The Principle of ImageCollection by Albert Ziegler University of Munich (LMU)

ZF versus CZF Ext, Pair, Union, Infty Foundation Separation Replacement Powerset Axiom Ext, Pair, Union, Infty  -Induction Separation for bounded formulae Strong Collection Subset Collection classical logicintuitionistic logic

Subset Collection Scheme...versus Fullness Axiom Complex Scheme Deals with Subsets More intuitive, concrete statement Single Axiom Deals with multivalued functions

Fullness Axiom...versus Exponentiation Asserts existence of a set of multivalued functions but such a set cannot be given Asserts existence of the set of functions which can be characterised uniquely but doesn‘t suffice for many applications i.e. C set of multivalued functions from A to B, such that every such function is an extension of an element of C

The Axioms so far

Crosilla, Ishihara & Schuster: Search weakenings of Fullness that still have its mathematical consequences (in particular: existence of Dedekind reals) Idea: Collect not a sub-mv-function for each mv-function, but only its premimages  Refinement Instead of

Refinement formally weaker than Fullness implies that Dedekind reals form a set implies that detatchable subsets of a given set form a set implies some instances of Exponentiation is equivalent to Fullness in the presence of Exponentiation

New Results about Refinement Refinement implies Exponentiation So Refinement is equivalent to Fullness Thus Refinement is no new principle

Proof-Sketch Consider For a function f:A->B, the statement that a pair belongs to f has a truth value in . Consider (mv-)function from AxB to  that maps (a,b) to the truth values of (a,b) in f. The preimage of any sub-mv-function of 1 is just the function f Therefore, all functions from A to B are in the Refinement of AxB and .

ImageCollection Idea: Collect not the mv-functions, but only certain properties of them (like preimages) Take the dual of Refinement: instead of preimages of elements, collect the images of elements  ImageCollection:

ImageCollection alone seems weak AC(A,B) implies ImageCollection(A,B) ImageCollection doesn‘t imply the existence of any uncountable sets

Consequences of ImageCollection ImageCollection with Exponentiation implies Fullness, more accurately: Let ImColl(A,B)=E mean that E is as postulated in ImageCollection. Then: ImColl(A,B)=E + Exp(A,E)  Full (A,B)

Proof-Sketch Suffices to show that the class C‘ of all mv-functions r such that is a set Its elements come from functions f from A to E, by mapping such a function on the mv-function This mapping is surjective If Exp(A,E), then this mapping has a set as domain and thus as image. But its image is the class C‘, which is full Note: A full set can be given uniquely in dependance of E

Exponentiation and Fullness Small step: ImColl(A,B)=E + Exp(A,E)  Full(A,B) Refine(A,B)=D + Exp(B,D)  Full(A,B) PA(A)=X + Exp(X,B)  Full(A,B) Fullness is slightly more than Exponentiation. Fullness is Exponentiation with a little choice.

Fullness divided into two parts ImageCollection can be viewed as a small choice principle that is implied by Fullness Thus the equation Fullness=ImageCollection + Exponentiation cuts Fullness into an concrete-set-existence- part and a choice-part.

Consider additional Variations The idea that we do not collect the whole mv-functions, but only aspects, is not exhausted. Consider e.g. collecting not the images of points, but of the whole set:  BigImageCollection:

BigImageCollection Similar to Subset Collection, but a single axiom BigImColl(A,AxB) is equivalent to Full(A,B) Its dual is trivial Some proofs work more smoothly with BigImageCollection

Application: Strongly adequate subsets Set S with two set relations,  (given by W) and . A subset M is called strongly adequate iff All elements of M are in  -relation to each other Each element of M has a  -predecessor in M If b  a, then there is c in M, such that b  c implies c=a

BigImColl(W,S) entails: The strongly adequate subsets form a set Let M be strongly adequate. Let R be This is a mv-function, so it has a sub-mv- function whose image is in the Collection. But by Lemma 56 [1], its image is R. So the strongly adequate sets are a subset of the BigImColl(W,S).

The End Questions? Comments?