[FOM] Re: Reflection and the Lucas-Penrose Argument

[Jeffrey Ketland]

Tim Chow:

>So let's turn to the interesting case, where we assume that it's actually
>possible to exhibit a Turing machine M and become unassailably certain
>that all the sentences M is generating---i.e., S---are true.  Then we
>can apply the usual construction to produce a sentence Con(S).  According
>to your reflective closure condition, Con(S) is also unassailably true.
>So by definition of S, Con(S) is in S.  But this contradicts Goedel's
>2nd theorem.  End of proof.

Just to check I understand your reconstruction right, I'll try and outline
the basic structure of the argument again. First, let S be the set of
accepted mathematical propositions in the end (in some idealized, Peircian,
sense). If a statement A is in S, then A will show up in human mathematical
inquiry at some point.
The philosophical theorem is then:

       1. Goedel's 2nd incompleteness theorem
       2. PA is a subset of S
       3. S satisfies some sort of reflective closure condition
       => non-axiomatizability of S

The detailed argument is by reductio: assume the human mind HM is
represented by some Turing machine M which enumerates S. By assumption, HM
has the reflective capacity, so S satisfies the reflective closure
condition. So, for any r.e. theory T, if T is accepted, then Con(T) is. Ex
hypothesi, S is r.e.. And S is accepted. By reflection, so is Con(S). But M
enumerates all elements of S, so Con(S) is in S. Furthermore, S contains PA.
So, S is an r.e. extension of PA containing Con(S). This contradicts
Goedel's 2nd incompleteness theorem.

I'm a little clearer now about this reconstruction of Lucas-Penrose based on
reflective closure capacities. But I'm still not 100% sure it works. The
internal conclusion that Con(S) is in S doesn't strictly contradict Goedel's
2nd. We might conclude just that S is inconsistent! So, the argument needs
to be reformulated,

       1. Goedel's 2nd incompleteness theorem
       2. PA is a subset of S
       3. S satisfies some sort of reflective closure condition
       => if S is consistent then S is non-axiomatizable

Perhaps that's right.

>Your description of Feferman's reflective closure and of theories of truth
>and so forth fail to get around this problem.

It's not really intended to get round any problems. It's merely that
Feferman defines a valuable notion of reflective closure of a formal system,
and provides a compelling justification for it. The construction S |->
Ref(S) is intriguingly ampliative, to use the philosopher's term, in that
you get "more out" than you "put in". I.e., Ref(S) is a non-conservative
extension of S.

My broader view is that it's of great importance for recent debates about
the notion of truth that (axioms for) the notion of truth can play this
crucial non-conservative role in extracting the reflective consequences of a
theory. It hardly needs stressing that mimimalist or deflationary theories
of truth are, at present, immensely popular amongst academic philosophers.
But minimalist or deflationary truth theories ought to be conservative, and
Ref(S) is a non-conservative extension of S. I also assume that this is only
part of the story of what is involved in developing a better understanding
of this kind of reflection. Subtle questions arise when one considers
iterating Ref itself, as Aatu Koskensilta mentioned.

>As I said before, Torkel Franzen has an excellent discussion of all this
>in his book on inexhaustibility.

I'm eagerly awaiting Torkel's book.

--- Jeff
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Jeffrey Ketland
School of Philosophy, Psychology and Language Sciences
University of Edinburgh, David Hume Tower
George Square, Edinburgh  EH8 9JX, United Kingdom
jeffrey.ketland at ed.ac.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~