Here’s one good online introduction to infinity and infinite sets.
This is an important distinction, when you try to define things like dimension, or topological equivalence. You can match the points in a line segment to the points in a square, but not in a way that preserves the relationship of those points to one another in the shape of a line.
However, a line segment and a line are both one-dimensional, so it seems to me you can talk about mapping one to the other. Tangent, as Andy L suggested, is one example; and what about something like 1/x from (0,1) to (1,infinity).
As I recall from set theory, a set, A, is equal to or smaller (in cardinality) than set B if you can match every item in A uniquely to an item in B, and a set A is equal to or larger in cardinality to B if you can match every item in B to an item in A, uniquely. If you can do both, then A has the same cardinality as B.
For A = (-pi/2,pi/2) and B = (-infinity,infinity), the function tangent provides the unique matching in one direction, and arc-tangent does the matching in the other direction, so the cardinality of A and B are the same.
My $0.02:
This thread shows the real danger of letting analogy stand as a substitute for an underlying principle.
The actual math of cardinality isn’t too much more complicated than initially discussed in this thread, yet there was already a misinterpretation from the get-go - the idea that these two sets are somehow “equal” when, instead, we are saying they share a property that is the same, their cardinality.
The actual principle is that ‘size’ is not as straightforward once you have sets with infinitely many elements, yet we still want to have some meaningful notion of ‘size’. So, in some sense, these two sets have the same ‘size’ but unlike with sets with a finite number of elements, this does not mean that if one set is a proper subset of another it is ‘smaller’ (however we define ‘smaller’ for infinitely large sets) - again, sets with infinitely many elements are no longer subject to standard human intuition.
As an aside, this kind of fuzzy wording also comes up with things like “pi is infinite”. No, the number pi is not infinite. It’s actually a bit bigger than 3 and definitely less than 4. Its decimal representation is infinite, but numbers are not their decimal representations. But the line gets repeated anyway and leads to all sorts of misconceptions. Ditto division by zero or other concepts.
As a further aside, there was the question about how many points are on the line segment connecting two circles. One potential answer is an infinite number of points. Another answer that I prefer is that it’s not really a meaningful notion in that sense. While it’s true you can find infinitely many points along a line segment, a line segment isn’t actually composed of an infinite number of points. That’s not how it’s defined. A line segment is some “thing” we define as existing between two points and has other properties based on other definitions and postulates. In math, definitions are important. Once you start getting fuzzy with definitions or their usage, it’s very easy to get into misconceptions, as in this thread.
In eg a lattice field theory, the quantum fields are defined on points of a discrete lattice. We do want the lattice spacing to get finer and finer in a continuum limit; is that what you mean?
If you don’t like that example, perhaps something like a quantum gravity where space or spacetime is a spin foam. Better yet, how about something like the Ising model? If that does not count as a serious physics model that is discrete, then what does?
Can anyone comment on mapping 1D to infinite dimensions? Mapping 1D to nD where n is finite is pretty easy. You just take the 1st through n digits of the 1D number as the first digit in the n dimensional space. n+1 through 2n as the second digit, etc.
That falls apart for infinite dimensional spaces. If it is possible, what is the mapping?
I guess the concept I was trying to convey, is if the theory believes that the actual physical world is made of discreet chunks of space time. Lattice field theory uses spacing to reduce the number of computer calculations to make calculations practical.
It probably depends if you accept the Axiom of Choice
This video will help provide a visual intuition why this results in sizeless sets.
If the number of dimensions is a countable infinity, it’s still no problem. The first digit of the number maps to the first digit of the first coordinate. The second digit to the second digit of the first coordinate, and the third to the first digit of the second coordinate. The fourth digit maps to the third digit of the first coordinate, the fifth to second digit of the second coordinate, and the sixth to the first digit of the third coordinate, and so on.
Of course, if the number of dimensions is uncountable, then it doesn’t work any more.
It may be easier to get at this more abstractly via the mappings underlying (2[sup]ℵ[sub]0[/sub][/sup])[sup]ℵ[sub]0[/sub][/sup] = 2[sup]ℵ[sub]0[/sub]ℵ[sub]0[/sub][/sup] combined with ℵ[sub]0[/sub]ℵ[sub]0[/sub] = ℵ[sub]0[/sub].
The last equation just says that there as many pairs of natural numbers as natural numbers, and the first equation is more obvious.
ETA you combine the indicated mappings to get a concrete description like Chronos’s.
When you use analogy, you make an anus out of u and me.
Thanks, this is clear to me.
ETA 2: in this vein, 2[sup]ℵ[sub]0[/sub][/sup] = ℵ[sub]0[/sub][sup]ℵ[sub]0[/sub][/sup] as well.
Whoa. Everybody skipped this, but it’s the critical factor that you’re misunderstanding.
No such thing as contiguous points exists on the real number line. The one overwhelmingly important factor to understand is that an infinity of points lie between any two stated points.
Take
0.0912385098134098-018579058790139850751048591275612457190735971 and 0.0912385098134098-018579058790139850751048591275612457190735972.
In between those are
0.0912385098134098-0185790587901398507510485912756124571907359711 and 0.0912385098134098-01857905879013985075104859127561245719073597111 and 0.0912385098134098-018579058790139850751048591275612457190735971111 and 0.0912385098134098-018579058790139850751048591275612457190735971112 and an infinity more.
You cannot find a place to stop adding numbers that lie between them. Wherever you look, no matter how many finite digits you add to the end, you haven’t even begun to scratch the surface of the endless supply of numbers.
For the same reason, there can be no fundamental limit of smallness. No matter how small your number is, no matter how many zeroes lie after the decimal point, you can always add more zeroes, or divide by a half, or divide by a googleplex on and on and on without end, and without ever reaching zero.
Infinity can’t be thought about like finite numbers. Special tools had to be developed to deal with its counterintuitveness. To be fair, most mathematicians of Cantor’s time blasted him and his set theory when he first laid it out. But that was 130 years ago. In the interim every great mind in math has worked to smooth out any bumps and spackle over any slight holes. Infinity works when mathematicians approach it properly with the proper tools. It’s getting non-mathematicians to understand that the stuff they learned in school about finite numbers doesn’t work for infinity that’s the hardest part.
This is the “infinite bisect,” one of the few things I actually remember from math.
Draw a line from 0 to 1.
Now draw a mark midway between 0 and 1 = 1/2
Bisect it = 1/4
Do it again = 1/8
And so on and so on. There will never be a fraction so small that you can’t cut it in half.
If all of this is true, then how can 0.999999999999… = 1?
ducks and runs
And don’t stop running until you reach infinity.
Hint. You’ll know it when you get there. It’s where the parallel lines meet.
And along the way he’ll pass both Achilles and the tortoise.
Mathematics has no generally accepted definition.[28][29]. Do we need some new math? If you’re talking about 1-1=0, then current thinking about infinity works. But this math doesn’t seem to translate very well to the physical world, where Infinity doesn’t exist.
Math is the language of physics and physics works extremely well.
What would your new math do to justify its existence?
More to the point, why should math change just because you choose not to understand it even when people patiently explain it to you?