Re: incompleteness and inconsistency

Per Freem wrote:
hello,

godel's first incompleteness theorem shows us true but unprovable
sentences, so that systems meeting certain conditions are incomplete.
does it follow from these that these systems (say arithmetic with the
usual operators required to achieve godel numbering, etc.) cannot ever
be known to be consistent? a friend of mine said that from
incompleteness it follows we cant know if arithmetic is consistent. my
impression was that this is not the case, but rather we know that
arithmetic is consistent, but if we could prove, INSIDE our system of
arithmetic that it was consistent, then it would be inconsistent (2nd
incompleteness) -- but we can still have proofs in the meta language
that arithmetic is consistent. am i missing something or is my
impression correct?

No one has actually given a direct, certain consistency proof, so
techincally, we don't "know" whether PA is consistent or not.
Nevertheless, there exist proofs called relative consistency proofs
which state that PA is consistent as long as some other system is. The
most famous relative consistency proof of PA is Gentzen's. Primitive
recursive arithmetic, or PRA, is a subsystem of PA in which the only
functions allowed are primitive recursive arithmetic. If
PRA+quantifier-transfinite induction upto epsilon_0 is consistent, so
is PA. So long story short, we don't really know whether arithmetic is
consistent or not.

I hope I was of help. Tell me if there's anything you didn't
understand.
finally, does this have anything to do with 'we don't know if zfc set
theory is consistent'?

thanks for your help.

.

Relevant Pages

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