adam ∞ wrote:If an apeirogon has mirror symmetry from top to bottom like a circle, it should have a vertex at the top and at the bottom. If there was a vertex at the top, and an edge at the bottom, this would be asymmetrical.
Thus, infinity is even.
Now that we know infinity is even, we can start to investigate what the digits of infinity must be:
All even numbers have to end in 0,2,4,6, or 8. So if there are digits of infinity, infinity should end in 0,2,4,6, or 8.
Since an apeirogon should have the symmetry of a circle, we should expect to find mirror symmetry not just from top to bottom, but also from left to right. This tells us that infinity should be a multiple of 4. We should expect to find vertices in between, as well, which gives us a multiple of 8, and in between those, which gives us a multiple of 16, and so on, like the series 2x2x2x2... and so on. With sides of polygons, this will eventually terminate when the edges meet together at 180°.
From this we are starting to get the picture that infinity must be a power of 2.
mr_e_man wrote:adam ∞ wrote:If an apeirogon has mirror symmetry from top to bottom like a circle, it should have a vertex at the top and at the bottom. If there was a vertex at the top, and an edge at the bottom, this would be asymmetrical.
Thus, infinity is even.
Now that we know infinity is even, we can start to investigate what the digits of infinity must be:
All even numbers have to end in 0,2,4,6, or 8. So if there are digits of infinity, infinity should end in 0,2,4,6, or 8.
Since an apeirogon should have the symmetry of a circle, we should expect to find mirror symmetry not just from top to bottom, but also from left to right. This tells us that infinity should be a multiple of 4. We should expect to find vertices in between, as well, which gives us a multiple of 8, and in between those, which gives us a multiple of 16, and so on, like the series 2x2x2x2... and so on. With sides of polygons, this will eventually terminate when the edges meet together at 180°.
From this we are starting to get the picture that infinity must be a power of 2.
But a circle also has mirror symmetry over a line at 60°; it has the symmetry of a triangle. Therefore infinity must be a multiple of 3.
And a circle also has the symmetry of a pentagon, so infinity must be a multiple of 5. And so on.
quickfur wrote: Which means that ...7437428736 cannot be the last digits of infinity, because none of the numbers in the sequence 2, 2^2, 2^2^2, ... are divisible by 3, 5, 7, 11, 13, ... etc..
quickfur wrote: Which means that ...7437428736 cannot be the last digits of infinity
adam ∞ wrote:quickfur wrote: Which means that ...7437428736 cannot be the last digits of infinity, because none of the numbers in the sequence 2, 2^2, 2^2^2, ... are divisible by 3, 5, 7, 11, 13, ... etc..
Assuming that infinity must be divisible by 3, 5, 7, 11, and 13, etc, is a false assumption. It is not possible for a number to have all properties of all numbers all at the same time, including contradictory ones.
If infinity was a multiple of 11, it would be far less symmetrical than what I have defined.
If you are claiming that infinity really is just all consecutive numbers up to some point multiplied together, which I don't think you actually believe by the way, it would result in an infinity sided polygon that lacks a lot of the symmetries attributed to circles. They can't both be right at the same time.
Also, this would contradict your earlier claim about not being able to reach infinity by modifying finite numbers.
Just because it is infinity doesn't mean that the law of non-contradiction goes out the window.
Even so, there is something known as a "benchmark" that is sometimes used for extending calculations to infinity to a well defined infinite sequence. If your "benchmark" for infinity is 1x2x3x4x5x6x7x8x9x10x11... up to some number (which one?), and you continued with trying to define digits and prime factorizations, you could probably construct a mostly coherent theory based on that, where, for example, infinity -1 would end in ...99999, infinity plus 1 might be prime, etc. Essentially you would be saying, if infinity is this number, then these properties should follow. Without an agreed upon benchmark, that might be as close to a consensus as people are going to get on something like this.
I have explored infinite numbers of this type before, but it is difficult to find patterns for something so irregular.
I don't think factorial numbers are anywhere near as regular as powers of 2, so there is a lot of stuff that you probably wouldn't be able to figure out that would be calculable for powers of two.
[...]
Anyways, I have been working on this for decades. It's easy to shoot things down with one sentence and move on. It's a lot harder to actually construct a consistent theory like this.
we could, for example, replace all the 3's in the construction of Graham's number with 2's instead, and we'd end up with a candidate for G that has many (perhaps even all!) of the properties your proposed "infinity" does.
adam ∞ wrote:[...]
It is provable, for example, that if infinity is not a power of 2, that an infinitely long line broken up into segments as edges cannot be split in half, then into quarters, then into eighths, etc, without splitting some of the edges themselves.
Maintaining this property of being able to split things up evenly like this without splitting edges, for example, is far more crucial than maintaining 11-fold and 13-fold symmetry, which you mistakenly think will make it more symmetric. I think you are not appreciating why it is that you cannot have all of these types of symmetries all at the same time without losing other ones.
