This chapter examines the expressive resources that would be needed to supply nominalistic paraphrases for the language of arithmetic. The upshot is that characterizing a suitable paraphrase-method is not as straightforward as one might have thought. One option is to avail oneself of potentially controversial logical resources, such as variables of very high type or infinitary operations. Another option is to make use of intensional operators. Intensional paraphrase-functions need to be treated with care, however, since they might be thought to shed more light on the truth-conditions of arithmetical sentences than they actually succeed in shedding.
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