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The Principle of ImageCollection

The Principle of ImageCollection. by Albert Ziegler University of Munich (LMU). Ext, Pair, Union, Infty Foundation Separation Replacement Powerset Axiom. Ext, Pair, Union, Infty -Induction Separation for bounded formulae Strong Collection Subset Collection. ZF versus CZF.

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The Principle of ImageCollection

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

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

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

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

  5. The Axioms so far

  6. 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

  7. 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

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

  9. 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 .

  10. 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:

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

  12. 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)

  13. 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

  14. 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.

  15. 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.

  16. 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:

  17. 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

  18. 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

  19. 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).

  20. The End Questions? Comments?

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