A short story based on an interesting premise that at some point in the (near) future, mankind will stop making interesting, fundamental discoveries because we have too much knowledge and too much apparatus around us. In the story, the Americans launch eight astronauts in a spaceship on a ten-year journey to Alpha Centauri to land on its habitable planet called “Alpha-Aleph”. The trouble is, the planet does not exist and the spaceship has only sufficient fuel to reach the star system. More damningly, the American President and a very small circle of science and political advisors have sanctioned this one-way suicide mission, fully aware that the planet does not exist.

The rationale for proceeding with this 40-billion dollar hoax is the strong desire to advance fundamental knowledge, backed by the theory that if you throw together a small group of very intelligent people in extended isolation, they will learn to use the meagre tools at their disposal along with raw brain power to make new discoveries in fundamental sciences, especially if such thought process is not aimed at making specific discoveries. So the astronauts are taught basic number theory and elements of human communication and told to think about these as “recreational puzzles” during the long voyage. A multi-billion dollar ivory tower in action.

The rag-tag bunch predictably starts tinkering around with basic number theory (“Ann has taken to binary arithmetic like fish to water”, “Ann has taken to some sort of statistical experimentation with flipping coins”, etc). They play with Fermat's Last Theorem, Goldbach Conjecture and Godel's theorems, in addition to chess problems and the structures of modes of communications we employ in order to come up with more efficient ways of conveying ideas and thoughts. In the process, they re-invent “Godelization”, an efficient way of coding up messages using prime numbers for transmission to earth, and use it to send back the proof of Goldbach, the secret of nuclear fusion, etc. (the message is:, “1973³⁵⁴ + 331⁸⁵² + 17²⁰⁰⁸ + 5⁴⁷ + 3⁹⁶⁰⁶ + 2⁸⁸ minus 78”)

The story contains a standard paragraph on Goldbach's Conjecture and nice one on Godelian statements. Pohl also explains the dilemma the scientists on Earth face, since decoding a gigantic Godelian number is nearly impossible for their existing technology (“The practical difficulties! You could not get even the first letter until you had the whole number, and IBM had refused even to bid on constructing a bank of computers to write that number out unless the development time was stretched to 25 years”)

The story ends up with the twist that the US starts going up in flames while the astronauts develop god-like powers to come back for the rescue.

I really liked the concept itself but the execution is completely far-fetched on multiple fronts. A focus on the process of discovery itself would have made for a far better story.

I read this as a boy in the 70s. it made such an impression on me that I always remembered it; found this site by Googling "science fiction short story alpha aleph". So for the 14 year old me this was good stuff.

When I read this story in Analog magazine in 1972, Tim Slack and I wrote a programme in assembler for Victoria university's IBM1130 to compute the number and factorise it. Tim had messed with "big numbers" using arrays to represent a number of thousands of digits, and I had been finding prime numbers, etc. The number proved to be about 4,000 digits long and had no factor less than 32749 so it was surely not a proper Godel message number. Oh well. A replublicised version in a paperback gave two versions of the number (a sign was changed, I think) - whatever the provenance, we tested both numbers and found the same result. The computations took only a few minutes as I recall, though writing checking and debugging the programmes took a few weeks.

In short, although the notion was good, the quoted number was just the result of splattering digits. We also had a go at encoding "What's in a code? A rose by any other word would smell as good." with ad-hoc choices for the non a-z symbols, but could find no compact representation. Our search was limited to +- 1:32767 as bases and exponents likewise and we found that typically, a**b where a and b were say four digit numbers, removed only about six digits from the message number if we were lucky. Our search method was not cunning. Produce an approximate floating-point representation of the message number, x, and calculate log(x) in base B, for various values of B, in the hope of finding a result that was N.0000000blah, whereupon calculate B**N exactly (in integers) and subtract/add it from the message number to produce a lesser message number; repeat.

Much later I wrote a prog. in Pascal that would calculate A Godel message number from a text file. I've just recompiled and run it on that example, which generated 1295 digits.

It may be that some message numbers could indeed be represented very briefly (compared to around six thousand decimal digits) but the majority of message numbers cannot be represented in lesser text sequence, since otherwise you would have a compression scheme that always produces a shorter string. If so, run it a second time on the result, for a still shorter string...

Cheers,
Nicky.

Please note that Pohl's "The Gold at the Starbow's End" has been anthologized in the book "Platinum Pohl-The Collected Best Stories" Orb NY 2005