In the preface of his book, Everything and More: A Compact History of Infinity, the late David Foster Wallace talks about the power of abstraction to suck you into an abyss of insanity. Mathematicians, who deal in pure abstraction, are more prone to insanity than poets and artists, he claims. Trying to define the mathematical properties of infinite sets is an extreme example and I have to wonder if some of the problems in this area of mathematics may arise from the category problem.
For example, Does it even make sense to consider “the set of all sets that do not contain themselves”, for example? I wonder sometimes. It certainly leads to paradoxes, and worse yet, it is this type of self-referential loop that is the source of Godel’s Incompleteness Theorem, which is a proof that mathematics is not free of contradiction (because there are truths that are not provable).
The following ideas need a lot of refinement, but here is my current thinking on this:
I think the failure has something to do with the characteristics of second-order logic but that is a guess. My very rudimentary understanding is that any “formal system” that is robust enough to make statements about itself (second-order logic?) fails David Hilbert’s Second Problem, which asks if mathematics, and logic, is free of contradiction.
Godel has proven that the answer to this question is in the negative.
But the source of the contradiction arises from a very specific case when a system of logic – be it mathematics or formal logic – is allowed to refer to, and operate on itself. Statements like, “I am lying” reveal the problem. If the statement is true then it is false. Only if the statement is false can it be true. And thus contradiction is inescapable, and unless you are quantum particle, this is just not allowed.
Thus we learn that when a system of logic is allowed to refer to itself, contradiction creeps in. To me, that is all the Incompleteness Theorem reveals. It is not an indictment of mathematics and logic as much as it is of self-reflective systems (second order logic?).
I am straining to make this distinction in order to prevent the misuse of the idea because I read again and again of Godel being invoked as a way to discredit mathematics and logic.
But I find it too much to allow it to hold that mathematics is incomplete. Set theory may be incomplete, but lets not allow this to tarnish all of mathematics. This is why I bring up the category problem.
The incompleteness lives specifically in set theory, not Logic or Mathematics. It is this “second order” stuff that fails the test, because of the unavoidable consequences that arise when a system applies itself to itself, like a snake eating its own tail, or an MC Escher hand drawing itself. Paradox and contradiction arise when a certain kind of rule is broken, like the category rule.
What we talk about when we talk about “the set of all sets that do not contain themselves” is non-sense (or at least redundant) because a set should never be able to contain itself. This is a gymnastics of words, misapplied. It is like saying the color red is ten miles long. It makes no sense. Neither does it make sense to discuss the set of all colors that have no length, or any other such nonsense.
Once this kind of idea – a set that contains itself – is allowed into formal logic, and ONLY when this is allowed, contradiction is a necessary result. For this reason alone, it should be clear that this should not be allowed.
But I am not mathematician or logician. Far from it. I would love for someone to correct my thinking on this. If nothing else, I would prefer if the Incompleteness Theorem could be more specific in identifying what it is that is incomplete. It seems to me that it is not “logic” that fails, but something else. Perhaps it is that “any logic (second order?) that attempts to apply itself to itself” fails.
At any rate, this failure is the same failure we see in metaphysical paradoxes, which usually seem to be victims of category mistakes. Can we really call a set that contains itself a “set”? Is it not some sort of meta-set, and thus separate from itself at that point? Suddenly we have jumped from logic to metaphysics!
These are the same kind of paradoxes that arise in fiction stories when a traveler goes back in time and runs into himself. How many of him are there? The logical conclusion is that there are infinite hims in time, or one for each nano second of his existence at least.
Well, if a set can contain itself, this begs the same question does it not? How many of this particular set is there to be contained? If there be but one set and that one set can contain itself, then why can’t it contain the set that contains itself as well?
See the problem here? It seems to me that we have made a mistake in our linguistics. The idea that a set can contain itself naturally leads to contradiction, therefore, like dividing by zero, it should not be allowed.
OK, so someone help me out here! I am confident that I am wrong about all of this, having never studied “formal logic”. I am merely a meek man of common sense so I need someone to show me where I have gone off the rails.
One last note on this…
there can be no set that contains all sets, for the same reason there can be no greatest number. Show me the set that contains all sets and I will show you the set that contains THAT set. ad infinitum. QED.