Re: About Consistency in 1st Order Theories.

Aatu Koskensilta wrote:

I don't recall if it was Friedman or some one else who conjectured that naturally occuring inconsistent theories have simple contradictions.

I searched the FOM archives and didn't find Friedman saying anything like this. In fact, in one post he writes

 It's a nice idea - that if something of a fundamentally set theoretic
 nature isn't very quickly seen to be inconsistent, then it is in fact
 consistent.

 The trouble is that the number of data points that seem relevant to
 this is very small.

 The applicability of the concept of evidence to mathematical contexts
 is well known to be a highly contentious matter.

So I was probably just imagining the whole thing.

The context of Friedman's reply is as follows. The following statement

 #) There exists an inaccessible kappa, s.t. there is an elementary
 embedding j: V_kappa --> V_kappa

is inconsistent with ZFC, as was shown by Kunen. Friedman suggests that it might be possible to work from the inconsistent premise Con(ZFC+#) to derive consistency of ZFC+P for some P in a non-trivial manner (i.e. without reasoning like Con(ZFC+#) is false, and hence trivially Con(ZFC+#) --> Con(ZFC+anything)):

 I could well imagine that we could perform the following experiment
 with some real creativity.

 The experiment is to stay well clear of Kunen's inconsistency proof,
 and try to use Con(ZFC + #), Con(ZFC + #+) to redo some of [Woodin's
 relative consistency proofs of form Con(ZF+#)-->Con(ZFC+P)], with
 perhaps easier arguments, and even where we casually use choice (even
 high up).

Kunen is said to have discovered the inconsistency of ZFC+# while trying to strenghten a result of Solovay (having to do with GCH above a strongly compact cardinal). This was one of the first attempts at doing anything subtantial with ZFC+#, and Steel remarks this might be evidence that the experiment suggested by Friedman simply cannot be done; one is bound to run into the inconsistency.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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