Whoa!

Whoa!
https://youtu.be/IANBoybVApQ

SpaceX lands rocket on barge

Originally shared by Colin Sullender

SpaceX lands rocket on barge

SpaceX successfully landed the booster stage of its Falcon 9 rocket today on the droneship Of Course I Still Love You in the Atlantic Ocean. This is the fifth attempt of an at-sea landing by the company and the first to not end with RUD (Rapid Unscheduled Disassembly). The Falcon 9 rocket launched the Dragon spacecraft to low Earth orbit to deliver critical cargo to the International Space Station as part of CRS-8.

Fun Fact: SpaceX has named its two floating landing platforms (Just Read the Instructions and Of Course I Still Love You) after spaceships from Iain M. Banks' Culture series.

Source: https://youtu.be/7pUAydjne5M (SpaceX)

#ScienceGIF #Science #GIF #SpaceX #ElonMusk #Rocket #Dragon #CRS8 #Spaceship #Booster #Landing #Barge #Culture #Sea #Ocean #Reusable

Computing the uncomputable

Originally shared by John Baez

Computing the uncomputable

Last month the logician Joel David Hamkins proved a surprising result: you can compute uncomputable functions!  

Of course there's a catch, but it's still interesting.

Alan Turing showed that a simple kind of computer, now called a Turing machine, can calculate a lot of functions.  In fact we believe Turing machines can calculate anything you can calculate with any fancier sort of computer.  So we say a function is computable if you can calculate it with some Turing machine.

Some functions are computable, others aren't.  That's a fundamental fact.

But there's a loophole.

We think we know what the natural numbers are:

0, 1, 2, 3, ...

and how to add and multiply them.  We know a bunch of axioms that describe this sort of arithmetic: the Peano axioms.  But these axioms don't completely capture our intuitions!  There are facts about natural numbers that most mathematicians would agree are true, but can't be proved from the Peano axioms.

Besides the natural numbers you think you know - but do you really? - there are lots of other models of arithmetic.  They all obey the Peano axioms, but they're different.  Whenever there's a question you can't settle using the Peano axioms, it's true in some model of arithmetic and false in some other model.

There's no way to decide which model of arithmetic is the right one - the so-called "standard" natural numbers.   

Hamkins showed there's a Turing machine that does something amazing.  It can compute any function from the natural numbers to the natural numbers, depending on which model of arithmetic we use. 

In particular, it can compute the uncomputable... but only in some weird "alternative universe" where the natural numbers aren't what we think they are. 

These other universes have "nonstandard" natural numbers that are bigger than the ones you understand.   A Turing machine can compute an uncomputable function... but it takes a nonstandard number of steps to do so.

So: computing the computable takes a "standard" number of steps.   Computing the uncomputable takes a little longer.

This is not a practical result.  But it shows how strange simple things like logic and the natural numbers really are.

For a better explanation, read my blog post:

https://johncarlosbaez.wordpress.com/2016/04/02/computing-the-uncomputable/

And for the actual proof, go on from there to the blog article by Joel David Hamkins.