How many kinds of infinity are there ?

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How many kinds of infinity are there ?

Perfect for a little Friday night (or weekend) contemplation...

For me this hints at the a problem with our concept of mathematics, or our perceptions of reality, and probably both.
And, as for the paradox of zero, that has always been a skeleton in the closet, I would love to see that one solved.


Duration 14:55


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  • octatonicoctatonic Frets: 33725
    Too many to count.
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  • From what I remember and its stretching a bit, there are an infinite number of aleph-numbers which are all well ordered so I think there is aleph^0 flavoured infinitely many "types" of infinity.
    ဈǝᴉʇsɐoʇǝsǝǝɥɔဪቌ
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  • monquixotemonquixote Frets: 17485
    tFB Trader
    There are infinite kinds of infinity. 
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  • There are infinite kinds of infinity. 
    Well that's a given...but what *kind* of infinite kinds of infinity?
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  • monquixotemonquixote Frets: 17485
    tFB Trader
    There are infinite kinds of infinity. 
    Well that's a given...but what *kind* of infinite kinds of infinity?
    I think the mathematical definition of a kind of infinity is that if you can't pair up all the terms via a mapping then it's a different type of infinity. 

    So the infinity of integers is (somewhat counter intuitively )  the same as the infinity of even numbers because you can pair 1-2 2-4 etc. However the infinity of infinite decimal expansions is not the same as the infinity of integers as there is no way the two can have a mapping.
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  • Right...aleph^0
    ဈǝᴉʇsɐoʇǝsǝǝɥɔဪቌ
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  • There are infinite kinds of infinity. 
    Well that's a given...but what *kind* of infinite kinds of infinity?
    I think the mathematical definition of a kind of infinity is that if you can't pair up all the terms via a mapping then it's a different type of infinity. 

    So the infinity of integers is (somewhat counter intuitively )  the same as the infinity of even numbers because you can pair 1-2 2-4 etc. However the infinity of infinite decimal expansions is not the same as the infinity of integers as there is no way the two can have a mapping.
    From what I remember the proof is essentially a proof by contradiction based on the fact that if you have any given 2 decimal expansions you can always find a decimal expansion in between them.
    ဈǝᴉʇsɐoʇǝsǝǝɥɔဪቌ
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  • SporkySporky Frets: 27569
    Six.

    Hope that helps. :)
    "[Sporky] brings a certain vibe and dignity to the forum."
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  • ICBMICBM Frets: 71952
    42

    "Take these three items, some WD-40, a vise grip, and a roll of duct tape. Any man worth his salt can fix almost any problem with this stuff alone." - Walt Kowalski

    "Just because I don't care, doesn't mean I don't understand." - Homer Simpson

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  • martmart Frets: 5205
    PolarityMan said:
    ....
    From what I remember the proof is essentially a proof by contradiction based on the fact that if you have any given 2 decimal expansions you can always find a decimal expansion in between them.
    It's a bit more complicated than that, since the rational numbers have the property that between any two of them you can find another and, yet, the rationals are countable.

    The argument for decimal expansions is that however you try to count them, you will miss some. This is dine by a clever construction to build a decimal that differs from every single one that you have counted. The really tricky bit is avoiding the problem that 1=0.999999...
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  • richardhomerrichardhomer Frets: 24793
    Loads - at least....
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  • mart said:
    PolarityMan said:
    ....
    From what I remember the proof is essentially a proof by contradiction based on the fact that if you have any given 2 decimal expansions you can always find a decimal expansion in between them.
    It's a bit more complicated than that, since the rational numbers have the property that between any two of them you can find another and, yet, the rationals are countable.

    The argument for decimal expansions is that however you try to count them, you will miss some. This is dine by a clever construction to build a decimal that differs from every single one that you have counted. The really tricky bit is avoiding the problem that 1=0.999999...
    Well it has been I think 14 years since I did this stuff.....I did enjoy this kind of maths though...unlike bastard analysis.
    ဈǝᴉʇsɐoʇǝsǝǝɥɔဪቌ
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  • HootsmonHootsmon Frets: 15924
    Image result for benny hill funny face
    tae be or not tae be
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  • scrumhalfscrumhalf Frets: 11262
    Well it has been I think 14 years since I did this stuff.....I did enjoy this kind of maths though...unlike bastard analysis.
    Is that something to do with Jeremy Kyle?
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  • speshul91speshul91 Frets: 1397
    I can see this thread going on forever. 
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  • mart said:
    PolarityMan said:
    ....
    From what I remember the proof is essentially a proof by contradiction based on the fact that if you have any given 2 decimal expansions you can always find a decimal expansion in between them.
    It's a bit more complicated than that, since the rational numbers have the property that between any two of them you can find another and, yet, the rationals are countable.

    The argument for decimal expansions is that however you try to count them, you will miss some. This is dine by a clever construction to build a decimal that differs from every single one that you have counted. The really tricky bit is avoiding the problem that 1=0.999999...
    yes, this is true. 

    other kinds of infinite forms are more interesting to me, rather than lists.

    1 + 1 + 1 + 1 + 1 + 1 + ... = infinity

    this is clear right? just like

    2 + 2 + 2 + 2 + 2 + 2 + ... = infinity

    etc. however, not all infinite sums are infinity. here are some sums which do not equal infinity

    1 + 1/10 + 1/100 + 1/1000 + 1/10000 + ... = 1.1111...

    1 + 1/2 + 1/4 + 1/8 + ... = 2

    1 + 1/2^2 + 1/3^2 + 1/4^2 + ... = pi^2 / 6

    now here are some infinite sums which look like they should not equal infinity, but they actually do

    1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ... = infinity

    WWHHAT? thats right boi. you have no clue how slowly this thing diverges.

    1 + 1/2 + 1/3 + 1/5 + 1/7 + ... (primes) = infinity

    I could go on forever 
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  • DesVegasDesVegas Frets: 4510
    One.... And it is rather large
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  • WolfetoneWolfetone Frets: 1479
    edited January 2017

    1 + 1/2 + 1/4 + 1/8 + ... = 2
    Surely that does not equal 2 as following the progression you will never be able to reach 2 as the number will become infinitely close to 2 but never reach it. 
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  • bodhibodhi Frets: 1334
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  • martmart Frets: 5205
    mart said:
    PolarityMan said:
    ....
    From what I remember the proof is essentially a proof by contradiction based on the fact that if you have any given 2 decimal expansions you can always find a decimal expansion in between them.
    It's a bit more complicated than that, since the rational numbers have the property that between any two of them you can find another and, yet, the rationals are countable.

    The argument for decimal expansions is that however you try to count them, you will miss some. This is dine by a clever construction to build a decimal that differs from every single one that you have counted. The really tricky bit is avoiding the problem that 1=0.999999...
    Well it has been I think 14 years since I did this stuff.....I did enjoy this kind of maths though...unlike bastard analysis.
    That's fair enough. It's fresh in my mind because I get to teach it every year or so. And it's probably the one topic I love teaching the most - from kindergarten ideas of counting to absolutely mind-blowing stuff in about two lectures. :)

    Sadly I don't know the answer to your question of how many infinities there are, but the answer has to depend on what axioms of set theory you take, given the resolution of the continuum hypothesis.
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