[FOM] Examples of paraconsistent systems

[Joe Shipman]

Can anyone provide an example of a formal system for some part of mathematics which is known to be inconsistent but not known to have a feasibly derivable inconsistency? (In other words, it is “known” to be inconsistent according to a proof relying on some other formal system which can formulate the proposition of its consistency, but also has a clear enough semantics that there is *some* statement it can express which is obviously false.)

— JS

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