[FOM] How hard can it be to detect an inconsistency?
dmytro at mit.edu
Wed May 6 16:08:32 EDT 2015
On 05/05/2015 07:35 PM, Abolfazl Karimi wrote:
> Given any natural number N,
> design a language (of non-logical symbols) and a finite set of
> contradictory axioms,
> such that any proof of a contradiction is of length greater than N.
PA + there is proof of not Con(PA) that has less than N symbols
is inconsistent but the proof of inconsistency cannot be done with much
less than N symbols. To get a large N, one can, for example, set
N=A(9,9) where A is Ackermann's function.
> If the problem has solutions for every N,
> how natural (i.e. similar to known mathematics) can the language and
> the axioms be?
I do not know any natural examples.
> This problem is in response to the argument that
> since for many years we haven't yet encountered
> any contradiction in some well-developed theory (e.g. ZFC)
> then the probability of its inconsistency must be low.
The hypothesis is that a natural axiomatic system with a well-developed
theory about what the axioms do and do not (assuming consistency) imply
is very likely to be consistent. Additional arguments are that
- ZFC is likely to be true.
- Con(ZFC) is implied by a variety of propositions (e.g every projective
set is measurable).
- There is an essentially well-ordered hierarchy of large cardinal
axioms that extend ZFC with no signs of inconsistency.
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