“The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers and more real numbers than rational numbers. However, this intuitive analysis is flawed; it does not take proper account of the fact that all three sets are infinite.”
Doesn’t QM demonstrate there is a lower limit to existence? So shouldn’t there be twice as many rational numbers between 0 and 2 as between 0 and 1? So zero wouldn’t be infinitely small. And you could divide by it.
That is like saying there can’t be an infinite number of integers as I would run out of bricks if tried to build an infinitely large tower. The fact there is a lower limit to the physics doesn’t have any bearing the maths.
Quantum Mechanics is a set of hypotheses about reality that, along with a lot of mathematics, makes very good predictions about what will happen with very small things.
That mathematics also exists divorced from reality as well. It’s just an axiomatic system in which you can prove things given a set of hypotheses. Mathematics only describes reality when we put in hypotheses that correspond with reality. It can also describe things that are completely unrealistic and that exist potentially only in the minds of mathematicians and their work. The existence of the real numbers as a set the way mathematicians describe it does not necessarily correspond to anything in reality. But they can definitely tell you what the various properties ascribed to the real numbers mean in terms of other properties they then must have.
Math isn’t the same as physics. There is no maximum integer, even though there is a maximum number of particles in the observable universe, and likewise, there’s no smallest positive rational (or real), even if there is a smallest possible distance in the universe.
I watched a math documentary, once, that explained that the infinite set of all whole numbers was equal to the infinite set of all fractions (IIRC) by pairing members of both sets and then pointing to the right (the number line’s left) and saying “then they both go on forever”. It’s that “they went that-a-way” that does it.
Now, if you ask how some infinite sets can be larger than other infinite sets, I’m sure that question’s been discussed here before.
The straightforward answer to the question posed by the thread title is: because there is a one-to-one correspondence between them.
Let A = the set of rational numbers between 0 and 1, and B = the set of rational numbers between 0 and 2. Every x in A matches up with a unique number 2x in B. Or, looking at it from the other direction, every y in B matches up with y/2 in A.
The OP seems to be suggesting that there are some numbers y for which y/2 is “too small to exist.” But there is no “lower limit to existence” in mathematics. The axioms on which the real number system is based guarantee that y/2 must exist in a mathematical sense—which is not the same as existing within the physical universe (echoes of the recent thread Is mathematics invented or discovered!).
There is no “lower limit to existence” in QM either.
The Planck scale where where the Heisenberg uncertainty principle restricts us from measuring positions. This doesn’t necessarily mean that space or numbers are discretized and or finite.
Think of the planck scale as a ruler with only marks for full inchs; you can’t measure 5/16" even if it exists.
Why can’t zero be defined as:
“In July 2017, the NIST measured the Planck constant using its Kibble balance instrument with an uncertainty of only 13 parts per billion, obtaining a value of 6.626069934(89)×10−34 J⋅s.[32] This measurement, along with others, allowed the redefinition of SI base units.[33] The two digits inside the parentheses denote the standard uncertainty in the last two digits of the value.”
There is a whole field called “Lattice quantum field theory” where several people work on problems that force us to consider distances below the Plank length.
The common, but misinformed claim that the plank scale is where “laws of physics break” is simply an artifact of oversimplification or misunderstandings. The Planck length is just the minimum measurable length. Another way to think of it is wave function cannot cycle, what energy is there? What will you measure?
Spacetime is continuous as far as we know right now and yes this is even true in the Quantum world. Non-perturbative gauge theory does calculations all the time in continuous spacetime. If Quantum Loop, String Theory, or some form of quantum gravity every become testable proven theory; we will see if they can prove if it is continuous or not.
Don’t confuse the assumptions of *completely theoretical *Quantum gravity research topics with our current generally accepted and tested theories.
Remember string theory is a guess that can’t even make predictions to test right now. While the media likes it, you cannot take anything it says as applying to the real world.
Is this true, though? I can see mapping a line segment onto another line segment, or a plane onto another plane, but isn’t this mapping a line segment onto an infinite line? A different order of infinity?
