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The illustration in the article doesn't start with the whole set of all real numbers, but with just "a" set containing an uncountable number of reals, so there

The illustration in the article doesn't start with the whole set of all real numbers, but with just "a" set containing an uncountable number of reals, so there is no contradiction in that sense. I'm assuming there's some unmentioned technical condition on which sets you are allowed to choose in order for the procedure to work, otherwise the argument would indeed seem to lead to a contradiction when choosing the set of all real numbers.

cevi 5y ago
This assumption is correct - the quanta article left out quite a bit of detail about how forcing actually works, while trying to give some hint of the philosophy behind it. As it turns out, you can still apply the forcing technique even if you start with the whole set of all real numbers. What happens in this case is a bit surprising: since there's no legitimate way to add new real numbers that weren't there previously, the forcing procedure cheats by pretending that real numbers have nonstandardly long decimal expansions. To make the way it cheats a bit more concrete, we start by first imagining that there are "nonstandard integers" n which are bigger than all of the "true" integers from our starting model. Once we assume that these integers exist, we have to commit ourselves to providing answers to questions like, "what is the nth digit of pi?", "what is the n+1th digit of pi?", and so on in a consistent way. By answering these questions, we embed all of the original real numbers from our starting universe into the new universe. But the point is that now that we have all of these extra digits to play with, all of the original real numbers only fill up a tiny subset of the potential "real" numbers in the new universe, and we can happily go back to forcing new ones in. Of course, adding nonstandard integers to our universe is quite a violent change, so we also have to worry about whether we've accidentally screwed up the way that the ordinals work in the process (for instance, the "first infinite ordinal" in the new universe now contains all of the nonstandard integers we added), and we need to worry about this again when we get around to forcing the new reals in. So a whole lot of technical details need to be ironed out carefully, using a few clever combinatorial arguments, to make the whole charade come together. All of this cheating happens under the hood in a way that is not obvious at all when you first go through the technical details of the forcing construction. I only learned about the full perspective from Joel David Hamkins's "naturalistic account of forcing".

cevi 5y ago

This assumption is correct - the quanta article left out quite a bit of detail about how forcing actually works, while trying to give some hint of the philosophy behind it. As it turns out, you can still apply the forcing technique even if you start with the whole set of all real numbers. What happens in this case is a bit surprising: since there's no legitimate way to add new real numbers that weren't there previously, the forcing procedure cheats by pretending that real numbers have nonstandardly long decimal expansions. To make the way it cheats a bit more concrete, we start by first imagining that there are "nonstandard integers" n which are bigger than all of the "true" integers from our starting model. Once we assume that these integers exist, we have to commit ourselves to providing answers to questions like, "what is the nth digit of pi?", "what is the n+1th digit of pi?", and so on in a consistent way. By answering these questions, we embed all of the original real numbers from our starting universe into the new universe. But the point is that now that we have all of these extra digits to play with, all of the original real numbers only fill up a tiny subset of the potential "real" numbers in the new universe, and we can happily go back to forcing new ones in. Of course, adding nonstandard integers to our universe is quite a violent change, so we also have to worry about whether we've accidentally screwed up the way that the ordinals work in the process (for instance, the "first infinite ordinal" in the new universe now contains all of the nonstandard integers we added), and we need to worry about this again when we get around to forcing the new reals in. So a whole lot of technical details need to be ironed out carefully, using a few clever combinatorial arguments, to make the whole charade come together. All of this cheating happens under the hood in a way that is not obvious at all when you first go through the technical details of the forcing construction. I only learned about the full perspective from Joel David Hamkins's "naturalistic account of forcing".