Returning to my own party a bit late, here. Thanks for the considered responses. I’ll try to address some of them.
I had come up with my own proof (which the margin below will certainly be large enough to contain) but I didn’t want to bias others before posting. SaintCad’s proof is very close to what I came up with.
In his pseudo-proof, he is assuming that 3 and 4 are finite. Also, he only asserts the the sum of a finite number of finite numbers is finite, so 1+1+1… is allowed to be (and is) infinite.
As ultrafilter says, Prof Rucker is a respected mathematician, and while this is intended to be a popular book, the mathematics is (are?) sound. I’m sure he’s offering the 4 + 3 = 7 proof tongue-in-cheek, just to get the reader to question his own assumptions.
No, because you may have counted infinitely many steps to get there without realizing it. Be careful of statements like, “Well, I obviously didn’t count infinitely many steps!” – those are dangerous in proofs. Your assertion implicitly assumes that 8 is finite. If your system of axioms includes one that “8 is finite”, and you have another that says “anything less than a finite value is finite”, then you would have proved it according to your axioms. But we try to have a minimal set of axioms, and in that case you’d need one for each natural number (“9 is finite”, “10 is finite” etc.)
But how many s’s are there in that equation? How do you know it’s a finite number? I’m not being cute or obtuse here, of course we all “know” it’s a finite number, but how do we “know” that? How do you know you didn’t use the successor function an infinite amount of times without realizing it?
He gives no justification for why 3 and 4 are finite, but takes them as given. Again, I’m sure this is tongue-in-cheek and not considered to be a formal proof of anything.
Very apt, given your username!
I disagree. In the set theoretic construction of the natural numbers, each number is precisely associated with a single set, namely the set of all numbers less than the number. Then we can say that the number is finite if the associated set has finite cardinality, and vice versa.
For the record, I’ll post the original “puzzle” verbatim, along with Prof Rucker’s answer. I was quoting from memory before but now I found the book.
His answer is as follows:
To answer’s ultrafilter’s question, the book is very set-theoretic in focus, though he gives plenty of time to the Peano axioms too. He is gently guiding the reader towards infinite ordinals/cardinals (can never remember the difference).
Here’s my “proof”, which I already emailed to Rucker to see what he thinks. I’m very much an armchair mathematician so it’s quite likely there’s a flaw in this.
How to show that 7 is finite, by using the set theory definition of 7:
(this is the shaky part) A set is finite iff it cannot be put in 1-1 correspondence with a proper subset of itself. (Is that true? I know it’s a sufficient condition for a set to be infinite that it can be put in 1-1 correspondence with a proper subset.)
Constructing 7 using the set approach:
{} = 0
{{}} = 1
{ {{}}, {} } = 2
… etc.
(lots of curly braces) = 7
Eventually you have a set theoretic definition for 7, which could be a set with infinity cardinality – we have neither proved nor assumed that yet. Enumerate the proper subsets, and show that the set representing 7 cannot be put in 1-1 correspondence with any of those listed subsets, so the set representing 7, and ergo 7 itself, cannot be infinite and are therefore finite.
I find it interesting that people have two kinds of reaction to this type of problem: either, “WTF…of course 7 is finite! What a stupid question!” vs. “Hmm…yes, how could we prove that while introducing the fewest number of axioims?”
A lot of results in the world of math seem intuitively obvious, but either can be proved from a much smaller set of axioms or turn out to be wrong. This is especially true in the world of infinite sets…e.g. there are “obviously” more natural numbers than there are prime numbers, and “obviously” more natural numbers than there are perfect squares, right?
I highly recommend the book, by the way, for anyone interested in math. Someone recommended it to me on the Dope years ago, and I’ve read through it many times.