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My 6 and 7 yo's call infinity the "endless number". Well, at least it is a NaN number :) PS: they seem to _know_ that endless*endless > endless but do not dare

My 6 and 7 yo's call infinity the "endless number". Well, at least it is a NaN number :) PS: they seem to _know_ that endless*endless > endless but do not dare to admit it

Dylan16807 3y ago
Well that fear is good because multiplication won't change the cardinality, you have to go exponential.
- smfjaw 3y ago
  That effectively does make it exponential x * x === x^2
  - ksarw 3y ago
    by going exponential, they mean 2^x; x^2 can still be the same "size" as x
  - NobodyNada 3y ago
    x^2 is not exponential; it's quadratic. 2^x is an example of an exponential function. The parent comment was alluding to the idea of set cardinality (https://en.wikipedia.org/wiki/Cardinality https://en.wikipedia.org/wiki/Cardinality). Two sets have the same cardinality if you can establish a bijection (a one-to-one mapping) between elements of one set and elements of the other. The set of all natural numbers is said to have a "countably infinite" cardinality. It turns out that for any countably infinite set S, the set S x S is also countably infinite (see Hilbert's hotel: https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...). For example, the set of 2-tuples of natural numbers (1, 1), (1, 2), (1, 3), ..., (2, 1), (2, 2), ... is the same size as the set of natural numbers. So in this sense, "endless*endless = endless". Whereas the set of infinitely-long tuples of natural numbers is "uncountably infinite;" it has a cardinality greater than that of the set of natural numbers. Thus, "you have to go exponential"; i.e. "endless^endless".
- danwills 3y ago
  Not an expert but I thought it needed the 'power set' (set of all subsets) but maybe that's kinda the same as exponentiation in the end?
  - bkallus 3y ago
    It is. The cardinality of the power set of a set S is 2^|S|.
    - dllthomas 3y ago
      You can see it as a binary "in or out" for each element of S.
thriftwy 3y ago
Infinite is just a fancy word for endless, anyway.
- short_sells_poo 3y ago
  Not necessarily:) A circle is endless, and yet certainly isn't infinite.
  - jdkee 3y ago
    A circle is made up of an uncountably infinite set of points.
    - selcuka 3y ago
      A line segment is also made up of an infinite set of points, but it's not endless.
      - davrosthedalek 3y ago
        You can also map the all reals to the reals between 0 and 1, so the infiniteness of the extend isn't that crucial.
      - hunter2_ 3y ago
        I can just imagine someone coming up with the infinity symbol, and arguing (as we are here) about whether a circle represents the idea. No, not quite; it needs something more... another circle should suffice, and connect them seamlessly. Yes, yes. This looks much more infinite than a mere circle.
  - jameshart 3y ago
    Infinite literally means ‘without end’. Finite means ‘something that finishes’. Infinite means ‘something that does not finish’.
    - dragonwriter 3y ago
MichaelMoser123 3y ago
there is a difference between infinity and not a number, NaN isn't event equal to itself, in nodejs you get: > Infinity + Infinity Infinity > Infinity - Infinity NaN > Infinity == Infinity true > NaN == NaN false The fun thing: NaN to the power of zero (because technically the type of NaN is a ... number) > NaN ** 0 1 > typeof(NaN) 'number' By the same logic, NaN is also even because > NaN + NaN NaN but i think it's better not explain that to your six year old...
- mikepurvis 3y ago
  Huh. Most of those make sense to me, but infinity == infinity being true definitely feels like risky business. Algebraic limits is full of even some pretty trivial scenarios where infinity divided by a lesser-infinity turns out to be a real number— those cases where the two infinities are definitely not equal to each other.
  - MichaelMoser123 3y ago
    Also: in javascript you can divide by zero > 1/0 Infinity But python throws a divide by zero error: >>> 1/0 Traceback (most recent call last): File "<stdin>", line 1, in <module> ZeroDivisionError: division by zero Still you do have infinity and nan in python - because these are part of the floating point spec. >>> float('inf') - float('inf') nan >>> float('inf') == float('inf') True >>> NAN=float('inf') - float('inf') >>> NAN == NAN False However that's not mathematics, it's computers (these are even stranger...) I have my own little programming language - PYX [1] - and i don't allow this madness (even if it is a violation of the floating point spec ;-) pyx > mathconst.Infinity - mathconst.Infinity Error: results in 'not a number' - that's not allowed here #(1) mathconst.Infinity - mathconst.Infinity |....................^ > 1/0 Error: Can't divide by zero #(1) 1/0 |..^ [1] PYX - https://github.com/MoserMichael/jscriptparse https://github.com/MoserMichael/jscriptparse - it's supposed to be an educational programming language, where I am trying to have detailed error messages, my side project.
    - pclmulqdq 3y ago

