Concept Audits by Rescher Nicholas;

Author:Rescher, Nicholas;
Language: eng
Format: epub
Publisher: Lexington Books/Fortress Academic

#23

Language Limits

FACTS ARE TRANSDENUMERABLE

A key consideration for the much controverted issue of the limitations of human knowledge lies in the circumstance that our knowledge of fact is linguistically mediated while the domain of fact itself transcends the limits of language.

Statements in general—and therefore true statements in particular—can be enumerated, and truths are consequently denumerable in number. But there is good reason to suppose that this will not hold for facts. On the contrary, there is every reason to think that, reality being what it is, there will be an uncountably large manifold of facts.

The reality of it is that facts, unlike truths, cannot be enumerated: no listing of fact-presenting truths—not even one of infinite length—can possibly manage to constitute a complete register of facts. Any attempt to register-fact-as-a-whole will founder: the list is bound to be incomplete because there are facts about the list-as-a-whole which no single entry can encompass.

We thus arrive at one of the key theses of these deliberations:

Thesis 1: The Transdenumerability of Facts. The manifold of fact is transdenumerably infinite.

The idea of a complete listing of all the facts is manifestly impracticable. For consider the following statement. “The list F of stated facts fails to have this statement on it.” But now suppose this statement to be on the list. Then it clearly does not state a fact, so that the list is after all not a list of the facts (contrary to hypothesis). And so it must be left off the list. But then in consequence that list will not be complete since the statement is true. Facts, that is to say, can never be listed in toto because there will always be further facts—facts about the entire list itself—that a supposedly complete list could not manage to register.

This conclusion can be rendered more graphic by the following considerations. Suppose that the list F

F: f1, f2, f3, …

were to constitute a complete enumeration of all facts. And now consider the statement

(Z) the list F takes the form f1, f2, f3, …

By hypothesis, this statement will present a fact. So if F is indeed a complete listing of all facts, then there will be an integer k such that

Z = fk

Accordingly, Z itself will occupy the kth place on the F listing, so that

fk = the list L takes the form f1, f2, f3, . . . fk, . . .

But this would require fk to be an expanded version of itself, which is absurd. With the kth position of the F listing already occupied by fk we cannot also squeeze that complex fk-involving thesis into it.

The crux here is simply that any supposedly complete listing of facts

f1, f2, f3 . . .

will itself exhibit, as a whole, certain features that none of its individual members can encompass. Once those individual entries are fixed and the series is defined, there will be further facts about that series-as-a-whole that its members themselves cannot articulate.

Moreover, the point at issue can also be made via an analogue of the diagonal

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