In JavaScript we're spoiled by having both. There's Number.NEGATIVE_INFINITY and Number.POSITIVE_INFINITY.
Don't forget -0 (negative zero)!
Floats are approximations of the reals and -0 is just the limit towards zero from the left.
Is infinity an odd or even number? (2011)
151–160 of 398 posts
Re: Is infinity an odd or even number? (2011)
#152So there seems like a glaring hole in the answer, but maybe I'm missing something. Because: > It is easy to prove from this definition by transfinite recursion that the ordinals come in an alternating even/odd pattern, and that every limit ordinal (and hence every infinite cardinal) is even. Sure, if we use the natural numbers and start at 1, then we can group: [1, 2], [3, 4], [5, 6], ... and prove infinity is even.…
The definition given was 'if there is another ordinal 𝛽 such that 2⋅𝛽=𝛼' [1], but the intuition is better explained by the post below:
> A set 𝑆 has even cardinality if it can be written as the disjoint union of two subsets 𝐴,𝐵 which have the same cardinality. [2]
In other words, a set is even if it can be paired up, by finding one grouping where it pairs. Finding alternative groupings that do not pair does not matter.
Re: Is infinity an odd or even number? (2011)
#153So there seems like a glaring hole in the answer, but maybe I'm missing something. Because: > It is easy to prove from this definition by transfinite recursion that the ordinals come in an alternating even/odd pattern, and that every limit ordinal (and hence every infinite cardinal) is even. Sure, if we use the natural numbers and start at 1, then we can group: [1, 2], [3, 4], [5, 6], ... and prove infinity is even.…
I would weaken the definition of even/odd to say that a set is even if /there exists/ a way to pair things off, and odd if /there is no way/ to pair things off (ie, not even). So the countable numbers would be even.
But that seems redundant with countable/uncountable sets, because then every countable infinite set would be even (e.g. rational numbers), and every uncountable infinite set would be odd (e.g. real numbers).
It's also not clear to me what justification there would be for a "preference" for the "even" category that way -- it seems arbitrary. Why not be odd if there exists a way to pair things off such that one is left over, and even if there isn't such a way?
Re: Is infinity an odd or even number? (2011)
#154Earlier quoted context omitted.
> it does make me think that infinity must be even since infinity can be divided into 2 pairs, each of which is of equal size since both are infinity. This is true, but the same is true of (infinity - 1) Therefor infinity must also be odd.
The concept of "infinity - 1" doesn't exist. Subtraction isn't defined for ordinals. Furthermore even if you try to define it, it doesn't work for limit ordinals. If you are thinking about the difference between [0,1,2,3,…] and 0, [1,2,3,4,…] Then I regret to inform you the former is omega and the latter is 1+omega which is the same as omega. In other words attempting to subtract one from infinity by removing from th…
> In other words attempting to subtract one from infinity by removing from the front results in infinity.
And I regret to inform you that if you read more carefully, you will find that my comment above makes use of that very same property of infinity. Not only do I already know it; that's the joke.
Specifically, that statements about omega are also statements about 1 + omega. The parent post saying "I think that infinity must be even" is such a statement. Regardless of if it's true or not, well-defined or not, coherent or not, it's equally all that about (infinity - 1).
Should I also spell out that an argument that "n - 1 is even" is also an argument that "n is odd" ?
Re: Is infinity an odd or even number? (2011)
#155Well, I asked ChatGPT: > Infinity is not a number, odd or even, but rather a concept or a mathematical idea that represents an unbounded or limitless quantity. Infinity is not a real number that can be used in ordinary arithmetic operations, but it is used to describe a quantity that is larger than any finite number. Therefore, the concept of odd or even does not apply to infinity.
Much more reasonable answer than the others in the thread
Re: Is infinity an odd or even number? (2011)
#156So there seems like a glaring hole in the answer, but maybe I'm missing something. Because: > It is easy to prove from this definition by transfinite recursion that the ordinals come in an alternating even/odd pattern, and that every limit ordinal (and hence every infinite cardinal) is even. Sure, if we use the natural numbers and start at 1, then we can group: [1, 2], [3, 4], [5, 6], ... and prove infinity is even.…
Omega is the lowest countable infinity. There's no parity within a countable infinite as you describe. It's only even or odd with respect to other infinities which the cardinal numbers can count based on the presence of a bijection or not. It's a kind of relative parity.
> There's no parity within a countable infinite as you describe.
That directly contradicts the quoted text I included from the original answer though, as far as I understand. It directly asserted that "every infinite cardinal is even".
> It's a kind of relative parity.
What is relative parity? The original question was whether infinity is even or odd... I don't know what you mean by relative parity.
Re: Is infinity an odd or even number? (2011)
#157[dead]
Re: Is infinity an odd or even number? (2011)
#158In my experience with children, one of the easiest-to-grasp concepts of infinity is provided by the transfinite ordinals, since it can be viewed as a continuation of the usual counting manner of children, but proceeding into the transfinite: 1,2,3,⋯,ω,ω+1,ω+2,⋯,ω+ω=ω⋅2,ω⋅2+1,⋯,ω⋅3,⋯,ω2,ω2+1,⋯,ω2+ω,⋯⋯ Presumably this person has no experience with 6 year olds? This explanation is horrendous haha
I explained basically this to my 4 year old nephew recently. He wanted to count to infinity. I asked him what is the biggest problem with counting to infinity? It's too slow. I said ok let's take bigger steps. We counted by 2's then 10's then hundreds and millions and then zillions and other ridiculous superlative numbers. It doesn't really matter because everything is still too slow. So then we said ok lets make up…
I taught my kid that the way to think of infinity is that it's like hugs, there's always one more, unlike candy, which is limited and can be counted, infinity cannot be counted.
Re: Is infinity an odd or even number? (2011)
#159So there seems like a glaring hole in the answer, but maybe I'm missing something. Because: > It is easy to prove from this definition by transfinite recursion that the ordinals come in an alternating even/odd pattern, and that every limit ordinal (and hence every infinite cardinal) is even. Sure, if we use the natural numbers and start at 1, then we can group: [1, 2], [3, 4], [5, 6], ... and prove infinity is even.…
The definition given was 'if there is another ordinal 𝛽 such that 2⋅𝛽=𝛼' [1], but the intuition is better explained by the post below: > A set 𝑆 has even cardinality if it can be written as the disjoint union of two subsets 𝐴,𝐵 which have the same cardinality. [2] In other words, a set is even if it can be paired up, by finding one grouping where it pairs. Finding alternative groupings that do not pair does not…
OK, so I guess I'm just understanding that mathematicians arbitrarily decided to prioritize "even" over "odd"?
Because as I stated in another comment, you could just as easily say odd cardinality exists if you can find two subsets with the same cardinality and there's one element left over, and otherwise we call it even.
So at the end of the day, what you're saying is that ultimately infinity would be even just because mathematicians arbitrarily defined 'even' that way -- not because there's any intuitive logic behind it, any deeper justification, or any necessary consistency with parity for finite sets.
Re: Is infinity an odd or even number? (2011)
#160I think the best answer is the first comment to the question:
Infinity is neither even nor odd, just like 1.4 isn't even or odd.