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In this paper paraconsistent second order arithmetic Z#2 with unrestrictedcomprehension scheme is proposed. We outline the development of certainportions of paraconsistent mathematics within paraconsistent second orderarithmetic Z#2.In particular we defined infinite hierarchy Berry's andRichard's inconsistent numbers as elements of the paraconsistent field R^#.

Paraconsistent second order arithmetic Z^#_2 based on the paraconsistent  logic LP^# with infinite hierarchy levels of contradiction. Berry's and  Richard's inconsistent numbers within Z^#_2