实在论视角下的大基数

Large Cardinals in the Perspective of Realism

为集合论寻找新公理并为其提供辩护是集合论哲学的重要课题,大基数公理是新公理的备选项之一。目前对大基数的辩护主要分为两种:外在辩护和内在辩护。外在辩护从丰富性和实用性角度出发为大基数做辩护,但它对大基数辩护的成效不足以令实在论者满意。接受高阶无穷的实在论者希望基于集合概念的本质为大基数做更充分的辩护,即为大基数做内在辩护。本文从一种实在论的逻辑观出发,讨论接受高阶无穷的实在论者为大基数做内在辩护的原因,并尝试说明实在论视角下为大基数辩护的具体可能性。

Finding and defending new axioms for set theory is an important topic in the philosophy of set theory,and the large cardinal axiom is one of the alternatives for new axioms.There are currently two main types of justifications for large cardinal axioms:the extrinsic justifications and the intrinsic justifications.The extrinsic justifications,from the point of view of richness and utility in mathematical practice,do not satisfy the realists.The realists wish to provide some justifications based on the nature of the concept of the set,i.e.,to provide intrinsic justifications for large cardinals.I discuss a view of logic from the realism perspective and why we should defend intrinsic justifications for large cardinals under this view,and give examples of how it is possible to justify large cardinals under the realism perspective.