[FOM] Paradox on Ordinals and Human Mind
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Sat Dec 18 00:20:57 EST 2004
Dmytro Taranovsky has asked about "König's Paradox" (=, for those who
like the prose of Russell's "Mathematical Logic as based on...," "the
contradiction concerning the least indefinable ordinal"). Suppose we
understand "define" or "specify" in a way that makes contact with
some, not TOO idealized, conception of the cognitive powers of human
beings (or beings of the same "epistemological type," if you prefer a
vaguer notion, as humans). On most reasonable precisifications of
that, it is at least plausible that there is some limit to the number
of definitions possible. [[[For example: Suppose you are a
philosophical physicalist and think that the humanly-understandable
concepts inject into physically distinguishable states of the brain.
Then it seems plausible to me that only finitely many concepts can
REALLY be understood, that with a moderate degree of idealization we
might go for a DENUMERABLE infinity, and that even immoderate
idealization is unlikely to take us beyond the first few
Beth-numbers.]]] So not all the ordinals in
the set-theoretic universe [[[I'm a -- moderate -- Platonist, so I
believe in such things; if you aren't at least willing to play along
with the Platonist assumption that there ARE ordinals, König's P.
isn't likely to interest you!]]] have "definitions" or "unique
specifications". But every non-empty class of ordinals has a least
member, so there must be a least indefinable ordinal ... and, oops,
haven't I just defined it?
Taranovsky suggests three resolutions:
>1. Infinite sets do not exist, but humans can define arbitrarily large
>integers.
>2. Word "identify" and certain other words are meaningless (at least in
>the sense they are used in the paradox).
>3. The potential of the human mind extends beyond the finite, and
>every ordinal can be identified by a human mind.
To which I would like to comment:
A) Resolution (1) doesn't look good. If you don't believe
in infinite SETS, you probably shouldn't be happy with idealizations
according to which humans can formulate infinitely many
"definitions," and the same basic logic comes back to you with
Berry's Paradox (the one about "the least integer not nameable in
fewer than nineteen syllables").
B) Gödel was attracted to resolution (3), which led him to
the notion of (what are now called) Ordinal Definable sets: cf. his
remarks to the Princeton Bicentennial. This is a beautiful and
well-motivated theory, but I can't help feeling that we shouldn't
give up the hope of finding SOME interesting theory of definability
based on a less wildly immoderate idealization of the human mind.
C) My own sympathies are with something like (2), but
"meaningless" is too strong. One can think of "define" or "possible
language" or "idealized human-like mind" as MEANINGFUL notions, but
ones with an ineliminable vagueness which makes it inappropriate to
reason CLASSICALLY about them. One can have a consistent theory
quantifying over, say, possible definitions (see SKETCH below) if
you use a formally intuitionistical logic: the inference from "Not
all ordinals are defined" to "THERE IS a least indefinable one" is
blocked.
D) There is, however, at least a 4th proposal out there:
DIALETHEISM, the view that some contadictions are true (and that we
must therefore use a logic not validating P, Not-P, therefore Q): cf.
Graham Priest, "The Logical Paradoxes and the Law of Excluded
Middle," in "Philosophical Quarterly," vol. 33 (1983), pp. 160-165.
(A critical reply by Ross Brady is in the same journal about two
years later.)
SKETCH: First-order theory with two sorts of variables,
ranging over SETS (inc. ordinals) and over CONCEPTS. Intuitionistic
logic, but law of excluded middle allowed for sentences in the purely
set-theoretic part of the language. ZFC or any other reasonable set
theory. (I don't know to what degree it is safe to allow
concept-theoretic vocabulary in instances of the set-theoretic axiom
schemes.) Two-place <concept,set> predicate HOLDINGOF. Nice
comprehension axioms for concepts. Two-place <concept,set> predicate
ENCODEDBY, axiom (embodying the idea that there are restrictions on
concept formation) that there is some set-- specify its cardinality
if you want-- such that every concept is encoded by a member of that
set and no member encodes more than one concept. Maybe axiom saying
ENCODEDBY is decidable. Define a DEFINITION as a concept HOLDINGOF
one set and NOT HOLDINGOF any other set. I ***believe*** that a
theory of this sort could be intellectually satisfying (though maybe
too weak to be of much interest), would allow a proof that not all
ordinals are defined, but would NOT allow the paradoxical
consequence that there is a least indefinable. For much the reason
that the "Least Number Principle" is independent of "Induction" in
intuitionistic arithmetic.
--
Allen Hazen
Philosophy Department
University of Melbourne
(About to leave town and not read e-mail for a while.)
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