I like paradoxes but I can’t wrap my head around the Richard’s paradox.
In: Mathematics
Not really sure how some have confused this with different infinities. Richard’s paradox is a different idea.
It’s an *apparent* contradiction which highlights how we must be very careful and precise in maths, especially when using maths to study maths (metamathematics). The idea is as follows:
English has a finite number of symbols, and thus we can define an ordering of finite-length sentences (in the mathematical sense – a string of symbols) by: 1. shorter sentences before longer sentences; 2. compare equal-length sentences alphabetically from the start. We now consider all (finite) sentences that can be used to unambiguously define *a* real number. Using the ordering to produce a list (let’s call it L), we can apply a diagonal argument to produce a new real number.
The diagonal argument here (to define a number, say, x) can be expressed as a finite sentence itself, in a form such as: Let d be the digit of the nth decimal place of the nth number in the list L. Define the number with 0 before the decimal point and in the nth decimal place, 2 if p is 1, and 1 otherwise.
x is different from all numbers in the list L, so it could not have been unambiguously defined, yet we produced a definition for it. This is the apparent contradiction at hand. The resolution to this is that the set of all unambiguously defined numbers itself is not well-defined (in the language of study rather than the metalanguage). Thus the definition of x has no defined meaning. Related to this is the fact that if such a set were well-defined, we would be able to tell whether any English sentence defines a real number. But this is a form of the halting problem, which we know is undecidable.
The ambiguity which produces Richard’s paradox has come from confusing the language and metalanguage (which in this case are both “English”). This same idea can be applied to formal mathematical systems, which once again highlights the importance of distinguishing language and metalanguage.
1. https://www.jamesrmeyer.com/paradoxes/richard-paradox
2. https://www.dpmms.cam.ac.uk/~wtg10/richardsparadox.html
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