I read everything I could get my hands on that Smullyan wrote when I was 12 or so. I still have many of the copies of articles, old books, and the notes from my attempts to work his puzzles in a box, and just opening Alice in Puzzle-Land, like the madeleine from Proust, instantly brings me back to that time. Highly recommended.

My daughter is eleven. When she asks me to read her Smullyan before bed...

I melt.

I met Raymond at a conference in the early 90s. I was an undergraduate in math then. We talked about philosophy, religion, and mathematics. Just the two of us. The next day he came up to me and said that he invented a religion for me.

God has a number in mind. If the sum total of good deeds minus bad deeds exceeds this number then everyone goes to heavan. If not then everyone goes to hell.

Then he said to me, “Imagine, something you do could send everyone to hell.”

Raymond just made up a religion for me. It was not meant by him to be serious. Our perceptions of reality appear to be very different.

Raymond Smullyan is one of the biggest reasons that I'm a programmer today, most of the reasons that I write about programming today, and nearly all of the reasons for what I choose to write about, when I write about programming.

He was, literally, who I wanted to be when I grew up. It turned out that I grew up to be Raganwald, not Raymond, and that is very alright with me.

But I am forever grateful for the influence he had on me.

This is the first time I read about his NPRNPR distillation of Gödel's proof (N=Not,P=Provable,R=Repeat). So clever. And so Smullyan.

I'll always remember his https://en.wikipedia.org/wiki/To_Mock_a_Mockingbird as a most delightful introduction to Combinatory Logic.

Funny enough, a similar trick can be used to give you the Y combinator.

Let comp be the 'compose' combinator

((comp f g) x) = (f (g x))

Let R be the 'repeat' or 'self-application' combinator

(R x) = (x x)

Then (Y f), the combinator obeying the equation

(Y f) = (f (Y f))

can be expressed as

(Y f) = (R (comp f R))

If our f is NP, then we'll have (NP (R (comp NP R))). So, in other words, NPRNPR is just (Y NP).

Funny enough, taking that definition of the Y combinator, you can get all of its special forms (normal order, applicative order, and polyvaradic normal & applicative order), just by changing the definition of the 'comp' function.

Here's a gist that shows how:

https://gist.github.com/mromyers/b6d7678bf7a04e106b3d7d5b649...

Then you must not miss his book "Satan, Cantor, and Infinity."

For the benefit of those skimming the comments and not reading TFA: https://amzn.to/2SvmTdX (affiliate link).

While his puzzle books obviously deserve a lot of praise, Smullyan's textbooks and papers definitely shouldn't be overlooked. There's a lot of wonderful gems to be found there.

Diagonalization and Self-Reference is the single book I would recommend the most to the HN crowd. There are a few sections on quotation and Quines that I've found endlessly useful, his 'Elementary Formal Systems' is my favorite presentation of computability, and there's a lot of really deep stuff in there about the interaction between incompleteness, uncomputability, and fixed-points.

Also, Logical Labyrinths is a pretty great textbook on formal logic. The first half is in the form of one of his puzzle books, introducing notions and building intuitions, while the second half builds off of them to provide a more formal perspective, while incidentally giving a kind of eye opening look at how he comes up with his puzzles and how they map to certain deeper properties of logic.

Smullyan's puzzle books are superbly entertaining and witty.

"Suppose there is a mapping from the natural numbers onto the decimals."

Infinity is a tricky subject.

There are infinite naturals, fractions and decimals but they are different kinds of infinities. If we define natural positives as N and both positive and negatives as the same infinities N * 2 (with a minus sign), we can safely say that decimals are simply all naturals multiplied by infinity N * N * N * 2 or N ^ 3 * 2 where for every single integer like 0, 1 or 2 there will be infinite decimals after the point prefixed by infinite zeroes too like .1 .01 .001 .0001 and even if N * 2 or N ^ 3 are equal to infinity (infinity because we can't measure it or at least not know its final boundary) both infinities are different.

And guessable, we're out of hell someday.

Cardinality is the name given as a measure of the size of a set. The cardinality of the rationals is the same as the integers. The cardinality of the reals is larger than that of the integers.

There is an arithmetic of cardinal numbers and it is well understood if you accept the axiom of choice. For instance, using your notation, N*N = N and N^2 = N. You can read more here

https://en.m.wikipedia.org/wiki/Cardinal_number

> The cardinality of the rationals is the same as the integers.

If you combine N as all possible numerators with N as denominators you get that cardinality of Q = N * N

Also, I don't accept the diagonal argument as proof. Given all possible combinations of numbers, any given number will occur in that set no matter what. If you add special rules of course it falls apart and Cantor's argument is just a special rule.

If we use fruits as an example, taking a diagonal from their letters won't form a fruit either.

    1. [A]PPLE
    2. O[R]ANGE
    3. MA[N]GO
    4. CHE[R]RY
    :
    N. ARNR ?

It might be worth your time to consider the possibility that it is likely that the whole of the mathematics profession is not wrong in this matter. Are there any professional mathematicians that agree with you? If pretty much the whole profession thinks you are wrong about something then it’s quite likely you are indeed wrong.

It’s worth pointing out that your logic on Q = N times N is a bit faulty too. Since you are counting things like 4/4 as different than 1/1. Even so you are correct that the cardinality of Q is N times N. This is because N times N = N.

It has been formally verified: http://us.metamath.org/mpegif/canth2.html

So, if you don't accept the proof, you have to reject some axiom used. Which one do you have a problem with?

> If we use fruits as an example, taking a diagonal from their letters won't form a fruit either.

It will, however, form a sequence of characters. The diagonalization argument requires all possible sequences to be valid, which isn't true for fruits.