The role of syntactic representations in set theory

In this paper, we explore the role of syntactic representations in set theory. We highlight a common inferential scheme in set theory, which we call the Syntactic Representation Inferential Scheme, in which the set theorist infers information about a concept based on the way that concept can be represented syntactically. However, the actual syntactic representation is only indicated, not explicitly provided. We consider this phenomenon in relation to the derivation indicator position that asserts that the ordinary proofs given in mathematical discourse indicate syntactic derivations in a formal logical system. In particular, we note that several of the arguments against the derivation indicator position would seem to imply that set theorists could gain no benefits from the syntactic representations of concepts indicated by their definitions, yet set theorists clearly do.

1. Many scholars contest the claim that “proven claims stay proven” by citing examples of proofs that were later retracted (e.g., Nathanson 2009).

2. In his more recent writings, Azzouni differed from other proponents of the Derivation Indicator position in that he did not require that the proofs be translatable to a formal logical system, but only a system that can be mechanically checked whose inferences are conservative with respect to other accepted inferential techniques (e.g., Azzouni 2016).

3. The quotation appears in Azzouni (2004).

4. Investigations of computability provide a sharp illustration of the distance between the objects being investigated and the discourse used to investigate these objects. The primary objects investigated in computability theory are Turing machines and their associated Gödel numbers. Yet actual Turing machines and specific Gödel numbers are almost never given. The Turing machines are indicated by informal descriptions of algorithms and a Gödel number for the machine is presumed to exist.

5. We can be pedantic and note that Jech’s definition of ω as “the least non-zero limit ordinal” cannot be characterized syntactically by replacing “least”, “non-zero”, “limit”, and “ordinal” with syntactic definitions that Jech provided earlier. This is because until this point in Jech’s volume, “least” is only defined on sets, where in “the least non-zero limit ordinal”, least is modifying a class. Jech does not discuss minimal elements of classes until 44 pages later. It is interesting to note that Jech’s definition is clearly comprehensible, even though the syntactic characterization indicated by it is unclear.

6. The syntactic characterization can be thought of as representing three conditions on x: (a) the empty set is in x, (b) if a is in x, then the successor of a is in x, and (c) if a is in x, then either a is the empty set or a is the successor of another element in x. This is enough to ensure that x = ω in ZFC because of the Foundation Axiom. It might not be sufficient in other systems without the Foundation Axiom, such as Peano arithmetic.

7. Jech (2000) did not provide such a representation, but the reader can construct the representation through direct substitution (see p. 164).

8. For simplicity, we omit the claim and proof that the set of Mahlo cardinals below κ is stationary in κ that Jech also stated in Corollary 17.19 and proved. The same phenomenon that we discuss in the current manuscript occurs in the justification of this claim as well.

9. Note here that the sentence that Jech asserted was \( \varPi_{1}^{1} \) referenced club sets, which we noted earlier were never explicitly provided with a syntactic representation.

10. We note that in principle, the set theorist could use closure theorems about the property in question (e.g., the conjunction of syntactic representations that are Δ₀ is also Δ₀). No doubt this plays a role in understanding why some relations are Δ₀. However, in most cases, the individual components that are composed are never written out formally.

11. http://us.metamath.org/mpegif/mmset.html#trivia.

12. Mathias (2002) claimed that Bourbaki’s definition of 1 exceeded four trillion characters. The situation is not so severe in ZFC; indeed, Matthias’ point was Bourbaki’s formalism was excessively cumbersome.

13. A further claim is that any hereditarily finite object can be represented in a sensible way in V_ω, a structure that is contained in and absolute for any model of ZFC.

14. Kunen (2011) treats theorems about the existence of formal derivations as finitistic statements on the grounds that derivations are finite strings of symbols and consequently finite objects. For example, “Finitists would agree that [ZFC proves 2^ω ≥ ω₁] since the formal proof is finite and is something that they can believe in” (p. 6) and “The basics of logical syntax and formal proofs… are at the same philosophical level as elementary arithmetic; they are both equally finitistic, so that Thesis I.1.2 applies equally to them” (p. 10).

15. Note that this can be used to strengthen Tanswell’s (2015) point that the same proof can indicate multiple derivations. If a proof is to indicate the same derivation, all of the facts appealed to in the proof must have themselves been proven in the same way. Many independence proofs about cardinal arithmetic use the relative consistency of V = L to show the existence of a model where the General Continuum Hypothesis (GCH) holds (assuming the consistency of ZFC). No matter how explicit these proofs are, they must point to different derivations, since the proof that L satisfies GCH depends on how L was defined.

I am indebted to Brendan Larvor, Colin Rittberg, and Fenner Tanswell, and the anonymous reviewers for providing extensive feedback on earlier drafts of this manuscript. The set theoretic ideas of this paper were refined through discussion with Joel David Hamkins.

Authors and Affiliations

1. Graduate School of Education, Rutgers University, New Brunswick, NJ, 08901, USA

   Keith Weber

Corresponding author

Correspondence to Keith Weber.

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