[FOM] Another easy solution does not work

Harvey Friedman friedman at math.ohio-state.edu
Fri Sep 13 08:28:30 EDT 2002

I wrote:
>  > The reason I said "or can be so well understood" is that I don't have
>>  such references. But it certainly is very well known that you can't
>>  do recursion along non well founded orderings. E.g., f(n) = f(n+1)+1
>>  if f(n+1) is a nonnegative integer; otherwise.
>>  The point is that this is a straightforward mathematical topic
>>  involving no delicate philosophical issues.

Bolander writes:

>But it seems to me that you can look at the Liar Paradox in the exact
>same way. (A variant of) Yablo's paradox is obtained by considering an
>infinite sequence of sentences S0,S1,... expressing the following
>This is also a recursive definition along a non-wellfounded ordering:
>the ordering ({0},<) given by 0<0 (the simple loop). From my viewpoint,
>if Yablo's paradox reduces to the well-known problem of recursive
>definitions on non-wellfounded orderings then the Liar Paradox does as
>well - in the exact same manner.
>If you disagree, where am I mistaking?

The difference is this. In the case of recursion along non well 
founded orders, there is an obvious totally satisfactory fix: just 
insist that recursive definitions be performed along well founded 
orders. Then recursion along non well founded orders just looks like 
a mathematical mistake. Making a definition by recursion along a non 
well founded order has the appearance of doing a mathematical 
construction, and so seems entirely dispensable.

Whereas in the case of the liar paradox, there is no appearance of 
doing something particularly mathematical, and given Kripke's 
Watergate example and other related examples, even involving Nixon's 
"I am not a crook", it has all the appearance of doing something that 
ordinary people do. So simply declaring that there shall be no self 
referential statements is not a satisfactory solution.

Put another way, the Liar Paradox is a very special and fundamental 
situation, unlike general recursion along non well founded orders, 
which explicitly involves, e.g., infinitely many objects.

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