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There's no overreach whatsoever and you can eliminate "there exists" altogether without any loss of expressivity. "There exists a number X such that P(X) is tr
by Kranar 4y ago
There's no overreach whatsoever and you can eliminate "there exists" altogether without any loss of expressivity.
"There exists a number X such that P(X) is true." is nothing more than a short hand for "It is false that for all numbers X, P(X) is false."
In other words, if you accept that it's possible to reason about "for all" then you must accept that it's possible to reason about "there exists" since all "there exists" does is assert that a property is not false for every number.
- thrown_22 4y agoI don't think OP was saying that "for all" is any more meaningful from a computational perspective than "there exits". The point is that both are higher order functions that return an answer for any function on all natural numbers (or inputs in a more general setting).
- Kranar 4y agoIn that case I'd argue the objection has nothing to do with "for all" or "there exists" but rather reasoning about infinite sets.
- thrown_22 4y agoA set doesn't need to be infinite for us to not be able to say something about all it's members. Each busy beaver number is very much finite but you'd run out of universe if you tried calculating BB value for a number as small as 12.
- Kranar 4y agoWhether something is practical to compute is a very different matter from what OP stated was their concern, namely logical knots, potential inconsistencies/contradictions of the "this sentence is false" type, and unbounded complexity.
- thrown_22 4y agoIt really isn't. Math proofs need to be not only finite but small enough to fit in the universe to be discovered. This strictly finite nonsense is a left over from a time when we could only do symbol manipulation with a human brain.
- Kranar 4y agoThat could be true, but it has nothing to do with what was being discussed.