91–100 of 226 posts
Earlier quoted context omitted.
> the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment
That a bijection exists is incredibly useful. But the idea of "measuring" infinite sets in the cardinality sense is not very interesting or useful.
Earlier quoted context omitted.
A computable real number is a real number for which a Turing Machine exists that can compute it to any arbitrary precision. So yes sqrt(2) is computable. Every BB(n) is computable since every every natutal number can be computed. It's the BB function itself that is not computable in general, not the specific output of that function for a given input.
Interesting point about BB(n)... Is it known that BB(n) is finite for every n?
Earlier quoted context omitted.
You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?
As in why bother using real numbers in physics? Mostly because you need them to make the maths rigorous. You can't do rigorous calculus (i.e. real analysis) on rationals alone.
I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result - you of course get the same result if you had simply used regular integration/differentiation, and it's a lot easier to use calculus than combinatorics.
But the point is the "rational", discretized approach will get you arbitrarily close to the answer.
It's why many analysis textbooks define a (given) real number as "a sequence of converging rational numbers" (before even defining what a limit is).
Earlier quoted context omitted.
A computable real number is a real number for which a Turing Machine exists that can compute it to any arbitrary precision. So yes sqrt(2) is computable. Every BB(n) is computable since every every natutal number can be computed. It's the BB function itself that is not computable in general, not the specific output of that function for a given input.
Interesting point about BB(n)... Is it known that BB(n) is finite for every n?
Earlier quoted context omitted.
You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?
> You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). Citation needed. Especially since there are well-established math proofs of irrational numbers.
The logic is circular, simply because mathematicians are the ones who invented irrationals. Of course they have proofs on them. They also have proofs on lots of things that don't exist in this universe.
And as I pointed out elsewhere, many analysis textbooks define a real number to be "a (converging) sequence of rationals". The notion of convergence is defined before reals even enter into the picture, and a real number is merely the identifier for a given converging sequence of rationals.
Earlier quoted context omitted.
That doesn’t sound right to me. What about the machines that don’t halt? You can’t compute whether or not to skip them directly. > A busy beaver hunter who goes by Racheline has shown that the question of whether Antihydra halts is closely related to a famous unsolved problem in mathematics called the Collatz conjecture. Since then, the team has discovered many other six-rule machines with similar characteristics. Sl…
What specifically doesn't sound right?
But given the answer, I suppose you could write a program that just returns it. This seems to hinge on the definition of “computable.” It’s an integer, so that fits the definition of a computable number.
My mistake.
Can't say that I'm completely in the headspace to follow the argument, but wanted to add my 2 cents from a few years ago. Integers come into existence long before god - as the only presumption required is a difference between one thing and another (or nothing). The integers also create infinite gaps. The primes. So no - I do not think reals are closer to the divine. They require we import infinity twice to be defined…
Depends on your view of God. If God existed before creation, there were not two things to compare. I'm not even sure "nothing" existed - maybe God was smart enough to avoid creating "null" values. Caveat: former Catholic; 50+ years of fervent atheism.
I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but…
Earlier quoted context omitted.
What specifically doesn't sound right?
The claim is that every bb(n) is computable but I don’t think you can compute bb(6) without knowing which machines won’t halt. That doesn’t seem like a finite calculation? But given the answer, I suppose you could write a program that just returns it. This seems to hinge on the definition of “computable.” It’s an integer, so that fits the definition of a computable number. My mistake.
So as you noticed, it only makes sense to talk about whether a function is computable, we can't meaningfully talk of computable numbers.