INTRODUCTION
Goedel’s “transfinite” constructs a vague and not an occassional “indeterminate” ontology.
The ontological perspective of mathematical logic places objects in a ‘domain’, where objects are particular to that domain only. I argue that Goedel selectively employs the object-domain distinction and so mutilates the traditional type-token distinction. In these short remarks I employ the traditional phrase “type” - “token” distinction, which, unfortunately, translates into the Goedellian “system” - “type” distinction.
Goedel launches the object-domain binary into a unifying supportive structure which he calls the “transfinite”. in the transfinite Goedel finds systems (types) and their particular expressions or “types” (tokens). But the transfinite also allows a type-token or Goedellian system-type relativism, where Goedellian systems can become Goedellian types (which translates as types becoming tokens).
The true reason for the incompleteness which attaches to all formal
systems of mathematics lies, as will be shown in Part II of this paper,
in the fact that the formation of higher and higher types can be
continued into the transfinite. (Go31, p28, footnote 48a. (See note 1.)
Despite this unification of object and domain as ‘types’ in the transfinite, Goedel restores the object-domain distinction in his description of objects appearing in a particular domain
“…while, in every formal system only countable
many are available.” ibid.
Here, “formal system” and “countable many” translate to types and tokens.
DISCUSSION
We must ask this question - if, for Godel, 1) objects subsist in a domain as tokens and, also for Godel, 2) domains, as types, subsist in a transfinite, then, does a domain subsist in a transfinite as a discrete object or token?
If the answer to that question is ‘yes’ - that domains (and their objects) subsist in a transfinite as discrete objects, then Goedel’s ontology is not threefold as object (the Goedellian “type”), domain (the Goedellian “system”), and transfinite, but rather consists entirely of heirarchically mutating ontological objects, where types (systems) mutate into tokens (Goedellian “types”). See note 3.
… one can show that the undecidable sentences
which have been constructed here always become decidable
through adjunction of suitable higher types (e.g. of the type w to
the system P. A similar result also holds for the axiom systems of
set theory. " ibid. (See note 2.,and 3.)
Goedel’s methodology is inconsistent, and the “transfinite” is an ontological reification of that methodology. Goedel uses the object-domain, token-type, type-system, distinction to present or frame the concepts of consistency and completeness, but he rejects that distinction and opts for an indeterminate type-token relativism when he “shows” that consistency and completeness cannot be had. It is the transfinite that, obscurely, forges this type-token, system-type, relativism (where types can mutate into tokens), and hence offers a vague, and not an indeterminate or ambiguous ontology. An objection to this summary can be made on the grounds that a token cannot be a token in the absence of a type. Well, it can’t, but that is Goedel’s formulation, and not mine. Further, if the transfinite is itself a bona fide system (and not merely an ontological playground, as I have argued) then it, itself, cannot be indeterminately placed.
[i]Notes
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What Goedel means by the ‘transfinite’ may be formally presented in the logical theories of the day, but the idea ought to be a familiar one with Kantian scholars. Kant argues against the idea that sequences of the conditioned can yield the unconditioned. Against Kant, Goedel would seem to be arguing for the idea that sequences of the conditioned (tokens or Goedellian “types”) can yield the unconditioned (types, or Goedellian “systems”). Goedel is employing an object relativism in a “transfinite”, where unconditioned and conditioned are relatively placed, and where the unconditioned may be passed off as the conditioned in a Goedellian “transfinite”. Goedel might put it as the idea that a theory and its objects can be viewed as objects in a higher type, thus undermining the type-token distinction, as I have argued, above.
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Goedel’s Note 48a is a natural language description of his mathematical project. It would be inappropriate, and unlikely, for Goedel to use his term ‘transfinite’ in that note in a technical sense, as if to partially postpone his descriptive intention. Let me add that Goedels’ natural language footnote is just the sort of discipline-neutral perspective that systematic structures require as their foundation, and as such is immediately accessible to any modest philosopher and reader of natural language.
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This might suggest why Goedel effortlessly compares his threefold system with that of sets, where the distinction between elements or objects and their supportive or manifesting framework is equally obscure.
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