On Σ1-Fixed Point Statements in KP

Silvia Steilaon-going work with Gerhard Jager

Universitat Bern

Arbeitstagung Bern–Munchen

MunchenDecember 8th, 2016

Let us start from a binary relation r

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.

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).

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?

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 ]

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 ]).

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).

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).

Σ1-separation

Σ1-separation

For every Σ1 formula ϕ in which x is not free and anyset a,

∃x(x = {y ∈ a : ϕ[y ]}).

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.

Σ1-separation implies Σ1-LFP

Does the viceversa hold?

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).

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

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.

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.

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

Σ1-fixed point principles in KP

Σ1-Sep

Σ1-MI Σ1-SBS

Σ1-BPI Σ1-LFP

Σ1-FP

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 .

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.

Σ1-fixed point principles in KP + (V=L)

Σ1-Sep

Σ1-MI Σ1-SBS

Σ1-BPI Σ1-LFP

Σ1-FP

(V=L)

(V=L)

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})

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.

Σ1-fixed point principles in KP

Σ1-Sep

Σ1-MI Σ1-SBS

Σ1-BPI Σ1-LFP

Σ1-FP

Thank you!