Can definitions of Operators and Relations lead to Contradictions?

Question

Operators and relation definitions are important in mathematics but could some definitions be inconsistent, leading to a contradiction either syntax wise and/or semantically? If so, is there any examples of relations and operators leading to a contradiction as a result of of being improperly defined?

Answer 1

If we're sloppy about how we make definitions, we can run into problems; but these problems emerge when our "definitions" implicitly make mathematical claims which are unjustified or even false.

Let's look at the case of defining a specific object - say, a function. When we say something like

"We define foo to be the function that blahs,"

we're really saying the following:

"There is a single function that blahs; we call this "foo.""

The second half of this claim is completely unproblematic; we're just giving a name to something. Really, we're just introducing an abbreviation. It's the first part that's a potential source of trouble. We're making two claims:

• Existence: There is some function that blahs.

• Uniqueness: There are no two distinct functions that blah.

Either of these claims could be incorrect. A definition needs to be preceded by proofs of existence and uniqueness. These proofs are often trivial and therefore omitted, but they're important.

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OK, let's see some examples of bad definitions of objects.

• "The set of all things which are not in this set." Here we have the implicit claim $$\mbox{"There is a unique set $S$ such that for all $x$, $x\in S\iff x\not\in S$."}$$ The existence part of this claim is clearly false, so we don't even need to talk about the uniqueness part. (Note that this is different from, and sillier than, Russell's paradox.)

• "The function that sends a real number $r$ to its seventh decimal digit." This is an interesting example, since there's a "depth-two" implicit claim: that "its seventh decimal digit" makes sense! This corresponds to the fact that our definition uses a phrase which needs defining, namely "its seventh decimal digit," so before we even start to talk about the whole function we need to think about decimals. The "depth-two" implicit claim is $$\mbox{"Every real number has a unique seventh decimal digit."}$$ The existence part of this is true, since every real number has a decimal expansion, but the uniqueness part fails since some real numbers have multiple decimal expansions: is the seventh decimal digit of $1$ the digit $0$ or the digit $9$? We could of course fix this by specifying that "the decimal expansion" disallows ones with trailing nines, but as written this is problematic.

On the other hand, here's a neat example of a definition which looks circular but is perfectly valid:

• "The set of all objects which are in this set if and only if they are not in this set."

Here I'm claiming $$\mbox{"There is a unique set $S$ such that for all $x$, $x\in S$ if and only if $(x\in S\iff x\not\in S)$."}$$ It's a good exercise to check that $(i)$ the emptyset has this property, and $(ii)$ no other set has this property. So self-referentiality does not immediately lead to invalidity! Psychologically, however, it's easier for us to see and be suspicious of the unique-existence-claim implicit in a self-referential definition than in a general definition.