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I was thinking about Godel's Incompleteness Theorem and was wondering if this is a fair conclusion: Godel seems to have demonstrated that within axiomatizations of arithmetic, that is of a sufficiently rich axiomatized system which could generate arithmetic, there are statements which can not be proven nor disproven as being theorems of the system in question but which are assumed nonetheless as being 'true.'

May I also conclude from this that as far as any notions of truth are concerned, Godel has effectively divorced the notion of justification (proof) from 'truth.?' If indeed there are statements which are true

but unprovable, then truth does not necessarily entail provability.

Does the assumption of truth for such theorems also maintain the logical consistency of those systems (ie. if we were to assume they were false, then that particular system would be inconsistent)?

Thanks and cheers, mrj

May I also conclude from this that as far as any notions of truth are concerned, Godel has effectively divorced the notion of justification (proof) from 'truth.?' If indeed there are statements which are true

but unprovable, then truth does not necessarily entail provability.

Does the assumption of truth for such theorems also maintain the logical consistency of those systems (ie. if we were to assume they were false, then that particular system would be inconsistent)?

Thanks and cheers, mrj

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