Re: defn of impredicativity (was Re: goals and language?)

["glen e. p. ropella" <***>]

Calvin Ostrum wrote:
> It is not the statement that the axiom of foundation entails "(a) a is
> not a member of a" which is grossly incorrect.   It is the statement
> that the axiom of foundation, because of that entailment, "apparently
> prohibit[s] impredicative definitions" that is grossly incorrect.

Excellent!  Thank you.  That was unclear from Torkel's emails.

Torkel Franzen wrote:
> This was indeed a misreading of your formulation on my part!

Whew!  OK.  Sorry for not making myself clearer.

> But why on earth do you have the idea that "for all a, a is not a
> member of a" - which is an immediate consequence of the axiom
> of foundation - is in any way incompatible with impredicative
> definitions?

Well I don't have that idea.  I don't think that B&M's prop 2.6 is in 
any way incompatible with impredicative definitions.

However, and I maintain my usual caveats of incompetence here, it seems 
to me that prop 2.6 says something _related_ to impredicative 
definitions (even as you've defined the term), just like the other stuff 
I've cited (like Kline's citation of Russell's definition of an -- 
coined by Poincare' -- impredicative definition) is _related_.

Your definition of the term: "a definition that defines an object by 
means of quantification over a totality to which that object itself 
belongs" contains a kind of circularity (X is used to define X).  Prop 
2.6 refers to a kind of circularity (a cannot be a member of itself). 
Kline's restatement of Russell refers to a kind of circularity.  Lazy 
evaluation in programming indirectly refers to a kind of cirularity (in 
the sense of a many-pass definition, "we'll come back to that later").  Etc.

The fact that these concepts of circularity (whether you maintain your 
strict separation of the terms and their referents or not) seem related 
to me.  And, because they're related, I believe they might hold clues to 
why so many people are mucking around with the term.

Specifically, I think that the relatedness of all these might help me 
understand what RR meant by the term.  (Or, it might even help me 
understand what Kercel or John M might mean by the term.)

Since this list is explicitly _for_ this type of thing, it seems totally 
reasonable for me to list all the above concepts together and try to 
tease out where one differs from the other.

And although I accept that you're not willing to relax your use of the 
term impredicativity from a rigid definition within what you call "logic 
and philosophy", it seems that in order to determine how it's being used 
on this list and in RR's writings, we should cover the relations between 
these concepts of circularity here, including Kercel's "impredicative 
property".

[grin]  By the way, I have more of these ill-advised correlations to 
cite... I'm just waiting for the right time.

>Is there any unclarity in my explanation of why there is no 
incompatibility?

I am unclear on why B&M would want to swap out the FA for what they call 
the "Anti-Foundation Axiom" in order to avoid the situation obtained in 
prop 2.6.  Just for completeness, here is all of prop 2.6:

"Assuming the Axiom of Foundation:
1) for all a, a is not in a.
2) there is no finite sequence a_1, a_2, ..., a_n so that a_1 is in a_2 
is in ... is in a_n is in a_1.
3) there are no a, b so that a is in TC(b) is in a.  (where TC(b) is 
transitive closure)
4) there are no a, b so that a is in TC(b) and b is in TC(a).
5) if c = <a,b>, then c != a, c != b, c is not in a, and c is not in b.
6) for all A, there are no non-empty X so that X = A x X.
7) the only solution of X = X x X is X = empty set.
8) there are no functions f so that f belongs to the domain of f.
9) in fact, it is impossible to find a finite sequence of functions

   f_1: A_1 -> A_2
   f_2: A_2 -> A_3
   f_n: A_n -> A_1

and an element a of A_1 so that

   f_n(f_n-1(...f_2(f_1(a))...)) = f_1."

These are all related to B&M's definition of "cycles" and circular 
relations, which are non-wellfounded.


To answer your question directly, _yes_ it is unclear why the 
circularity you refer to in your definition of impredicative definitions 
as used in logic and philosophy is _different_ from the circularity 
referred to (sometimes by "impredicativity" and sometimes not) by these 
other authors.

--
glen e. p. ropella              =><=                Hail Eris!
H: 503-630-4505                       http://ropella.net/~gepr
M: 503-971-3846                        http://tempusdictum.com