aa r X i v : . [ m a t h . L O ] M a r The Nuisance Principle in Infinite Settings
Sean C. Ebels-DugganSubmitted 1 July 2015
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This is the pre-peer-reviewed version of the following article:Sean C. Ebels-Duggan. The Nuisance Principle in Infinite Set-tings,
Thought: A Journal of Philosophy
Note
The final version has additional remarks and corollaries; namelythese: First corollary: SOL + Nuisance Principle (NP) prove that thereis no pairing injection of the universe. Second corollary: SOL+ HP + NPprove that the universe is uncountable. Remark: Even if NP does not implythe finitude of the universe, it is still deductively non-conservative.)
Submitted Article
Abstract
Neo-Fregeans have been troubled by the Nuisance Principle (NP),an abstraction principle that is consistent but not jointly (second-order) satisfiable with the favored abstraction principle HP. We showthat logically this situation persists if one looks at joint (second-order)consistency rather than satisfiability: under a modest assumptionabout infinite concepts, NP is also inconsistent with HP. consistent ? The further question arises be-cause satisfiability (having a standard model) and consistency (not provinga contradiction) are not the same in second-order logic. The question waspartially answered in [8, 21–22]; the present note moves us further, but notfully, towards a complete answer.The principle HP, attributed loosely to Hume by Frege [4, § F s (denoted F ) is identical to the Number of G s( G ) just if there is a bijection from the F s to the G s—that is, a functionassociating all of the objects falling under F with all of the objects fallingunder G , such that no two objects falling under F are associated with thesame object falling under G . In second-order logic the existence of such abijection can be represented, and is demonstrably an equivalence relation.Thus HP can be represented by:( ∀ F )( ∀ G )( F = G ↔ F ≈ G )where ‘ ≈ ’ is shorthand for the second-order formula asserting the existenceof a bijection.The Nuisance Principle is a simplification due to Crispin Wright [9] of aprinciple introduced by George Boolos [2]. One can express in second-orderlogic the following equivalence relation: N ( F, G ) iff there are finitely many objects falling under F butnot G , and finitely many falling under G but not F The Nuisance principle is then the claim that the
Nuisance of F (denoted ‡ F ) is identical to the Nuisance of G ( ‡ G ) if and only if N holds of F and G .2sing our abbreviation N ( F, G ) we can represent this in second-order logicby ( ∀ F )( ∀ G )( ‡ F = ‡ G ↔ N ( F, G ))Notice that NP and HP are both abstraction principles in virtue of havingthe same form: equality between objects on the left, an equivalence relationbetween concepts on the right.That NP and HP are jointly unsatisfiable can be seen by deploying fea-tures of cardinal numbers in set theory to show that the former is satisfiableonly if there are finitely many objects. Since HP proves there are infinitelymany objects, the two are not jointly satisfiable. But one cannot adapt thisproof to a deductive setting. The complicating factor is that outside of stan-dard models, being “infinite” can mean many things. Typically, concepts are“infinite” if they are
Dedekind infinite : there is a function from all the objectsfalling under the concept to not all of the objects falling under the concept,such that no two objects are sent to the same object by that function. (Thatis to say, there is an injection from the concept to a proper subconcept ofitself.) In standard models of second-order logic, Dedekind infinite conceptsbehave like infinite sets behave in set theory. But this isn’t guaranteed innon-standard models (and this is why in this note we use “concepts” ratherthan “sets” to indicate the semantic correlate to second-order variables).In this note we show that NP is inconsistent with the Dedekind infinity ofthe universe in the presence of a natural and relatively modest strengthening of the assertion that the universe is Dedekind infinite. Such a strengtheningis a conditional describing the behavior of Dedekind infinite concepts.This is a significant improvement over what was shown in [8]. The proofin that paper used two versions of the Axiom of Choice: a global well-ordering GC to get Dedekind infinite concepts to behave like infinite sets, and a uniformmeans of selecting representatives for each equivalence class, AC . So what wasshown in that paper is that, if one’s second-order logic includes these versionsof the Axiom of Choice, then NP is not consistent with the universe beingDedekind infinite. Thus HP and NP are jointly inconsistent, as in the proofthat they are unsatisfiable.Our improvement is that we can obtain this result by appeal to an ostensi-bly weaker principle. The principle in question is the following strengtheningof infinity: For such a proof that NP is unsatisfiable, see [1]. Pairing)
