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New set-theoretic axioms derived from a lean metamathematics

I will formulate in this paper three set-theoretic axioms, (A1), (A2), and (A3), which appear natural and settle some well-known questions, and I will give some metamathematical evidence supporting these axioms. I build upon a distinction between pure mathematics and applied metamathematics which views the first as an art dealing with imaginary objects, where following Poincaré we can say to exist is to be free of contradiction, and the second as a science describing the phenomenon of mathematics, where science means a description of some physical reality (in this case the reality of thoughts in our brains which underlie spoken or written mathematics). Of course this distinction is not new, but it has been disregarded by many mathematicians and philosophers who wrote about the nature of mathematics, e.g. by the Platonists, and even by some empiricists who thought that mathematics is a science. Since applied metamathematics is a science, unlike pure mathematics it has to be lean, i.e., to obey Ockham's principle of economy of concepts (entia non sunt multiplicanda praeter necessitatem). In this lean metamathematics we describe pure mathematics as a finite structure of thoughts in our brains, and we think that written or spoken mathematics is an abstract description of this structure. (I think that this is the view which Poincaré expressed informally in his discussions with the logicians and set theorists of his times, although his idea has to be modified to some extent; see Remark 4 at the end of this paper.) We claim that all mathematical objects which are imagined when we develop a (first-order) theory T can be represented by means of terms of the language of a Skolemization of the set of sentences consisting of the axioms of T and of all theorems of logic.