321–330 of 398 posts
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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?
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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
The parent comment was alluding to the idea of set 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_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".
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Infinite is just a fancy word for endless, anyway.
Not necessarily:) A circle is endless, and yet certainly isn't infinite.
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well, if you claim omega is odd, are you willing to claim omega + 1 is even? There is no ordinal B such that 2 * B is omega + 1, so it fails that definition. So you have to say omega is odd and omega + 1 is also odd, which is... odd.
But that "oddness" is precisely my whole point. I'm arguing that because it's just as easy to say that omega is odd as to say that it's even, that the whole concept breaks down and loses and all meaning. Because if you want to divide omega + 1 in half to show that it's even, we can do that. If we denote the set element inside of the "1" of "+ 1" by the symbol "a", then we can write out: [1, 3, 5, 7, ...] [a, 2, 4, 6,…
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No math instruction I had ever discussed infinity with any rigor until calculus -- and even then, it was only infinity as a limit. Infinity as a concept was brushed off in the same way that the square root of negative one was brushed off until we were actually taught about it.
On the one hand I get why that is - the calculus notion of infinity is the one that tends to be useful in applied math - on the other hand it's a shame because the set theoretic notion of infinity has more to offer to someone trying to ponder the nature of the infinite. Or put another way, "what's ∞ + 1" basically invites the non-answer "that's not a well-formed question" whereas "what's ω + 1" gives you a whole inte…
...and that the intellectual thread you are pulling on is a (more) artificial notion, constructed by set theorists for the sake of set theorists, not for the sake of counting or measuring in any real sense.
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What? Where does that follow from?
It follows from the way addition is defined on top of set theory. "a + b" is implemented as "increment a (the set that represents a) b times". A number is represented in set theory as a set that contains all of the numbers before it. 0, 1, 2 is {}, {{}}, {{} {{}}}... SO! If you start with a finite "a" and increment it infinite times, you still have infinity; you haven't broken out. But if you start with Infinity, the…
In that context the first quantity that you refer to above is nonsensical because you can't "increment infinitely many times".
Secondly, I'm not sure your construction is correct, since your Infinity+1 set cannot be a singleton (it must contain all the numbers less than Infinity).
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.
Sensei: Grasshopper, lecture over today.