4 ms·

I always love this topic. > Nat = 1 + Nat. This equation is clearly inconsistent, since x=1+x is false for all possible values of x. Ah, but no, my friend. Th

by cvoss 2y ago

I always love this topic. > Nat = 1 + Nat. This equation is clearly inconsistent, since x=1+x is false for all possible values of x. Ah, but no, my friend. That equation is false merely for all possible finite values of x. But why impose such a constraint? We've already been identifying a correspondence between certain types and the number of inhabitants they have. Nat has countably many inhabitants. And one more than countably infinite is countably infinite!

empath75 2y ago
Yeah because aren't the numbers associated with the types related to cardinality?
Infinity315 2y ago
> Nat has countably many inhabitants. And one more than countably infinite is countably infinite! Two sets having the same cardinality (number of inhabitants) is insufficient to prove that two sets are equal. There is an additional condition that they all share the same elements. I read 1 + Nat as: map each element in the Naturals to its successor (i.e. Nat - {0}). 0 is in Nat, but 0 does not exist in "1 + Nat" since it got mapped to 1. This would be true if the set we were working with were the Integers instead of the Naturals.
- srcreigh 2y ago
  This is incorrect.. the equation represents the number of elements in Nat based on its definition. Integers could be Int = Nat | Neg Nat. So Int = 2 * Nat. Your logic would presumably interpret this as even naturals.. but ya
  - Infinity315 2y ago
    > Your logic would presumably interpret this as even naturals.. but ya My logic comes from a Math background. Take any Abstract Algebra class and this notation is common. For example: https://math.stackexchange.com/questions/1378174/how-to-denote-an-even-number-in-mathematics#:~:text=The%20integers.,set%20of%20all%20even%20numbers https://math.stackexchange.com/questions/1378174/how-to-deno.... To say the least I take numerous issues with the notation used, but I accept it. Since the integers and the naturals have the same cardinality, we have that Int = Nat.
zdragnar 2y ago
I'm always confused as to why people want to be able to do arithmetic with infinity. Infinity isn't a number. > countably infinite Countably infinite means you can count to any particular element of a set in finite time, even though there is no end to the pattern of elements in the set. You cannot count to infinity plus one in finite time.
- Smaug123 2y ago
  That's not the attitude which leads to inventing the negative numbers, for example! The ordinals are useful and it's very reasonable to call them "numbers"; and I'm told that the surreal numbers are useful, although I've never actually used them.
- gnulinux 2y ago
  Infinity is not a number but transfinite numbers [1] like omega [2] or aleph-null [3] are. They are just as number as 0, 1, -5, 6/7, pi, i, or quaternion numbers. What is a "number" is not well-defined, any mathematical algebraic object can be called a number with some justification. Mathematicians just don't do it anymore, they just call it "X Algebra" instead of "X Numbers" but there is no fundamental distinction. From this perspective, arithmetic with infinity is not only possible but also extremely useful for Programming Language Theory, e.g. for Dependent Type Theory. [1] https://en.wikipedia.org/wiki/Transfinite_number https://en.wikipedia.org/wiki/Transfinite_number [2] https://en.wikipedia.org/wiki/Ordinal_number#Ordinals_extend_the_natural_numbers https://en.wikipedia.org/wiki/Ordinal_number#Ordinals_extend... [3] https://en.wikipedia.org/wiki/Aleph_number#Aleph-null https://en.wikipedia.org/wiki/Aleph_number#Aleph-null
- empath75 2y ago
  A set is "countably infinite" if you can construct a one to one mapping from the set to the natural (counting) numbers. The natural numbers are of course countably infinite, but so are even numbers, prime numbers, fractions, algebraic numbers and lots of other kinds of numbers that people want to do arithmetic with. "Uncountably infinite" sets can't be put into a one to one correspondence with the naturals, no matter how you produce to produce such a mapping, it's always possible to produce a number in the set which wasn't part of it. Real numbers being the prototypical example of that, but the power set (the set of all the subsets) of the naturals is, also. People do arithmetic with all kinds of things that "aren't numbers". Math isn't restricted to working with numbers.

by cvoss 2y ago

empath75 2y ago

Yeah because aren't the numbers associated with the types related to cardinality?

Infinity315 2y ago

> Nat has countably many inhabitants. And one more than countably infinite is countably infinite! Two sets having the same cardinality (number of inhabitants) is insufficient to prove that two sets are equal. There is an additional condition that they all share the same elements. I read 1 + Nat as: map each element in the Naturals to its successor (i.e. Nat - {0}). 0 is in Nat, but 0 does not exist in "1 + Nat" since it got mapped to 1. This would be true if the set we were working with were the Integers instead of the Naturals.

srcreigh 2y ago
This is incorrect.. the equation represents the number of elements in Nat based on its definition. Integers could be Int = Nat | Neg Nat. So Int = 2 * Nat. Your logic would presumably interpret this as even naturals.. but ya
- Infinity315 2y ago
  > Your logic would presumably interpret this as even naturals.. but ya My logic comes from a Math background. Take any Abstract Algebra class and this notation is common. For example: https://math.stackexchange.com/questions/1378174/how-to-denote-an-even-number-in-mathematics#:~:text=The%20integers.,set%20of%20all%20even%20numbers https://math.stackexchange.com/questions/1378174/how-to-deno.... To say the least I take numerous issues with the notation used, but I accept it. Since the integers and the naturals have the same cardinality, we have that Int = Nat.

zdragnar 2y ago

I'm always confused as to why people want to be able to do arithmetic with infinity. Infinity isn't a number. > countably infinite Countably infinite means you can count to any particular element of a set in finite time, even though there is no end to the pattern of elements in the set. You cannot count to infinity plus one in finite time.

Smaug123 2y ago
That's not the attitude which leads to inventing the negative numbers, for example! The ordinals are useful and it's very reasonable to call them "numbers"; and I'm told that the surreal numbers are useful, although I've never actually used them.
gnulinux 2y ago
Infinity is not a number but transfinite numbers [1] like omega [2] or aleph-null [3] are. They are just as number as 0, 1, -5, 6/7, pi, i, or quaternion numbers. What is a "number" is not well-defined, any mathematical algebraic object can be called a number with some justification. Mathematicians just don't do it anymore, they just call it "X Algebra" instead of "X Numbers" but there is no fundamental distinction. From this perspective, arithmetic with infinity is not only possible but also extremely useful for Programming Language Theory, e.g. for Dependent Type Theory. [1] https://en.wikipedia.org/wiki/Transfinite_number https://en.wikipedia.org/wiki/Transfinite_number [2] https://en.wikipedia.org/wiki/Ordinal_number#Ordinals_extend_the_natural_numbers https://en.wikipedia.org/wiki/Ordinal_number#Ordinals_extend... [3] https://en.wikipedia.org/wiki/Aleph_number#Aleph-null https://en.wikipedia.org/wiki/Aleph_number#Aleph-null
empath75 2y ago
A set is "countably infinite" if you can construct a one to one mapping from the set to the natural (counting) numbers. The natural numbers are of course countably infinite, but so are even numbers, prime numbers, fractions, algebraic numbers and lots of other kinds of numbers that people want to do arithmetic with. "Uncountably infinite" sets can't be put into a one to one correspondence with the naturals, no matter how you produce to produce such a mapping, it's always possible to produce a number in the set which wasn't part of it. Real numbers being the prototypical example of that, but the power set (the set of all the subsets) of the naturals is, also. People do arithmetic with all kinds of things that "aren't numbers". Math isn't restricted to working with numbers.