My 6 and 7 yo's call infinity the "endless number". Well, at least it is a NaN number :) PS: they seem to _know_ that endless*endless > endless but do not dare to admit it

Dylan16807 3y ago

Well that fear is good because multiplication won't change the cardinality, you have to go exponential.

smfjaw 3y ago
That effectively does make it exponential x * x === x^2
- ksarw 3y ago
  by going exponential, they mean 2^x; x^2 can still be the same "size" as x
- NobodyNada 3y ago
  x^2 is not exponential; it's quadratic. 2^x is an example of an exponential function. The parent comment was alluding to the idea of set cardinality (https://en.wikipedia.org/wiki/Cardinality https://en.wikipedia.org/wiki/Cardinality). Two sets have the same cardinality if you can establish a bijection (a one-to-one mapping) between elements of one set and elements of the other. The set of all natural numbers is said to have a "countably infinite" cardinality. It turns out that for any countably infinite set S, the set S x S is also countably infinite (see Hilbert's hotel: https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...). For example, the set of 2-tuples of natural numbers (1, 1), (1, 2), (1, 3), ..., (2, 1), (2, 2), ... is the same size as the set of natural numbers. So in this sense, "endless*endless = endless". Whereas the set of infinitely-long tuples of natural numbers is "uncountably infinite;" it has a cardinality greater than that of the set of natural numbers. Thus, "you have to go exponential"; i.e. "endless^endless".
danwills 3y ago
Not an expert but I thought it needed the 'power set' (set of all subsets) but maybe that's kinda the same as exponentiation in the end?
- bkallus 3y ago
  It is. The cardinality of the power set of a set S is 2^|S|.
  - dllthomas 3y ago
    You can see it as a binary "in or out" for each element of S.

thriftwy 3y ago

Infinite is just a fancy word for endless, anyway.

short_sells_poo 3y ago
Not necessarily:) A circle is endless, and yet certainly isn't infinite.
- jdkee 3y ago
  A circle is made up of an uncountably infinite set of points.
  - selcuka 3y ago
    A line segment is also made up of an infinite set of points, but it's not endless.
    - davrosthedalek 3y ago
      You can also map the all reals to the reals between 0 and 1, so the infiniteness of the extend isn't that crucial.
    - hunter2_ 3y ago
      I can just imagine someone coming up with the infinity symbol, and arguing (as we are here) about whether a circle represents the idea. No, not quite; it needs something more... another circle should suffice, and connect them seamlessly. Yes, yes. This looks much more infinite than a mere circle.
- jameshart 3y ago
  Infinite literally means ‘without end’. Finite means ‘something that finishes’. Infinite means ‘something that does not finish’.
  - dragonwriter 3y ago

MichaelMoser123 3y ago

there is a difference between infinity and not a number, NaN isn't event equal to itself, in nodejs you get: > Infinity + Infinity Infinity > Infinity - Infinity NaN > Infinity == Infinity true > NaN == NaN false The fun thing: NaN to the power of zero (because technically the type of NaN is a ... number) > NaN ** 0 1 > typeof(NaN) 'number' By the same logic, NaN is also even because > NaN + NaN NaN but i think it's better not explain that to your six year old...

mikepurvis 3y ago
Huh. Most of those make sense to me, but infinity == infinity being true definitely feels like risky business. Algebraic limits is full of even some pretty trivial scenarios where infinity divided by a lesser-infinity turns out to be a real number— those cases where the two infinities are definitely not equal to each other.
- MichaelMoser123 3y ago
  Also: in javascript you can divide by zero > 1/0 Infinity But python throws a divide by zero error: >>> 1/0 Traceback (most recent call last): File "<stdin>", line 1, in <module> ZeroDivisionError: division by zero Still you do have infinity and nan in python - because these are part of the floating point spec. >>> float('inf') - float('inf') nan >>> float('inf') == float('inf') True >>> NAN=float('inf') - float('inf') >>> NAN == NAN False However that's not mathematics, it's computers (these are even stranger...) I have my own little programming language - PYX [1] - and i don't allow this madness (even if it is a violation of the floating point spec ;-) pyx > mathconst.Infinity - mathconst.Infinity Error: results in 'not a number' - that's not allowed here #(1) mathconst.Infinity - mathconst.Infinity |....................^ > 1/0 Error: Can't divide by zero #(1) 1/0 |..^ [1] PYX - https://github.com/MoserMichael/jscriptparse https://github.com/MoserMichael/jscriptparse - it's supposed to be an educational programming language, where I am trying to have detailed error messages, my side project.
  - pclmulqdq 3y ago