Lambda the Ultimate

The issue of consistency has come up in many LtU posts.

Some readers might be interested in the following abstract from Mathematics self-proves its own Consistency:

                   Mathematics self-proves its own Consistency
                          (contra Gödel et. al.)

                             Carl Hewitt
                         http://carlhewitt.info

That mathematics is thought to be consistent justifies the use of Proof by Contradiction.
In addition, Proof by Contradiction can be used to infer the consistency of mathematics 
by the following simple proof:
   The self-proof is a proof by contradiction.
     Suppose to obtain a contradiction, that mathematics is inconsistent.
     Then there is some proposition Ψ such that ⊢Ψ and ⊢¬Ψ¦.
     Consequently, both Ψ and ¬Ψ are theorems
     that can be used in the proof to produce an immediate contradiction.
     Therefore mathematics is consistent.
The above theorem means that the assumption of consistency
is deeply embedded in the structure of classical mathematics.

The above proof of consistency is carried out in Direct Logic [Hewitt 2010]
(a powerful inference system in which theories can reason about their own inferences).
Having a powerful system like Direct Logic is important in computer science
because computers must be able to carry out all inferences
(including inferences about their own inference processes)
without requiring recourse to human intervention.

Self-proving consistency raises that possibility that mathematics could be inconsistent
because of contradiction with the result of Gödel  et. al. that
if mathematics is consistent, then it cannot infer its own consistency. 
The resolution is not to allow self-referential propositions
that are used in the proof by Gödel  et. al.
that mathematics cannot self-prove its own consistency.
This restriction can be achieved by using type theory
in the rules for propositions
so that self-referential propositions cannot be constructed
because fixed points do not exist. 
Fortunately, self-referential propositions
do not seem to have any important practical applications.
(There is a very weak theory called Provability Logic
that has been used for self-referential propositions coded as integers,
but it is not strong enough for the purposes of computer science.) 
It is important to note that disallowing self-referential propositions
does not place restrictions on recursion in computation,
e.g., the Actor Model, untyped lambda calculus, etc.

The self-proof of consistency in this paper
does not intuitively increase our confidence in the consistency of mathematics.
In order to have an intuitively satisfactory proof of consistency,
it may be necessary to use Inconsistency Robust Logic
(which rules out the use of proof by contradiction, excluded middle, etc.).
Existing proofs of consistency of substantial fragments of mathematics
(e.g. [Gentzen 1936], etc.) are not Inconsistency Robust.

A mathematical theory is an extension of mathematics
whose proofs are computationally enumerable.
For example, group theory is obtained
by adding the axioms of groups to Direct Logic.
If a mathematical theory T is consistent,
then it is inferentially undecidable,
i.e. there is a proposition Ψ such that
¬⊢TΨ and ¬⊢T¬Ψ,
(which is sometimes called incompleteness)
because provability in T
is computationally undecidable [Church 1936, Turing 1936].

Information Invariance is a
fundamental technical goal of logic consisting of the following:
  1. Soundness of inference: information is not increased by inference
  2. Completeness of inference: all information that necessarily holds can be inferred
Direct Logic aims to achieve information invariance
even when information is inconsistent
using inconsistency robust inference.
But that mathematics is inferentially undecidable (incomplete)
with respect to ¬ above
does not mean incompleteness
with respect to the information that can be inferred
because it is provable in mathematics that ¬⊢TΨ and ¬⊢T¬Ψ.

The full paper is published at the following location:
Mathematics self-proves its own Consistency