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 ...
They need all the numbers they can find to state our national indebtedness.
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.
"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.
What you did there. We saw it.
"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.
And is the rounded number from 41.7
WOW! This sounds even to big for even AOC to solve!
Dr. Edward de Bono invented the word PO as intermediate between YES and NO
I think in Japanese its HA
Dr. Edward de Bono invented the word PO as intermediate between YES and NO
I think in Japanese its HA
Ping
Bump for later
I wrote out an infinite string of numbers once but I showed it to my wife and she added one and messed it up.
Not to mention the imaginary ones.
You are a very lucky man to have such balance in your life, and to help with spelling too!
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.
The answer could be anything.
More research money is needed.
I think I understand the distinction. It seems a little bit like semantic hair-splitting, but I’ll concede the point.
Stephen Hawking said, "God created the integers. Floating point belongs to Satan".
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