Free Republic
Browse · Search
General/Chat
Topics · Post Article

Skip to comments.

Mathematicians Discovered a Computer Problem that No One Can Ever Solve
livescience.com ^ | January 11, 2019 08:08am ET | By Rafi Letzter,

Posted on 01/12/2019 5:15:03 AM PST by BenLurkin

The trouble is, math is sort of broken. It's been broken since 1931, when the logician Kurt Gödel published his famous incompleteness theorems. They showed that in any mathematical system, there are certain questions that cannot be answered. They're not really difficult — they're unknowable. Mathematicians learned that their ability to understand the universe was fundamentally limited. Gödel and another mathematician named Paul Cohen found an example: the continuum hypothesis.

The continuum hypothesis goes like this: Mathematicians already know that there are infinities of different sizes. For instance, there are infinitely many integers (numbers like 1, 2, 3, 4, 5 and so on); and there are infinitely many real numbers (which include numbers like 1, 2, 3 and so on, but they also include numbers like 1.8 and 5,222.7 and pi). But even though there are infinitely many integers and infinitely many real numbers, there are clearly more real numbers than there are integers. Which raises the question, are there any infinities larger than the set of integers but smaller than the set of real numbers? The continuum hypothesis says, yes, there are.

Gödel and Cohen showed that it's impossible to prove that the continuum hypothesis is right, but also it's impossible to prove that it's wrong. "Is the continuum hypothesis true?" is a question without an answer.

In a paper published Monday, Jan. 7, in the journal Nature Machine Intelligence, the researchers showed that EMX is inextricably linked to the continuum hypothesis. It turns out that EMX can solve a problem only if the continuum hypothesis is true. But if it's not true, EMX can't.. That means that the question, "Can EMX learn to solve this problem?"has an answer as unknowable as the continuum hypothesis itself.

(Excerpt) Read more at livescience.com ...


TOPICS: Science
KEYWORDS: continuumhypothesis; kurtgodel; paulcohen; rafiletzter; stringtheory
Navigation: use the links below to view more comments.
first previous 1-20 ... 121-140141-160161-180181-183 next last
To: BenLurkin

They need all the numbers they can find to state our national indebtedness.


141 posted on 01/12/2019 9:26:22 AM PST by Joe Bfstplk (How unfortunate tarring and feathering was abandoned.)
[ Post Reply | Private Reply | To 1 | View Replies]

To: BenLurkin

A lot of good answers here. I think “infinity” is an invention of the mind. If there was a true infinity, an infinity with anything physical that has volume, we wouldn’t exist, because we would limit infinity by our presence. In other words, we are taking up space where what we are measuring should exist in, which means that infinity is lacking. Infinity is less and, therefore, not infinite. It’s the same with God. To say that God exists infinitely everywhere, then there is no space or room for us.


142 posted on 01/12/2019 9:32:25 AM PST by BEJ
[ Post Reply | Private Reply | To 1 | View Replies]

To: BenLurkin
Interesting. I really with that when I was younger someone had pointed out to me some of the weird things that can happen with math. Things like Collatz systems and the like. It might have made me more interested in it.
143 posted on 01/12/2019 9:46:47 AM PST by zeugma (Power without accountability is fertilizer for tyranny.)
[ Post Reply | Private Reply | To 1 | View Replies]

To: JohnG45
And in February, 2002, at a Department of Defense briefing, Donald Rumsfeld agreed:

"There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don't know. But there are also unknown unknowns. There are things we don't know we don't know."

That wasn't original to him. I never understood the crap he took for making the statement. It's absolutely true. It's the unknown unknowns that really bite you in the ass.

144 posted on 01/12/2019 9:48:15 AM PST by zeugma (Power without accountability is fertilizer for tyranny.)
[ Post Reply | Private Reply | To 15 | View Replies]

To: Chengdu54
I have a proof for this, but there’s insufficient space to show it here.

What you did there. We saw it.

145 posted on 01/12/2019 9:49:05 AM PST by zeugma (Power without accountability is fertilizer for tyranny.)
[ Post Reply | Private Reply | To 16 | View Replies]

To: coloradan
Integers are countably infinite

"Countably infinite" is a contradiction in terms. If it's countable, it's not infinite. By definition.

That reals are different than integers does not make either of them countable, or one of a "greater" infinity than the other.

Infinity is not a number. Thus, no arithmetic comparators apply to it. Terms like "greater" or "lesser," "more" or "fewer" are meaningless.

146 posted on 01/12/2019 9:58:50 AM PST by IronJack
[ Post Reply | Private Reply | To 139 | View Replies]

To: Bitman
It has been solved. The answer is, of course, 42.

And is the rounded number from 41.7

147 posted on 01/12/2019 10:03:16 AM PST by spokeshave2 (https://www.gofundme.com/TheTrumpWall.... $20,248,521 of $1.0B goal by 338,785 people in 26 days)
[ Post Reply | Private Reply | To 13 | View Replies]

To: BenLurkin

WOW! This sounds even to big for even AOC to solve!


