Infinity

Discussion in 'Physics & Math' started by wegs, May 30, 2019.

  1. wegs Matter & Pixie Dust Valued Senior Member

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    Hmm, I hadn’t thought of it that way.
     
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  3. sculptor Valued Senior Member

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    infinitesimal

    outside of maths (.00000000000000000000000000000000000000000000000000000... etc...)
    can anything be infinitesimally small?

    after an infinity of .0000000000000000000000000000000000000000000...etc... we have a what? a 1?
     
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  5. Seattle Valued Senior Member

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    Infinity is usually where math breaks down isn't it?
     
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  7. DaveC426913 Valued Senior Member

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    You are probably thinking that a point has to have a minimum length, and therefore an infinite number of them will have an infinite length.

    Look at it the other way around. Start with the yardstick of finite length. You can easily place a point at the 18" halfway mark, and then another at each 9" quarter length. Keep adding points at the midpoint of other point pairs. Do you ever reach an end? No.

    No matter how close any two points are, you can always add another point between them. Notice that - despite adding an unlimited number of points - not one of them need be beyond the ends of the yardstick.
     
  8. Michael 345 New year. PRESENT is 70 years old Valued Senior Member

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    Does not Planck length prevent that?

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  9. RainbowSingularity Valued Senior Member

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    that is a mathematical point which requires an ability to be quantified as a known value of measure ?
    a theoretical point of mathematics ... ?
    does the "x" used to show an unknown value equal a "point of no measurable distance" ?
    or does the "x" have to have an end-point calculatable value before it can be labelled with the 'x' ?

    my impression was that mathematics had/s a symbol used to define a value of something that is unknown ?

    is that wrong ?
     
  10. wegs Matter & Pixie Dust Valued Senior Member

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    This is where I get confused. How can something finite, like a yardstick, have infinite points? Infinite meaning endless?
     
  11. wegs Matter & Pixie Dust Valued Senior Member

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    dupe...
     
  12. Michael 345 New year. PRESENT is 70 years old Valued Senior Member

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    I am with you

    I think Planck Length would prevent infinite points

    The Planck length is 1.6 x 10-35metres. (That's 0.000000000000000000000000000000000016 meters.) To give you an idea, let's compare it with the size of an atom, which is already about 100,000 times smaller than anything you can see with your unaided eye (an atom size is about 0.0000000001 meters)

    Google Planck Length

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  13. Neddy Bate Valued Senior Member

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    Imagine a square which is one plank length on each side. What is the diagonal distance across the far corners? It is not a whole number of plank lengths. Therefore, I win the internet.
     
  14. Michael 345 New year. PRESENT is 70 years old Valued Senior Member

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    Before you jump for joy and take your prize

    show us your working out calculations

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  15. DaveC426913 Valued Senior Member

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    Points are not physical things; they are geometrical abstractions.
    They are not limited by physical constraints.

    Maybe I shouldn't have mentioned the yardstick itself.
    Simply start with two points one yard apart.

    As a note: the Mandelbrot set - a mathematical object - has infinite regression, as far as we know. There is no reason why you could not zoom in without limit - well beyond the Planck length.
     
  16. Neddy Bate Valued Senior Member

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    Please correct me if I am wrong, but, the diagonal distance across the far corners of a square of 1 x 1 (planck length) square would surely be √(1² + 1²) = 1.414... planck lengths which is not a whole number of planck lengths, correct?

    https://en.wikipedia.org/wiki/Square_root_of_2
     
  17. DaveC426913 Valued Senior Member

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    Sure. Physically, it might be a problem. But points are not constrained by the Planck length.
     
  18. Neddy Bate Valued Senior Member

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    Yes, points on my square would not be constrained by the Planck length, (the sides are constrained to it by the 1x1 definition, but not the diagonal). But what do you mean it might be a problem physically? Does the concept of Planck length contradict Euclidean geometry in some physical way?
     
    Last edited: Jun 2, 2019
  19. Write4U Valued Senior Member

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    According to Renate Loll et al, the universe unfolds fractally

    https://en.wikipedia.org/wiki/Causal_dynamical_triangulation
     
  20. AlphaNumeric Fully ionized Registered Senior Member

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    The use/nature of infinity, within mathematics, requires careful definition. Even something like "Is infinity a number?" is open to multiple answers.

    First, how are we defining a number? Most of us are actually thinking about Real Numbers : https://en.wikipedia.org/wiki/Real_number . This set of numbers has specific rules/properties selected so they are a consistent framework to work within. If a and b are Real then so is a+b. If a is Real then -a is also a Real and a + (-a) = 0. If a is a Real and not 0 then 1/a is also a Real and a*(1/a) = 0. Notice that 0 is explicitly excluded from having an inverse.

