Those dumb robots can do anything! Anything at all, that any computer can do.
Archive for logic
One of the great discoveries of the twentieth century is that mathematics can describe the limits of mathematical thought! We’ll discuss some of these ideas from time to time in coming weeks. In this segment, we consider Alan Turing’s insightful question:
Can the answer to any mathematical question be computed?
We catch up with Raymond Smullyan, author of many fantastic books on logic, puzzles and paradoxes at this year’s Gathering for Gardner!
We discuss, among other things, whether all mathematicians are liars.
Send us your favorite paradoxes of this kind and we’ll report back on April 15.
We consider that perennial spring conundrum: Would a woodchuck chuck her own wood if she would chuck wood for exactly those woodchucks who would not chuck their own wood?
What follows after 0, 1, 2, … , once you’ve managed to list every counting number?
Around 1875, Georg Cantor created — or discovered if you like — the transfinite ordinals : the list continues 0, 1, 2, …, then ω , ω + 1, ω + 2, etc, for quite a long long way. John H. Conway tells us about his Surreal Numbers , which add in such gems as
1 / √ ω
Amusingly, this problem has exactly the same solution as the proof that there are as many rational numbers as there are counting numbers. And the proof generalizes: one stork can catch three frogs, or ten or fifty.
Here are some bonus problems:
- The stork can catch the frog even if it can start at any rational number and hop any fixed rational distance each step.
- However, if the frog can start at any real number or hop any real distance, the stork has no strategy that guarantees a catch. This is, in effect, the same as proving that the real numbers are not countable.
A contestant for our Million-Dollar-Give-Away sent in Rayo’s Number, hitherto the largest number ever used for any real purpose: to wit, winning the