If the universe is Dedekind infinite, then there is a binary function f defined on all pairs of objects such that for any z and any x, y, x ′ , and y ′ , if z = f ( x, y ) and z = f ( x ′ , y ′ ), then x = x ′ and y = y ′ .In other words, if the universe is Dedekind infinite, then there is an injectionfrom pairs of objects into the universe. In effect, this strengthening saysthat universe-sized concepts can be broken-up into universe-many disjointsubconcepts, each of universe-size.We now sketch a deductive argument showing that NP and Pairing areinconsistent with the assertion that the universe is Dedekind infinite. Forif the universe is Dedekind infinite and Pairing holds, we can associate aDedekind infinite concept with each concept, whether the latter is finite orinfinite . For given a concept F , let U [ F ] be defined by z falls under U [ F ] ↔ there is an x falling under F , and a y such that f ( x, y ) = z In other words, U [ F ] is the image of F when projected (on the right) withthe universal set V (on the left): U [ F ] = f ( F, V ).Now we show that if concepts F and G are extensionally distinct (if some-thing falls under one that doesn’t fall under the other), then the equivalencerelation N , described above, does not hold between U [ F ] and U [ G ]. For if a falls under F but not G , then by fixing a we obtain a one-to-one map f ( a, y )from the universal set into U [ F ] − U [ G ], the part of U [ F ] that does not overlap U [ G ]. Since the universe is Dedekind infinite, by Pairing, so is U [ F ] − U [ G ].An identical argument can be made for any element falling under G but not F . Thus N does not obtain between U [ F ] and U [ G ]. Clearly, of course, if F and G are not extensionally distinct, then N ( U [ F ] , U [ G ]), since U [ A ] and U [ B ] will not be extensionally distinct either. In other words,( ∀ F )( ∀ G )( N ( U [ F ] , U [ G ]) ↔ ( ∀ x )( F x ↔ Gx ))Suppose now that NP obtains; we then have( ∀ F )( ∀ G )( ‡ U [ F ] = ‡ U [ G ] ↔ ( ∀ x )( F x ↔ Gx )) It is worth reiterating the remark of [8] that Pairing is a consequence of GC . It is alsoworth the separate remark that in ZF set theory, a version of Pairing implies the (settheoretic) Axiom of Choice (see [7, Theorem 11.7]). Because equivalence in ZF is not thesame as equivalence in second-order logic, we here treat these these principles as distinct. R is theconcept defined by y falls under R ↔ there is a concept Y such that y = ‡ U [ Y ]and y does not fall under Y , the usual argument shows that ‡ U [ R ] falls under R if and only if it doesn’t.So Pairing and NP imply that the universe is not Dedekind infinite. Thus inthe presence of Pairing, HP and NP are not jointly consistent. References [1] G. Aldo Antonelli. Notions of Invariance for Abstraction Principles.
Philosophia Mathematica , 18(3):276–292, 2010.[2] George Boolos. The Standard Equality of Numbers. In
Meaning andMethod: Essays in Honor of Hilary Putnam , pages 261–277. CambridgeUniversity Press, Cambridge, 1990. Edited by George Boolos. Reprintedin [3], page numbers refer to the reprinted version.[3] George Boolos.
Logic, Logic, and Logic . Harvard University Press, Cam-bridge, MA, 1998. Edited by Richard Jeffrey.[4] Gottlob Frege.
The Foundations of Arithmetic: A Logico-MathematicalEnquiry into the Concept of Number . Northwestern University Press,Evanston, second edition, 1980. Translated by J.L. Austin.[5] Bob Hale and Crispin Wright.
The Reason’s Proper Study . Oxford Uni-versity Press, Oxford, 2001.[6] Richard G. Heck, Jr., editor.
Language, Thought, and Logic: Essays inHonour of Michael Dummett . Oxford University Press, Oxford, 1997.[7] Thomas Jech.
The Axiom of Choice , volume 75 of
Studies in Logic andthe Foundations of Mathematics . North-Holland, Amsterdam, 1973.[8] Sean Walsh and Sean Ebels-Duggan. Relative Categoricity and Abstrac-tion Principles.
The Review of Symbolic Logic , FirstView:1–35, 5 2015. Acknowledgments removed for blind review.
59] Crispin Wright. On the Philosophical Significance of Frege’s Theorem. In [6][6]