6/1/2023 0 Comments Absolute infinityNon-standard models of the Peano Axioms subvert our usual notions of the natural numbers, so that we could not even say with any confidence that there is an absolute notion of a natural number. But of course, if you do not fix any theory whatsoever, nothing whatsoever can be absolute. Because one can always extend beyond any one model in which there is a largest cardinality, one can say that in the subject of mathematics - not in any one model, but in the discipline as a whole - there is no absolute infinite. (This is essentially what you point out with inaccessible cardinals). One criticism that you might make is that you could always make a model for NBG in some other set theory in which there are no proper classes. But one could certainly make a statement, in NBG theory, about whether or not a class has a bijective mapping to the universal class, which would certainly indicate that it had the largest cardinality. Of course, this cardinality wouldn't exist as an element in the cardinality function using any conventional set- or class-theoretic construction of functions, because this would require the infinite cardinality to be an element of a class, which is impossible for proper classes by definition. If you use NBG set theory - a conservative extension ZFC - then there is an absolute infinity: the cardinality of proper classes, which in the usual construction of these things would be identified with the proper class of all von Neumann ordinals. So even though not all infinities would necessarily be comparable, there would always be one cardinality which was demonstrably larger. If you use ZF set theory (with or without the Axiom of Choice), you can for any infinite cardinal construct one which is larger by diagonalization. There is no "absolute answer" as to whether there is an absolute infinity, because whether or not you can have an absolute infinity is a function of what mathematical formalism you use. (Without an accepted definition of large cardinal property, it is not subject to proof in the ordinary sense).Saharon Shelah has asked, "s there some theorem explaining this, or is our vision just more uniform than we realize?" Woodin, however, deduces this from the Ω-conjecture, the main unsolved problem of his Ω-logic." "However the observation that large cardinal axioms are linearly ordered by consistency strength is just that, an observation, not a theorem. To be more explicit, the Large Cardinal Axioms extend ZFC by postulating ever higher 'Cardinalities'. But just as we view the naive infinite as bigger than any finite number, is there an 'infinite' greater than any infinite?īut then the same pattern can carry on, in which case we are actually no nearer the absolute infinite than we were to begin with, that is Cantors mathematics of the infinite cannot do real justice to the idea of the Infinite. After Cantor, mathematicians realised that infinities can be graded by size (cardinalities).
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