Santa and Contradictions

Perhaps you just happened to notice that my previous "proof" that Santa exists could be used for proving other things, too. For example, that Santa doesn’t exist. Or your children have already proved that they don’t need to go to bed.

OK, so what’s the catch? It seems possible to prove anything using this method, even contradictions. Is Mathematics inconsistent, after all? Well, no. It’s just that we’re not used to definitions causing contradictions. This is something that mathematicians realized at the beginning of the 20th century, when they investigated the foundations of mathematics.

For example, Bertrand Russell found a "paradox", when he postulated a set X that contained all sets that didn’t contain themselves. Which leads to a contradiction: Suppose X contains itself. Then X can’t contain itself, since it’s a member of X. Then suppose X doesn’t contain itself. Then X contains itself, by definition! So both cases give us contradictions. The conclusion Russell didn’t draw from this (I think) is "So X isn’t a set". Just to be extreme, suppose that S is a set that does contain 0 and does not contain 0. Anyone surprised that we get a contradiction from that? I guess not.

So, let’s check where S in the Santa example leads us. S is defined as "If S is true, then Santa exists". If S is to make sense, it must have a well-defined truth value; either true or false. We’ll check: can S be true? OK, then it seems to be the case when Santa exists, because if S is true, certainly Santa exists. It’s a logical possibility. But if S is false? Then the left-hand-side of the logical implication described by S will be false, and the implication itself will be true. Which means that S is true. But S is false! So the definition of S implies that S has to be true, and that Santa exists!

But if we view definitions as equations, things make sense. The definition of S is really an equation, which only has the solution "S is true". Other definitions have no solutions (like the set of all sets not containing themselves), and other might have several.

2 thoughts on “Santa and Contradictions

  1. Marcus September 27, 2005 / 4:48 pm

    Since the definition can have a truth value, it can be entirely written in propositional logic.

    The statement “S iff (S implies Santa Exists)” has a single propositional model in which S and Santa Exists are true. So the existence of Santa is postulated, which typically makes proving it quite easy.

    I actually never liked the “Santa Exists” argument as an introduction to Gödel, but I love the fact that you reminded me of the beautiful Boolos&Jeffrey.

    It is sad that most modern books on Mathematical Logic just juggle syntax.

  2. Thomas Drakengren September 27, 2005 / 8:02 pm

    Right, your “proof” is simpler than mine!
    And I agree, the book by Boolos and Jeffrey’s is just so very beautiful. That is, if you’re able to see through all the ugly details! 😉

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