    Mathematicians define sets/collections/systems to have specific applications/uses. It's illuminating to see how we arrive at The Reals :

    The notion of 'whole numbers' is something quite natural (they are actually called The Natural Numbers). 3 sheep. 5 fingers. 2 children. So mathematics started by considering problems like "If I have 3 sheep and I buy 4 more sheep how many sheep have I now got?". So to make this mathematically meaningful we'd want 3, 4 and whatever 3+4 is to be 'numbers'. This gives rise to the set N = {1,2,3,4,.....} . If a and b are in N then so is a+b. But what if a sheep does? What number do I had to 3 sheep to make 2 sheep? No element of N solves that problem. Aha! I'll DEFINE some more numbers to solve this problem. I define -1 to be the number which solves 3 + x = 2. Other negatives are definable likewise. Now I have the sets {1,2,3,4,...} and {-1,-2,-3,-4,....}. But what if I have 1 sheep and it dies? How many sheep have I now? I'll DEFINE another number to solve that, let x=0 be the thing which solves a + (-a) = x for any a in {1,2,3,...}. So now I have a set, called The Integers (labelled by Z in mathematics) where I can solve any x+a = b for any a,b in Z. Hurray!

    But what if I have 10 sheep, I die and I want to share them between my 2 children? What x solves 2x = 10? I know what 2x means, it is x+x but I can't solve this by adding -x or -2x or +3x etc, even though I know what 2 and 10 mean. Maybe if I'm clever and have the time I can work out, by trial and error, that x=5 solves this, each kid gets 5 sheep. Okay, not 'nice' but workable. But suppose I have only 1 sheep. Now I need to solve 2*x = 1. No amount of trial and error leads to a solution. Once again I'll DEFINE x = (1/2) to solve the problem 2*x =1. Now I can solve problems like 2x = y because I know that 2x=1 is solved by x = (1/2) and so 2x=y is solved by y*(1/2), which I'll write as y/2.

    So far, so good. Each time I hit a problem I define a new object to solve the problem in a unique manner. But what happens when I consider the general problem of a*x=b for a=0? After all a is in Z so I'd want a*x=b to be solvable for ALL pairs of (a,b) in Z.

    Well given I only just defined 0 I should probable work out it's properties (ie its axiomatic structure). After all, it isn't enough to just define a solution, you have to check the solution doesn't break previously essential rules. So 0 can be written as a+(-a) = 0 for any a in {1,2,3,...}. So what is 0*b? Well replace it : 0*b = (a+(-a))*b. Now I'll use another rule which is important that I want to ALWAYS be true : (x+y)*z = x*z + y*z. This is essential for me working out how to divide up my sheep, so it cannot be violated now. 0*b = (a+(-a))*b = a*b + (-a)*b = a*b - (a*b) . But now I'm adding c = ab to it's additive inverse, so I must get 0. Therefore 0*b = 0 for ALL b in Z. Right, so if I wanted to solve 0*x = y then this simplifies to 0 = y. But what if I want to solve 0*x = 5? I'm stuck, I can't solve it! But why don't I just DEFINE something, called (1/0) (or 'Infinity') to solve this, just like before? Okay, let's define it : 0*(1/0) = 1. Solved right? Using similar thinking to before 1/0 solves 0*x = 1 and 5/0 solves 0*x = 5 etc. Okay, so 0*(1/0) = 1. Multiply both sides by 5 : 5*0*(1/0) = 5. But we know 5*0 = 0 so 0*(1/0) = 5. So 1/0 solves 0*x = 1 AND it solves 0*x = 5. And every other possible multiple. So 0*x = y has no UNIQUE solution even if we define a new number into existence. Even if we set y=0 we can still have 0*1 = 0 and 0*5 = 0, no unique solution.

    This is by no means 'rigorous' but it is the general principle by which you cannot divide by zero or use infinity when working with Real numbers. It annoys the hell out of me when laypeople think mathematicians haven't thought about Infinity. Actually, we have. A LOT. And there's EXTREMELY important reasons it isn't in The Reals.

    It is in other things. The extended Reals specifically include 'Infinity' as a single extra element but you have to give up some nice rules like a*x = b always having a unique solution. The extended Complex Plane, aka the Riemann Sphere, uses it. Complex analysis is built around the careful control of singularities, where functions can 'go infinite'.

    The hyperreals go even further, defining an infinite hierarchy of infinite and infinitestimals, where an infinitesimal x is non-zero but is smaller than ALL Reals y. In the usual Reals there is no such number. Again, a property someone found of use to have and important enough to violate some of the nicer structures of The Reals (many mathematical proofs rely on there being no non-zero smallest positive number).

    Questions like "How big is infinity" or "Can I add 1 to it" arise because human language is VERY imprecise and vague, especially for such carefully defined concepts. "Is Infinity a number?" presumes we all have agreed on the definition of 'number' and 'infinity'. When a mathematician says "The infinite cardinality of The Reals is strictly greater than the cardinality of the Integers" they are able to give give laser sharp definitions of those terms so it can be used in a meaningful manner.

    TL

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    R A flippant summary of the above is - if you have to ask the question then you aren't familiar with enough relevant, precisely defined mathematical terminology to know what you're actually referring to.
     
  21. Willem Banned Banned

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    They are actually both of cardinality: aleph-zero.
     
  22. wegs Matter & Pixie Dust Valued Senior Member

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    I think where I'm going wrong, is conflating math theory with the reality of physics. Trying to conceptualize an infinite universe using math, thus I'm misusing the term, ''infinity.'' I might be causing my own confusion.

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  23. kx000 Valued Senior Member

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    Infinity is necessarily the largest integer, just uncountable.
     

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