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If $\phi$ is any formula of set theory with just one free variable $x$, the abstraction term $A_{\phi}=\lbrace x | \phi(x) \rbrace$ is either a set or a proper class. Assume that ZFC is consistent, or any large cardinal axiom you like. Then my question is, are there two formulas $\phi$ and $\psi$ such that ZFC+($A_{\phi}$ is a set) is consistent, ZFC+($A_{\psi}$ is a set) is consistent also, but ZFC+($A_{\phi}$ and $A_{\psi}$ are both sets) is not?

UPDATE 09/15/2011 : to avoid "cheating" as in François Dorais' answer, we may introduce the following additional constraint : if $T$ is any theory extending $ZFC$, say that the abstraction term $A_{\phi}=\lbrace x | \phi(x) \rbrace$ is small in $T$ if $T$ proves that $A_{\neg \phi}$ is not a set ; for example, if $\phi(x)$ is "x is an accessible ordinal" or "all cardinals below the ordinal $x$ are not measurable" or "all cardinals below the ordinal $x$ are not Mahlo" then $A_{\phi}$ will be small, but this will not be the case if $\phi$ is an undecidable statement independent of $x$ as in Francois Dorais' answer.

The question then becomes, are there two formulas $\phi$ and $\psi$ such that $A_{\phi}$ and $A_{\psi}$ are both small in $ZFC$, ZFC+($A_{\phi}$ is a set) is consistent, ZFC+($A_{\psi}$ is a set) is consistent also, but ZFC+($A_{\phi}$ and $A_{\psi}$ are both sets) is not?

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What prevents me from taking a completion of ZFC for T? –  François G. Dorais Sep 15 '11 at 3:50
    
(My trick still works for the new version, unless maybe if the final sentence was supposed to involve T rather than ZFC.) –  François G. Dorais Sep 15 '11 at 3:55
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At least (as of this writing), it is an edit ahead of its time. Gerhard "What Day Is It Again?" Paseman, 2011.09.18 –  Gerhard Paseman Sep 15 '11 at 5:03
    
@Francois : you're right, your trick still works in my updated version and I should have thought more before posting it on mathoverflow. As I found a new idea, I indulged in a breach of mathoverflow etiquette by erasing the old update's content and replacing it with a new content. Hopefully this new update will be more resistant to simple attacks. –  Ewan Delanoy Sep 15 '11 at 18:02

2 Answers 2

up vote 5 down vote accepted

If by "consistent" you mean "consistent relative to the consistency of ZFC," then there is a simple example. Let $\phi$ be the Rosser sentence for ZFC, and let $\psi$ be its negation. Then, in any model of ZFC, $A_\phi$ is either $V$ (if $\phi$ is true) or $\varnothing$ (if $\phi$ is false); similarly for $A_\psi$. However, we cannot have both $A_\phi = \varnothing$ and $A_\psi = \varnothing$ since one of the two sentences must be true.

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For a philosophical perspective on this, see Alan Weir’s “Neo-Fregeanism: An Embarrassment of Riches”: “The embarrassment of riches objection is that there is a plurality of consistent but pairwise inconsistent abstraction principles[;] thus not all consistent abstractions can be true.”

Closer to my own work, Oberschelp [1973] and I [forthcoming] have consistency proofs for set theories with a universal set in which the singleton function is a set. This would lead to an immediate contradiction in Quine’s New Foundations, whose consistency is an open problem. Another open problem is constructing a model in ZFC for the combination of Church’s set theory with a universal set (which has a sequence of Frege-Russell cardinals for equivalence relations generalizing equinumerosity), with Mitchell’s variant, which lacks the Frege-Russell cardinals but has an unrestricted axiom of power set. My conjecture is that such a construction would be impractical with current techniques, but we’d all be very disappointed if the combined theory were inconsistent.

Bibliography

• Alonzo Church 1974a. “Set Theory with a Universal Set,” Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics XXV, ed. Leon Henkin, American Mathematical Society pp. 297-308. (Delivered 24 June 1971.)

• Emerson Mitchell 1976. A Model of Set Theory with a Universal Set, unpublished Ph.D. thesis, University of Wisconsin at Madison.

• Arnold Oberschelp 1964a. “Eigentliche Klasse als Urelemente in der Mengenlehre,” Mathematische Annalen 157 pp. 234-260. [MR 31#2136]. (Delivered 20 August 1962)

• Arnold Oberschelp 1964b. “Sets and Non-Sets in Set Theory” (abstract), received 3 June 1964, Journal of Symbolic Logic XXIX p. 227

• Arnold Oberschelp 1973. Set Theory over Classes, Dissertationes Mathematicæ (Rozprawy Mat.) 106. [MR 42 #8300].

• Ulf Friedrichsdorf 1979. “Zur Mengenlehre über Klassen,” Zeitschrift f. Mathem. Logik 25, pp. 379-383. Contains helpful summary of [Oberschelp 1964a & 1973].

Note that the crucial part of the consistency proof in both [Friedrichsdorf 1979] and [Oberschelp 1973] is merely a reference to [Oberschelp 1964a], which uses a very different formalism.

• Alan Weir 2003. “Neo-Fregeanism: An Embarrassment of Riches,” Notre Dame Journal of Formal Logic volume 44, Number 1 (2003), pp. 13-48, http://projecteuclid.org/euclid.ndjfl/1082637613

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