[FOM] Paradox on Ordinals and Human Mind

[A.P. Hazen]

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.)