A Paradox of Infinite Knowledge

For every proposition we know, we can reflect upon it to create a new piece of knowledge. For example, if I know that I exist, I can likewise say that I know that I know I exist. Again, if I comprehend this statement, then I could truthfully say that I know that I know that I know I exist. Because I can always add a further “I know that” statement to the previous piece of knowledge, this can continue ad infinitum. In this way, even if I only started with a single proposition I know to be true, it is possible to infinitely multiply the number of propositions I know through self-reflection.

The infinitely many propositions would take this form:

  1. I know I exist.
  2. I know that I know I exist.
  3. I know that I know that I know I exist.
  4. I know that I know that I know that I know I exist.
  5. I know that I know that I know that I know that I know I exist.
  6. I know that I know that I know that I know that I know that I know I exist…

…And on it goes.

Infinite knowledge would be the knowledge of this entire series from 1 through ∞.

But this may be cheating. Adding “I know that” statements may not even substantially increase our knowledge. Even assuming “I know that” statements do in fact add to our knowledge, it is much less clear how it an individual could grasp an infinite series.

Let us suppose, for the time being, that “I know that” statements are not merely vacuous and do increase our knowledge. Let us also assume it is possible to grasp the infinite series which arises from adding these statements ad infinitum. Hence, you know an infinite series of propositions, but of course this does not mean you possess all knowledge. This only means you know infinitely many things relative to the starting proposition. Given these assumptions, something strange happens.

What if this was our starting proposition: “I know an infinite series of propositions”?

This starting proposition seems to entail a paradox. The paradox arises from the fact that, in order to first know an infinite series of “I know that” propositions, we would need to begin with a proposition that does not include knowledge of an infinite series. Knowledge of the infinite series only arises once we begin with a proposition that does not reference the infinite series. Propositions like “I exist” and “Water is H2O” both do this because they are finite. These finite propositions go on to entail knowledge of infinitely many “I know that” propositions, but to say “I know an infinite series of propositions” turns the procedure on it head by asserting infinite knowledge to begin with.

This starting proposition devolves into a paradoxical story of the chicken and the egg; which comes first? For the statement to be true we would have to begin with an infinite series, but in order to get the infinite series we must begin with the finite proposition…

…And on it goes.

This is just one example of the many paradoxes that can arise from self-reference. Here’s another example of a paradox: “This statement is false.” If it is true that it is false, then it is false and not true. But, if it is false, then it is true.

Paradoxes also arise in set theory. Set theory, to put it simply, is the study of the qualities of sets and their implications. For instance, the set of all cats contains all cats in the world. Sets are like theoretical categories containing things in the world that fall under that category. But, sets can also contain sets. There is ‘the set of all individual sets of every mammal species’, which would contain the set of dogs, cats, lions and every other mammal. There can also be ‘the set of all sets’. In this case, the set of all sets would contain itself, because the set of all sets is itself a set!

But then, how would we make sense of ‘the set of all sets that does not contain itself’? This does not seem possible, for the set of all sets that does not contain itself would have to contain itself in order to be a set. But it couldn’t be a set because then it would contain itself. It must be a set, it can’t be a set, it must contain itself, but it can’t contain itself…

And on it goes.

Bertrand Russel and Alfred Whitehead tried to solves these paradoxes that would arise in set theory by making each set non self-referential. The sets could not refer to themselves, because these paradoxes would arise when they mentioned themselves.

Then Kurt Gödel came along with his incompleteness theorem and complicated matters even more.

At any rate, my paradox of infinite knowledge takes the same form as many other paradoxes which arise from self-reference. If we begin with a finite proposition of possessing knowledge of infinite propositions, we would first need the infinite series. But to get the infinite series, we need the finite proposition.

And on it goes.

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