Proof of the Consistency of HP with Second-Order Logic

Logic

An immediate question arises for the neo-logicist program: how can we know that HP and second-order logic will not lead to another contradiction like Basic Law V and second-order logic did? How is one assured that the resultant system is consistent? As Boolos vividly writes, “How do we know that some Super-Russell of the 22nd Century won’t find some ingenious derivation of a contradiction from the number principle, the way our Russell derived a contradiction from the set principle” [7, p. 151]?

Boolos provides a proof that consistency of his Frege Arithmetic(FA)—the system within which Boolos is able to reconstruct Frege’s program in the Grundlagen—is equiconsistent the strictly weaker theory of analysis as constructed out of the natural numbers via the rationals.16 _{That is, Boolos proves that FA is consistent if and only if}

analysis is consistent. I will focus on the proof in which Boolos provides a constructive account of how a proof of ⊥ in FA can be directly translated into a proof in analysis of ⊥, since analysis is not the theory in question.

The proof proceeds by first demonstrating a model M for HP that also satisfies the principles of second-order logic. We must give a characterization of FA and then interpret those notions within ZF, and then finally translate this interpretation into an interpretation in analysis. This final step must be done by changing the domain from U = {0,1,2, . . . ,ℵ0} in ZF to U = {0,1,2, . . .} in analysis, while rejiggering

16_{It should be noted that Boolos’s proof builds on ideas in Hodes [32] and Burgess [9], but I will}

some of the related notions.

FA, recall, is the system generated by the conjunction of HP with second-order logic, so the fundamental objects of FA will be nearly identical to the fundamental objects of second-order logic. There are three types of variables: object variables (a, b, c, . . . m, n, o, . . . x, y, z), unary-predicate variables (F, G, H, . . . , R, S, T, . . .), and binary-predicate variables (ϕ, ψ, . . .). These variables are simply the standard com- ponents of second-order logic, with an added restriction of predicate variables to only two places. The only non-logical component of FA is dubbed by Boolos as ‘η’, “a two-place predicate letter attaching to a concept variable and object variable. (η

is intended to be reminiscent of ∈ and may be read ‘is in the extension.’ ”[3, p. 185] Atomic formulas are the standard ones in second-order logic, plus formulas of the form ‘F ηx’, and these formulas can be combined in the usual ways by the usual connectives. The final part of FA is the only non-logical axiom:17

∀F∃!x∀G(Gηx ↔F eq G) (Numbers)

Notice that Numbers is a formal analogue of Hume’s Principle. Numbers is the claim that there is the unique object x that is the extension of some higher level concept that G falls under. In the case of Hume’s Principle, that higher level concept is marked out by a unary function sign ‘N’, which has as its domain concept variables and its range object variables. Thus, ‘N F’ should be read as ‘the number of Fs’.

So, to demonstrate a model of Hume’s Principle and second-order logic in ZF set theory, let the domain of the model be U = {0,1,2, . . . ,ℵ0}. We can interpret the

concept variables as subsets ofUand similarly interpret the binary predicate variables as sets of ordered pairs.18 η is easily rendered as ∈. The last to be interpreted is the

17_{‘eq’ here means equinumerous.}

function letter ‘N’, which we are able to interpret as the cardinality of a set. Thus, the number of things falling under a concept in FA becomes the cardinality of the set in ZF. The proof of the truth of Hume’s Principle in Mfollows almost immediately thereafter.19

The key to the success of this model is the fact that the cardinality of any of the subsets of U is in U itself. Because this fact fails to be true for the restricted domain of the natural numbers, we were forced to includeℵ0. However, this presents

a problem for the attempt to give a model in analysis, as the domain of analysis is only the subsets of the natural numbers. We cannot use ℵ0 as a point in the model.

The trick, however, is to reconsider ℵ0 as zero—what Boolos calls “coding”—and

the remaining numbers as their successor (that is, n as n+ 1). Boolos outlines the translative procedure by which to convert any proof of ⊥ in FA into a proof of ⊥ in analysis[3, p.190], but he presents a more dialectically clear—if less rigorous— version in “Gottlob Frege and the Foundations of Arithmetic”[7] In §5.3.2, I will present in detail that argument to highlight where a Poincar´ean petitio seems viable. This objection will be equally applicable to both the more rigorous and less rigorous versions. As such, I want to give a sketch of the rigorous proof of Boolos here.

The basic picture of the coding into analysis is that by interpreting ℵ0 as 0 and

every finite n as n+ 1, one can give an expression of the relation “exactlyz natural numbers belong to the set F” through the analysis relation “there exists a one-one correspondence between the natural numbers less thanz and the members of F”[3, p. 190]. This expression becomes a way of expressing the FA relation ‘η’ as a notion of analysis. Then, we are able to form the analysis analogue of Numbers by replacement:

∀F∃!x∀G(Eta(G, x)↔F eq G) (Analogue)

19_{I pass over this proof to look more in detail at the analogous proof in analysis. Boolos performs}

By providing this translation of Numbers into analysis, combined with the fact that the basic comprehension principles of second-order logic are provable in analysis, we have a direct translation procedure for any proof of ⊥ in FA into analysis.

The contradiction arising from Basic Law V and second-order logic meant that Frege forsaked his ambition of deriving mathematics from logic. The neo-logicist looks to replace Basic Law V with HP and derives in FA an analog to Peano’s Postulates. Finally, because of the threat of contradiction, the neo-logicist provides a proof of the equiconsistency of FA with analysis. Having achieved this result and presuming that the original argument of §5.1.2 is supported, the neo-logicist has achieved his goals. We turn not to examine these goals in more detail.