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Without getting too technical, what is "forcing" in mathematical logic?


In mathematical logic, ‘forcing’ means refuting a conjecture by showing that it holds only relative to domains that do not include objects they should include.

Here is a trivial example. Smith does not believe that there are any numbers larger than 10, and he knows that 7+8, if it exists, is larger than 10. So he conjectures that some numbers cannot be added together. Jones refutes him by showing that, given a domain that includes all natural numbers, there is no pair of numbers that cannot be added.

This method was singled out for attention when Paul Cohen used it to prove that both the Continuum Hypothesis and its negation are consistent with the basic principles of set-theory. CH is the contention that, if N is the number of an infinitely large set, there is no number intermediate between N and 2^N. (2^N is always larger.) Godel showed that there are models of mathematics relative to which CH is true. Paul Cohen showed that, relative to admissible enlargements of those models, CH is false. In other words, Paul Cohen ‘forced’ CH out, in much the way that Jones forced out Smith’s conjecture.

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