[...] For more than 100 years, mathematicians have known that there are different kinds, and sizes, of infinity. This was first shown by the 19th-century genius Georg Cantor. Cantor's discovery was that it makes sense to say that one infinite collection can be bigger than another. Infinity resembles a ladder, with the lowest rung corresponding to the most familiar level of infinity, that of the ordinary whole numbers: 1,2,3… On the next rung lives the collection of all possible infinite decimal strings, a larger uncountably infinite collection, and so on, forever.
This astonishing breakthrough raised new questions. For instance, are there even higher levels which can never be reached this way? Such enigmatic entities are known as "large cardinals". The trouble is that whether or not they exist is a question beyond the principles of mathematics. It is equally consistent that large cardinals exist and that they do not.
At least, so we thought. But, like gods descending to earth to walk among mortals, we now realise their effect can be felt among the ordinary finite numbers. In particular, the existence of large cardinals is the condition needed to tame Friedman's unprovable theorems. If their existence is assumed as an additional axiom, then it can indeed be proven that his numerical patterns must always appear when they should. But without large cardinals, no such proof is possible. Mathematicians of earlier eras would have been amazed by this invasion of arithmetic by infinite giants. Read more
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