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Is infinity an odd or even number? (2011)

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Re: Is infinity an odd or even number? (2011)

#311

Post by gopher_space »

justinator wrote: post #285
gopher_space wrote: post #283

Earlier quoted context omitted.

Carlos Castaneda is at his most interesting when he wrestles with "what's outside of the universe" paradoxes since his informants seem like they're able to not only hold mutually exclusive concepts but exist in a relationship between them. They'd have an internally consistent idea about what's North of the North Pole and could explain it to you in terms you might understand. He's given me quite a bit to think about i…

I was so confused at your glowing review until the redemption of the last sentence.

The first one is basically "Fear and Loathing on the Campaign Trail '72" for anthro majors, and then he gets less focused somehow. He'd be my personal Kilgore Trout if we didn't have contemporary science fiction.

Re: Is infinity an odd or even number? (2011)

#312

Post by smfjaw »

Dylan16807 wrote: post #257
tsoukase wrote: post #252

Earlier quoted context omitted.

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

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

That effectively does make it exponential

x * x === x^2

Re: Is infinity an odd or even number? (2011)

#313

Post by selcuka »

jdkee wrote: post #308
short_sells_poo wrote: post #299

Earlier quoted context omitted.

Not necessarily:) A circle is endless, and yet certainly isn't infinite.

A circle is made up of an uncountably infinite set of points.

A line segment is also made up of an infinite set of points, but it's not endless.

Re: Is infinity an odd or even number? (2011)

#314

Post by selcuka »

JKCalhoun wrote: post #179
bhk wrote: post #72

Earlier quoted context omitted.

I thought that most of us learn at an early age, as a result of this kind of exchange, that "infinity" is not "the biggest number" or even a number at all, as far as the ordinary notion of "number" goes.

My child mind conflated infinity and God. Or maybe I was correct, I have no idea now.

Historically, anything that can't be easily comprehended has been attributed to a higher power.

Re: Is infinity an odd or even number? (2011)

#315

Post by analog31 »

zw123456 wrote: post #202

The way I would explain it to a 6 year old would be like this: Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.

I would start with a game: For any number, I can name a bigger number.

For any collection with a certain number of things (such as n potatoes), I can always name a collection with more things, such as n+1 potatoes.

"Infinity" is a word that we use for games like that.

Re: Is infinity an odd or even number? (2011)

#316

Post by raldi »

zw123456 wrote: post #202

The way I would explain it to a 6 year old would be like this: Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.

Is “up” an even or odd number of meters?

Re: Is infinity an odd or even number? (2011)

#317

Post by danwills »

Dylan16807 wrote: post #257
tsoukase wrote: post #252

Earlier quoted context omitted.

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

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

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?

Re: Is infinity an odd or even number? (2011)

#318

Post by contravariant »

smallnamespace wrote: post #152
crazygringo wrote: post #119

So 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…

It might be a better explanation but those two are very much not equivalent.

Actually the fact that splitting it into pairs is the same as splitting into two equal sets of equal cardinality is itself non-trivial. The reason why shows up when you try to get the two definitions closer together.

Splitting an ordinal into pairs is essentially splitting it into ordered pairs (a_i, b_i) such that the map i to a_i is monotonic and for no iSplitting a set into pairs is splitting it into sets {a_i, b_i} such that for no i != j the two sets {a_i, b_i} and {a_j, b_j} overlap.

These two are note the same, you can split pretty much any infinite set into two disjoint sets of equal cardinality.

It's hard to get the definitions general enough to get one definition for both ordinals and sets. Mostly because products of ordinals are a bit weird. For sets (and most other types of mathematical objects) it doesn't matter which way around you pair things up, but for the ordinals you end up with a completely different object if you do it the other way around and this is apparently the more interesting definition of the two.

Re: Is infinity an odd or even number? (2011)

#319

Post by sreekotay »

FabHK wrote: post #101
alexb_ wrote: post #12

Earlier quoted context omitted.

No it isn't. If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).

> Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". Or is it "Two Times Infinity"? (Hint: It isn't, because "Two Times Infinity" = "Infinity", while "Infinity Times Two" = "Infinity + Infinity". Not sure every kid knows that.)

Genuinely lol'ed! This is such a perfect child + 1 argument - well played :)

Re: Is infinity an odd or even number? (2011)

#320

Post by gus_massa »

squidsoup wrote: post #248
gus_massa wrote: post #58

The problem with transfinite is that you lose commutatively. Flowing the standard notation, where the usual infinite in the integer or the real line is "ω = ∞ = 1,2,3,..." ω+1 = ω+1 , i.e. "the next thing after infinity" 1+ω = ω , i.e. "the same infinity as before" 2ω = ω , i.e. "the same infinity as before", so it's even 1+2ω = ω , i.e. "the same infinity as before", so it looks odd, but don't fall in that trap ω2 =…

> ω+1 = ω+1 , i.e. "the next thing after infinity" I find this concept perplexing. To me this implies that "infinity" has a value. How can you add 1 to a thing that by definition has no value?

Symbols in math are overloaded, like in C++.

Imagine the + in C++ when you have to add two complex numbers. They are just a struct with x and y, and some magic to make all operations work as intended.

The use of the + in this example is more like the concatenation of strings, like "Hello " + "World!" is "Hello World!". But in this case, the content of the string doesn't matter so "Hello " == "World!" and there are some magical strings that are infinite.

The idea is that anyone can overload the symbol + and sum whatever they want. It's not necessary to use + with numbers. Obviously, most silly overloads are ignored, and nobody use them. In this case it's a popular overload so it is teach in an advanced math course and has it's own Wikipedia article.

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