81–90 of 275 posts
I like the idea of encoding answers to all questions, or for that matter all books written so far (or both, while we're at it), in one real number between 0 and 1. My favourite number, really.
Unrelated, but I read a article a while back which said something similar, but it was based on the fact that our entire mathematical system is designed around "base 10" and as such is only relevant in many constructs to our specific human 'ten fingered', interpretation of math.
Ironically the publisher of that article used a variety of radices to make the claim that 10 is the only radix! 2 - to enter and transmit data on the machine 10 - radix in question 16 - color description on the page etc.
Note that the probability of randomly picking a rational number from [0,1] is exactly 0. That is, with P=1 you will end up with an irrational number, not representable in any modern computer.
The event set is non-empty (i.e. there are rational numbers between 0 and 1) but the probability of picking one at random is zero.
The continuity of real numbers provides a clean theoretical basis for continuity of functions. I personally view it as more of a theoretical tool that seems to do pretty well and instead steer clear of the philosophical questions. One thing that I think is important to note though is that the jump from real numbers to complex numbers is nothing compared to the jump from integers/rational numbers/etc. to real numbers.…
Earlier quoted context omitted.
I'm not sure that I'd agree with you on the Standard Model. To me, it's view of the universe is pretty discrete. (That is, presuming that you're speaking of the particle physics Standard Model, not something in cosmology.)
The Standard Model has discrete energy levels but continuous spacetime.
> "the halting probability Ω, which is irreducibly complex (algorithmically random), maximally unknowable, and dramatically illustrates the limits of reason" I really enjoyed the beauty of this statement.
http://www.mathrix.org/liquid/archives/the-history-of-the-ch...
"God has chosen that which is the most simple in hypotheses and the most rich in phenomena. But when a rule is extremely complex, that which conforms to it passes for random."
"Everything can be summarized in one thing, but the thing itself cannot be reached."
"Mathematical facts are true by chance."
"To make all things from nothing, unity suffices."
Earlier quoted context omitted.
Consider the set of all real numbers which you can actually specify IN ANY FASHION AT ALL, so that the person writing a paper about it and the person reading the paper are talking about the same number. This includes easy ones like "3" or "Square root of Pi". It includes oddballs like Chaitin's constant -- the probability that a randomly constructed program will not get stuck in an infinite loop (for some particular…
> Consider that great big set. It's still countable. You didn't prove that statement, and actually Cantor's diagonal arguments [1] proves you wrong: Consider the set of «all real numbers which you can actually specify IN ANY FASHION AT ALL». If you can count it, you can order them in a certain fashion: n0, n1, n2, n3 … It's easy to specify a number X which is not part of this set (which is in contradiction with the d…
How do you guarantee that you can compute the n-th decimal of the n-th number in the list? In fact, from what I understand, this paper[1] shows how to specify exactly a number can't be computed like that.
Earlier quoted context omitted.
Even though you defined the number, you can't really do usual stuff with it: e.g. you can't compare it to other numbers. IMO, these kind of numbers are no more "real" than infinitesimals: https://en.wikipedia.org/wiki/Hyperreal_number
Oh it's definitely more real, as in it belongs to R and not to any of its extensions. You guys realize that the definition of R is non-controversial in modern math, right? There are fringe theories like constructivist logic and other groups that reject all infinite constructions, but this is not the consensus view among practicing mathematicians... The way you defined that number makes it a perfectly valid element of…
In constructive mathematics, there is a perfectly well-defined set of real numbers. The usual diagonalization proof that this set is not a countable set applies.
"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…
Think about a real number with digits: 0.a_1 a_2 a_3 ... in which a_i encodes whether Turing Machine with index i (or program i) halts or not. Since Halting problem is not computable, we will never be able to compute the digits of this real number. Never in our universe.
However, you might be interested in the effective topos. This is an alternate mathematical universe in which one can't define construct this real number. It has a number of curious properties:
* The statement "1 + 1 = 2" is true in that topos, just like it is in the ordinary topos.
* The statement "any number is either prime or not prime" is true in that topos, just like it is in the ordinary topos, however its truth is not trivial.
* The statement "any real number is either equal to zero or not equal to zero" is false in that topos.
* The statement "any function is computable" is true in that topos (and wildly false in the standard topos).
* The statement "any function is continuous" is true in that topos (and wildly false in the standard topos).
Details are in this set of slides:
https://rawgit.com/iblech/mathezirkel-kurs/master/superturin...
Questions are welcome!
The continuity of real numbers provides a clean theoretical basis for continuity of functions. I personally view it as more of a theoretical tool that seems to do pretty well and instead steer clear of the philosophical questions. One thing that I think is important to note though is that the jump from real numbers to complex numbers is nothing compared to the jump from integers/rational numbers/etc. to real numbers.…
Rationals are conintuous, but not a continuum. you don't need real numbers to have continuous functions. http://math.stackexchange.com/a/672151
No. In fact, your claim is contradicted by your own link. As stated by Asaf Karagila in the comments:
> Continuity is a property of functions. You seem to ask why the rational numbers are not connected (or path connected).
But you are correct that you don't need real numbers to describe continuous functions. As you link points out, one is a property of spaces (or domains or whatever you want to call it) and the other is a property of function.
However, talking about continuous functions between complete spaces (basically "complete" is what makes the real numbers "real") is extremely natural and basically goes hand in hand with continuous functions. It really ties together a lot of the theory if you're talking about metric continuity.
Regardless, you also don't _lose_ anything by talking about real numbers. You can of course use it as a tool to develop a theory and then choose to apply the results to the rational numbers (or algebraic, etc.).