Give it some honest consideration, can you not see why the apeirogon cannot have 3-fold symmetry without seriously limiting other types of symmetry? Do you not see why some symmetries involved in this necessarily have precedence over others, [...]
It's interesting, that one of your objections is that you don't think regular polygons converge to a circle.
[...]
I am interested to hear your interpretation of what the sequence of regular polygons does or does not tend to or converge to. Do you think it is something that is in any way related to a circle?
Given the conventional interpretation of the series of regular polygons converging to a circle, there is a deep connection between circles and infinite lines demonstrated by the apeirogon, and there are far reaching consequences of this!
quickfur wrote:But, if you would indulge me, a little footnote: if you construct a n-polygon where n = G, a very large finite number like I describe in my other post, then it could indeed have the properties that you ascribe to the apeirogon. At least, it would have properties sufficiently close to an apeirogon that, for the most part, we could assume without contradiction that it's an apeirogon. Consider it. It's one way of constructing a fully-consistent, contradiction-free system that has virtually all the properties you desire in your constructions.
Finite numbers, as defined above, include SO MUCH MORE than most people's intuition of "infinity" that it's almost laughable to read about supposed "infinities" that can be reached by elementary arithmetic operations. Such "infinities" are so small they're obviously finite, and rather low down among the smaller finite numbers at that, that calling them "infinity" frankly sounds like a joke.
adam ∞ wrote:If you conceptual followed the blue path, to the right, starting at the black square-
Do you just think of it as always staying finite?
Suppose there is a black square infinitely far away.
Do you think of it in such a way that, if you followed the blue path toward the black square, you could in some sense never get there (even conceptually)? That you could only get there by being magically transported, skipping the space in between? I don't get it.
adam ∞ wrote:Like, there is notion that you can (conceptually) "count" to infinity in a short period of time, if you just speed up each time you "say" a number, like a converging infinite series 1 + 1/2 + 1/4 + 1/8... etc.
How do you make sense of something like that, either the counting thing, or the converging series, if there is a discontinuity between the finite and the infinite?
If there is no continuity between finite and infinite, can it ever converge? In what sense does it "get" to infinity if there is not some kind of continuity or transition from the finite to the infinite?
For any positive rational number ε, there is some natural number M (generally large, generally depending on ε), such that the inequality -ε < x_{n} < ε is true for all natural numbers n > M.
mr_e_man wrote:I'll take the position that "actual infinity" or "completed infinity" doesn't exist. There's only "potential infinity". (Though, I'm not sure my ideas of these are the same as yours.)
mr_e_man wrote:Yes, it always stays finite.
There's no such thing as "infinitely far away". The Euclidean plane is potentially infinite. Given any line segment, we can attach to it another line segment, extending in the same direction, thus increasing the length of the segment; still the length is finite, but there's no upper limit on the length.
For any two black squares in the plane, there is a finite distance between them.
adam ∞ wrote:Quickfur,
Right, I believe infinity is successive, you believe it's non-successive, and therefor you think that something that is successive must be finite.
But why do you believe this, other than just accepting that as the definition, "by definition". Where does this definition come from? Why do you think infinity can only mean that one particular definition and not something else?
Obviously a huge amount of people believe that 1+1+1+1... equals infinity, by just adding ones until it gets there.
So the definition you gave is not the only one that is widely accepted.
I was trying to establish some terminology in my original post so that we could at least discuss it in a way where we make distinctions between these concepts without having to accept someone else view on it.
You are convinced that a nonsuccessive infinity has to be the definition of infinity. I see it as being meaningless, contradictory, and a major problem that needs to be resolved in any theory of infinity. It's like a broken part of the theory.
It's interesting that you don't see a discontinuous gap as a major foundational problem.
I'm talking about infinity in the sense mathematicians define it
Any number X for which a regular polygon with X vertices has a 180°angle between adjacent edges must have at least one infinitely-long "gap" that cannot be bridged by the successor function.
steelpillow wrote:It is generally held that un-countables are larger than countables, but I am not sure if that can be proved without assuming something like it in the first place, rendering the proof circular.
It does not convince me. Cantor assumes that you can reach the end of the infinite series and declare something is missing. But you can never reach the end of an infinite series, so you can never make such a declaration. This kind of fallacy, that by considering finite steps you can prove something about infinity, also trips up the Leibniz/Newton "proofs" of the mechanics of infinitesimals (the differential and integral calculi): just because a sequence of finite approximations appears to converge on a particular result does not prove that, at the limit where the difference between terms reaches zero, the apparent result is valid. I recall someone claiming ca.1970 that they had fixed this flaw, only to have that claim rebutted shortly afterwards. There has since been another fix offered, which is generally accepted. Frankly, it has never made any better sense to me than the last fix. But what do I know.PatrickPowers wrote:Cantor's proof is both convincing and very simple.
I used to think there were no infinities in nature until it became popular to suppose our Universe is infinite. Maybe it is, maybe it isn't, but it doesn't make much practical difference.
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