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The paper examines Dummett's argument for the indefinite extensibility of the concepts set, ordinal, real number, set of natural numbers, and natural number. In particular it investigates how the indefinite extensibility of the concept set affects our understanding of the notion of real number and whether the argument to the indefinite extensibility of the reals is cogent. It claims that Dummett is right to think of the universe of sets as an indefinitely extensible domain but questions the cogency of the (...) further claim that this fact raises an issue as to what sets or real numbers there are. (shrink)
A central theme in the foundational debates in the early Twentieth century in response to the paradoxes was to invoke the notion of the indefinite extensibility of certain concepts e,g. definability (the Richard paradox) and class (the Zermelo-Russell contradiction). Dummett has recently revived the notion, as the real lesson of the paradoxes and the source of Frege's error in basic law five of the Grundgesetze. The paper traces the historical and conceptual evolution of the concept and critices Dummett's argument that (...) the proper lesson of the paradoxes is that set theory is a theory of indefinitely extensible domains. (shrink)