incompleteness and inconsistency

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?

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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  • Re: My talk about Godel to the post-grads.
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  • Re: incompleteness and inconsistency
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  • Re: incompleteness and inconsistency
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