given any set a and any F as in Σ1-LFP, define by Σ-recursion:
Iα = F (⋃{Iβ : β < α}).
I Define by Σ1-Separation, the set
z = {x ∈ a : ∃α(x ∈ Iα)}.
I Σ-Reflection and monotonicity yield z = Iγ for some ordinal γ.
I z is a set and it is the least fixed point.
![Page 12: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/12.jpg)
Σ1-separation implies Σ1-LFP
Does the viceversa hold?
![Page 13: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/13.jpg)
Σ1-bounded proper injections
Σ1-BPI
Given any set a and any Σ1-function symbol F such that
I ∀x(F (x) ∈ a),
there exist x and y such that
x 6= y ∧ F (x) = F (y).
![Page 14: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/14.jpg)
Σ1-Subset Bounded Separation
Σ1-SBS
For every ∆-formula ϕ and sets a and b,
{x ∈ a : ∃y ⊆ b(ϕ[x , y ])}
is a set.
Σ1-SBS
Σ1-BPI Σ1-LFP
![Page 15: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/15.jpg)
Σ1-SBS implies Σ1-BPI
I Given F and a as in Σ1-BPI define by Σ1-SBS the set
X = {x ∈ a : ∃z ⊆ a(F (z) = x)}.
I Suppose by contradiction that
∀y , z ⊆ a(F (y) 6= F (z)).
I Define h : X → V such that
h(x) = the unique z ⊆ a(F (z) = x).
I We can prove that ∀z(z ⊆ a ⇐⇒ z ∈ h[X ]).
I We can conclude with the usual Cantor’s argument.
![Page 16: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/16.jpg)
Σ1-SBS implies Σ1-LFP
I Given F and a as in Σ1-LFP, define
ClF [y ] ⇐⇒ F (y) ⊆ y .
I By Σ1-SBS we can define
z = {x ∈ a : ∀y ⊆ a(ClF [y ] =⇒ x ∈ y)}.
I We can prove that F (z) = z .
I Since every fixed point is closed under F , we have leastness.
![Page 17: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/17.jpg)
Σ1-Maximal Iteration
Σ1-MI
Let a be any set and F be any Σ1-function symbol suchthat
I ∀x(F (x) ⊆ a) (i.e, bounded).Then there exists α and f such that
I fun(f ) ∧ dom(f ) = α + 1;
I ∀β ≤ α(F (⋃
γ∈β f (γ)) = f (β))
I⋃
γ∈α f (γ) ⊇ f (α)
Σ1-MI
Σ1-BPI Σ1-LFP
![Page 18: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/18.jpg)
Σ1-fixed point principles in KP
Σ1-Sep
Σ1-MI Σ1-SBS
Σ1-BPI Σ1-LFP
Σ1-FP
![Page 19: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/19.jpg)
Working with the Axiom of Constructibility (V=L)
Godel’s constructible universe is defined as follows:
I L0 = ∅,I Lα+1 = P(Lα) ∩ C(Lα ∪ {Lα}),
I Lα =⋃{Lβ : β < α} for α limit,
I L =⋃{Lα : α ∈ On}.
Where C (x) is the closure under the Godel’s functions F1, . . . ,F8 of x .
![Page 20: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/20.jpg)
Working with the Axiom of Constructibility (V=L)
In KP + (V=L) the following implications hold:
I Σ1-BPI implies Σ1-Separation.
I Σ1-FP implies Σ1-SBS.
We can conclude that all our principles are not provable in KP + (V=L)since all of them are equivalent to Σ1-Separation in this setting.
![Page 21: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/21.jpg)
Σ1-fixed point principles in KP + (V=L)
Σ1-Sep
Σ1-MI Σ1-SBS
Σ1-BPI Σ1-LFP
Σ1-FP
(V=L)
(V=L)
![Page 22: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/22.jpg)
Are all these statements equivalent also in KP?
Beta
For any well-founded relation r on some set a thereexists a function f such that:
∀x ∈ a(f (x) = {f (y) : (y , x) ∈ r})
![Page 23: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/23.jpg)
Are all these statements equivalent also in KP?
I Mathias proved that KP + Pow does not imply Axiom Beta
And we have
I Pow implies Σ1-SBS,
I Σ1-MI implies Beta.
![Page 24: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/24.jpg)
Σ1-fixed point principles in KP
Σ1-Sep
Σ1-MI Σ1-SBS
Σ1-BPI Σ1-LFP
Σ1-FP
![Page 25: On 1-Fixed Point Statements in KPlogicatorino.altervista.org/steila/slides/Munich2016.pdf · resume the axioms of KP. I extensionality,pair,union,foundation,in nity, I 0-Separation:](https://reader033.vdocuments.site/reader033/viewer/2022051812/602cc5588d481d2ab6304abd/html5/thumbnails/25.jpg)
Thank you!