1*3*4*5*… = ?

I suggest ℵ1.

It’s at least 2 to the power of ℵ0 – the number of factors in the title. And by definition, 2 to the power of some aleph is the next aleph. As every finite set with N elements also has its power set of 2^N elements.

This is the lower bound.

The next reason is, that countably many sided cuboid 1 by 2 by 3 by 4 … has a bijective mapping between the real numbers on the interval [0,1] and its consisting hypercubes 1 by 1 by 1 …

And this is also the upper bound.

ℵ1 is therefore the most natural definition for such a product. That the product of all naturals, is equal to the number of all reals.


Why is 1+2+4+8+… = -1 ?

In a 8 bit CPU register -1 is written as 11111111.

In a 16 bit CPU register -1 is written as 1111111111111111.

And so on.

Say, that our register has an infinite number of bits. Then -1 is written as ….111111.

With the infinite number of 1s on the left.

32 is written as …0000100000.

With the infinite number of 0s on the left.

So we have:

+…000000001 (1 decimal)

+…000000010 (2 decimal)

+…000000100 (4 decimal)

+ ….

+ ….


+…111111111111 (-1 decimal)

It’s not a joke. At least not more of a joke than the title is.


Points of Interest


Robin Hanson’s book The Age of Ems (haven’t read it) has a very interesting premise. That Universal (General) AI is more difficult than Whole Brain Emulation.

This might even be true. Not very likely, but still possible.


A new theory about cancer is out. That it is essentially the last line of defense. Like sinking the ship, if you’ll have to abandon it soon anyway! Mining the fortress before leaving it, because you don’t want the pathogens to use your body as a lab, and you also don’t want to spread your already damaged DNA further.


Adam 60 000 years ago, our common male ancestor is fake. Some African tribes are 200 000 years, maybe even 300 000 years away from Europeans or East Asians. They were lying to us.


Some (reactionaries) developed an interesting theory of how our society works. Essentially people defect at “who is holier” game. Instead of maintaining a realistic level of holiness, they often claim to be “holier than thou”. And then we have almost an arm race, who is going to present himself as more holy. This leads to a danger spiral. Once Pope washed the feet of cardinals, now he is washing anybody’s feet so as to present himself as a more holy. And it’s not only the Pope, most people are racing here.

Alice: I want the elephants of Africa to survive and flourish!

Bob: I am holier than thou, so I want their stomach parasites to survive and flourish, too!

Alice: I am even holier than thou. I’m not going to remove my tapeworm from my intestines, because they are endangered species!

Very soon, all of society is looking pretty retarded and is decadent.


If I was harvesting wind energy, I would stick to the fact that the power of wind is proportional to the cube of its speed. To the cube, ladies and gentlemen! So stopping the windmills at a certain wind speed means only one thing. They are no good, yet.


Someone suggested that we could mine for cold. And to trade with it. It costs a lot of money to cool some large building with electricity. Maybe, we should catch an iceberg and pump the needed cold out from it. People have done it before. Ice trade flourished before the era of refrigerators, perhaps we should reinvent it using modern technologies.


Speaking about old ways of doing things. Pre-ancient-Greek mathematics was conjectural. No formal logical proofs of “truths” but empirical observation that they are likely, as they were observed frequently in different circumstances with no known exceptions. Ramanujan usually operated this way, leaving the proofs for his formulas to others. Currently, we have more and more of mathematics based solely on the assumption that the Riemann conjecture is true, for example.


Recently, I had a strange dream about calculators. I urgently needed to do some calculation, the specifics of which I don’t remember. So I tried and I did not succeed. Because my brains are not strong enough to emulate even a simple calculation when sleeping. But they are smart enough to stage a calculator malfunction and they did it. The dream calculator stopped working. I managed to find another one, which failed me also. The third calculator was printed in a newspaper and was not even meant to work. Of course. Mister Sandman is such an opportunist!