148 posted on 01/12/2019 10:08:51 AM PST by chickenlips
[ Post Reply | Private Reply | To 1 | View Replies]

To: imardmd1
For all practical purposes, the answer is “No.” Or is it?

Dr. Edward de Bono invented the word PO as intermediate between YES and NO

I think in Japanese its HA

149 posted on 01/12/2019 10:28:08 AM PST by spokeshave2 (https://www.gofundme.com/TheTrumpWall.... $20,248,521 of $1.0B goal by 338,785 people in 26 days)
[ Post Reply | Private Reply | To 31 | View Replies]

To: imardmd1
For all practical purposes, the answer is “No.” Or is it?

Dr. Edward de Bono invented the word PO as intermediate between YES and NO

I think in Japanese its HA

150 posted on 01/12/2019 10:28:08 AM PST by spokeshave2 (https://www.gofundme.com/TheTrumpWall.... $20,248,521 of $1.0B goal by 338,785 people in 26 days)
[ Post Reply | Private Reply | To 31 | View Replies]

To: JoeProBono

Ping


151 posted on 01/12/2019 10:30:56 AM PST by BenLurkin (The above is not a statement of fact. It is either satire or opinion. Or both)
[ Post Reply | Private Reply | To 149 | View Replies]

To: BenLurkin

Bump for later


152 posted on 01/12/2019 10:42:23 AM PST by Hegemony Cricket (< < Wandering aimfully > >)
[ Post Reply | Private Reply | To 1 | View Replies]

To: BenLurkin

"But even though there are infinitely many integers and infinitely many real numbers, there are clearly more real numbers than there are integers."

Fail. This statement attaches "magic" to the decimal point. The decimal point is arbitrary. Without it there is only an infinite series of integers. If we had brains capable of understanding the true smallest possible undivided unit, and perform maths at levels infinitely larger than this, we would not require decimal places at all. Only because of the limited observational and computational power of mankind are decimals used.


153 posted on 01/12/2019 11:13:49 AM PST by so_real ( "The Congress of the United States recommends and approves the Holy Bible for use in all schools.")
[ Post Reply | Private Reply | To 1 | View Replies]

To: P.O.E.
OTOH, there are infinity-deniers:

I wrote out an infinite string of numbers once but I showed it to my wife and she added one and messed it up.

154 posted on 01/12/2019 11:45:22 AM PST by itsahoot (Welcome to the New USA where Islam is a religion of peace and Christianity is a mental disorder.)
[ Post Reply | Private Reply | To 10 | View Replies]

To: HangnJudge
That's not without getting into Complex Numbers

Not to mention the imaginary ones.

155 posted on 01/12/2019 11:47:30 AM PST by itsahoot (Welcome to the New USA where Islam is a religion of peace and Christianity is a mental disorder.)
[ Post Reply | Private Reply | To 46 | View Replies]

To: super7man

You are a very lucky man to have such balance in your life, and to help with spelling too!


156 posted on 01/12/2019 12:39:01 PM PST by Susquehanna Patriot (Evolution is the long term solution to Global Warming. So let's party while we can!)
[ Post Reply | Private Reply | To 137 | View Replies]

To: IronJack

You are mistaken. There are infinitely many integers, because, for any integer, one can always obtain a larger one. However, they are countable in that they can be ordered, giving each one a place, with a definite “next one” and a definite “previous one” - in the case of every single integer.

In particular, for any given integer, there is always a “next one” and so the set of integers is infinite. (Indeed, even the set of “positive integers” is infinite.)

Just because something is countable doesn’t mean it isn’t infinite. In order to be not infinite, there has to be a finite number of them - a last one of them. The number of ways a deck of cards can be ordered is finite. It’s a huge number, but, it is a definite number, and there are no other ways the deck can be ordered other than as one of those ways. But there is no last integer; one can always get a larger integer by taking the factorial, by squaring it, or even, modestly, by just adding 1.


157 posted on 01/12/2019 1:05:38 PM PST by coloradan (The US has become a banana republic, except without the bananas - or the republic.)
[ Post Reply | Private Reply | To 146 | View Replies]

To: BenLurkin

The answer could be anything.
More research money is needed.


158 posted on 01/12/2019 1:15:00 PM PST by right way right (May we remain sober over mere men, for God really is our only true hope.)
[ Post Reply | Private Reply | To 1 | View Replies]

To: coloradan

I think I understand the distinction. It seems a little bit like semantic hair-splitting, but I’ll concede the point.


159 posted on 01/12/2019 1:19:25 PM PST by IronJack
[ Post Reply | Private Reply | To 157 | View Replies]

To: BenLurkin
Throw in floating point, and nothing is truly accurate.

Stephen Hawking said, "God created the integers. Floating point belongs to Satan".

160 posted on 01/12/2019 1:20:06 PM PST by GingisK
[ Post Reply | Private Reply | To 1 | View Replies]


Navigation: use the links below to view more comments.
first previous 1-20 ... 121-140141-160161-180181-183 next last

Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.

Free Republic
Browse · Search
General/Chat
Topics · Post Article

FreeRepublic, LLC, PO BOX 9771, FRESNO, CA 93794
FreeRepublic.com is powered by software copyright 2000-2008 John Robinson