Quantum mechanics is a rich source of mathematics, but this has nothing to do with it. Simply put, the property of a set having the same number of elements as a proper subset of itself (call this Dedekind-infinity) characterizes all infinite sets in ZFC set theory. (Note you definitely need some sort of formal set theory even to compare how many elements various sets contain.) This all dates back to around 1888.
Choose any of the left over points on the larger circle and draw a line back to the circles’ center. That line intersects with a unique point on the smaller circle. So, no points can get left over on the larger circle. (I must ask, though, if this argument would apply in all non-Euclidean geometries).
You’re not mapping a line segment, but individual points, as in Karl Gauss’ circle. The methods vary slightly, but the mapping can be done to any number of dimensions. (I believe even to an infinite number of dimensions, but I may be getting in over my head.)
EastUmpqua, mathematical points have no dimension. They are not physical objects. Physics has to deal with the real; math can be abstract and ideal. No physical limitations can be placed on mathematical objects. That’s how there can be perfect circles, infinite lines, and all the non-physical structures of advanced math.
I am pretty sure under ZFC this is consider impossible to prove or disprove without the Axiom of Choice.
But this is a bit different than the traditional form of Hilbert’s problem #1
DOH!!! I forgot the C in ZFC meant the Axiom of Choice was included…
Please ignore.
Does this circle ‘argument’ hold in non-Euclidenan geometry(ies)?
I ask because it seems to me that the ‘proof’ depends on the assumption that radial lines never intersect.
What about the other way around. Any two lines from the center passing through any two contiguous points on the smaller circle would create and angle passing through two points on the larger circle. How many points are in that arc length?
Q1: Are there serious physics models that are not continuous, i.e. discrete?
Q2 (not directed at anyone): There could be a mathematical system with eg 1/a google[1] being the smallest unit, right? It wouldn’t include most real numbers between 0 and 1, but you could toss out paradoxes associated with infinity, right?
ETA
[1] or 1.8 x 10^308, which is the highest double precision value, if you want a built-in application for this quixotic system.
The arc lengths are the same; their number of points is an equal infinity (unlike that of some infinities which are ‘larger’ or ‘smaller’ than others).
EastUmpqua - How about this:
take any infinite line segment.
place a zero on some point on it.
then place a 2 where you think it should go.
Oops, you really meant to place a 1 there. Rub out the 2 and re-label the point as 1
As we currently lack a believable theory for quantum gravity, no.
Superstring theory and loop quantum gravity may need either minimal length or discretization of space-time but it depends on who you talk to on the nature and obviously neither of those theoretical constructs are merely thought experiments at this stage.
Quantum chromodynamics, being based on SR works with curved time so it is the gravity question that is driving this. But to be honest even if the gravitation existed and the standard model was correct designing an experiment to test it is a problem just due to how weak gravity is.
Superstring theory typically adds strings that have an average diameter of the Planck length but this still results in quantized time (and thus space).
People just don’t like numerical solutions like Lattice QCD, but they will gratefully consume the products of it. Dr. Frank Wilczek would have had his ideas end in a m Landau’s catastrophe and not a Nobel prize without it though.
This is a pretty iconic animation from his prize lecture and is a product of that method.
http://www.physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/Nobel/
String theory is analytical and “pretty” and somehow has a good publicist seems to one of the big reasons why the perception that space and time are quantized seems so common, despite a complete lack of evidence to show it is (or isn’t).
No, it doesn’t. All it hypothesizes (not “demonstrates”) is that certain items have a certain amount which happens to be supertiny but not equal to zero. Items whose amount is supertiny but not equal to zero would include, among others, the mass of a single photon or the charge of an electron. Note that the charge of a neutron is still equal to zero.
Various lattice models, for example. Also examples in non-commutative geometry.
Real computers can and do only work with some sort of floating-point numbers, not actual real numbers.
FYI, that page needs some serious attention.
The lattice gauge link is typically exponentiated as a finite jump which creates a phase factor. As exponentiation typically ends up being SU(3) matrices which allows for gauge invariance. Plus it gets rid of the hairy ball theorem.
It is still a numerical solution and can have problems but that is exactly why it lets you avoid the infinitesimals. The Lagrangian is invariant under transformations.
While there are discrete gauge groups, but neither space nor time are treated as discrete.
While experimentally we only have lower limits, the mass of a photon is generally considered to be zero even in quantum theory.
Quantum mechanics doesn’t generically say that there is a lower limit to every quantity. Indeed, even position, the example usually used, is continuous for a free particle (although that does get you in a bit of mathematical trouble). The Planck length’s physical significance is purely conjectural at this point.
And the charge of an electron isn’t the tiniest around; quarks can have charges of a third of that (although good luck catching a quark on its own).
I think it’s important to pin down, terminologically, what we’re talking about here. It is not true that the set of rational numbers between 0 and 1 is equal to the set between 0 and 2, nor that the set of whole numbers is equal to the set of all fractions. These statements are trivial to disprove by counterexample: the number 3/2 is in the set of rational numbers between 0 and 2, but not in the set of rational numbers between 0 and 1. And likewise, 3/2 is in the set of fractions (i.e., rational numbers), but not in the set of whole numbers. If one set contains an element that another set lacks, then the two sets cannot possibly be equal.
What we’re talking about is not the equality of sets (i.e., whether two sets have exactly the same elements) but the cardinality of sets—that is, whether the sets contain the same number of elements.
Although it was originally written in the 1950’s, George Gamow’s One Two Three…Infinity has an excellent discussion of infinities. To a mathematician, not all infinities are created equal. I recommend reading this book.
Hilbert’s Hotel is a classic way to introduce counting small infinities.
Hilbert’s Hotel has an infinity of rooms numbered {1, 2, 3, 4, 5, … } and is full up — no empty rooms at all. A bus pulls up, and the driver announces. “I got all the rational numbers aboard. Can you put us up?”
“No problem,” says the Night Manager; he’s been here before. He gets on the P/A system and announces “All guests, pack your bags. If you’re in room N, move to room 2N. You’ll find a pocket calculator next to Gideon’s Bible to help with the arithmetic.”
He then turns to the busdriver. “All set. Tell your number a/b to go to room number 3[sup]a[/sup]5[sup]b[/sup]; it won’t matter whether he’s in lowest terms or not. Did you bring the negative rationals? Tell -a/b to go to room number 3[sup]a[/sup]5[sup]b[/sup]7.” The Night Manager goes back to his porn show, happy that he’s now got an infinite number of empty rooms in case another batch of guests arrive.
The closest one I know of is causal set theory, where one tries to view spacetime as a very large set of discrete events that may or may not be causally related to each other. But that one is still pretty speculative.
Do you mean the smallest nonzero number, or the smallest number altogether?
If the latter, what’s 1–1 (or any number minus itself)? If 1–1 = 1/(1.8 x 10^308), then 1 = 1 + 1/(1.8 x 10^308), and so 1 (or any other number) isn’t quite equal to itself, which, it seems to me, would lead to all sorts of paradoxes.
If the former, I don’t see how it “tosses out paradoxes associated with infinity.”