Infinity can have a starting point. I.e. Start at zero and count up forever.
Infinity can go both ways. I.e. Start at zero and count up and down forever.
Are both these infinities equivalent? (My brain can’t kick the idea that counting up and down is a bigger infinity than just up.)
2) You’re about to start a Universe (pre-Big Bang). There are an infinite number of initial states that can be randomly chosen from. Only 1 in 100 septillion of the initial states can result in a Universe where life is possible.
Aren’t there then an infinite number of initial states where life is possible? If there are an inifnite number of states where life can exist and an infinite number where life doesn’t exist how can random chance favor one or the other? Or, does the 1:100,000,000,000,000,000,000,000,000 chance really equate to a 50/50 chance?
Cantor did a lot of work with this, you should look him up if you really want to read more about it.
Basically, two sets are the same size (even if they are infinite) if you can pair up the members of one set with the members of the other set, no member of either set is used twice, and no member of either set is left over.
So if we compare the natural numbers {0,1,2,…} with the integers {…,-2,-1,0,1,…}, we can pair them up:
Pair 0 with 0,
Pair the odd natural numbers with the negative integers,
Pair the (nonzero) even natural numbers with the positive integers.
In a way, this amounts to putting the integers in the order:
0,-1,1,-2,2,-3,3,-4,4,…
and (I hope) it’s more or less clear, then, that the set of natural numbers is the same size as the set of integers.
This gets into measure and probability theory. Random chance can favor one over the other, even if they’re both infinite.
The easiest example to think of is this: If you pick an integer at random, what’s the probability that it will be divisible by 3? 1/3, because of the “density” of the numbers divisible by 3–they’re kind of spread out, in other words. It’s 1/3 even though the set {integers divisible by 3} is the same size as {integers not divisible by 3}.
Both of those are countable infinities(cardinal) so they are equal, but there are non-countable infinities which are considered to be larger sets. The difference is whether or not you can describe an algorythm that will eventually cover all of them.
The continum hypothesis says that the set of reals has cardinality of aleph-1. The generalized continum hypothesis says that the power set of a set with cardinality aleph-a has cardinality aleph-a+1.
CH and GCH are independent of the axioms of set theory. That is, they cannot be proven or disproven. They can be added as axioms with no problem. You can also add as axioms that they are false without introducing a contradiction.
I’ve read Infinity and the Mind. It is a good non-technical book about infinity and the philosophy of infinity.
That tip on the Rudy Rucker book is a great idea, but you really should read Rucker’s book “White Light, or, What is Cantor’s Continuum Problem?” It specifically addresses the issue of cardinal infinities, and the premise of the book is that even mathematically untrained people can “access” higher infinities directly, and even experience these higher mathematical states.
An example anecdote from the book. The protagonist checks into “Hilbert’s Hotel” which is an infinitely large hotel with an infinite number of rooms. But it also has an infinite number of guests. Since there are an infinite number of rooms, and you can always add 1 to infinity, there is always an free room. But it is infinitely far away on the infinite-th floor of the hotel, it would take an infinite time to get to your room from the lobby. So how does a guest check in to the hotel? He checks into room 1. Guest in room 1 moves into room 2, room 2 checks into room 3, ad infinitum. No guest has to move an infinite distance, although everyone is inconvenienced a bit. The guest doesn’t like the idea of having to inconvenience all the guests, making an infinite amount of trouble for an infinite amount of guests. So the clerk tells him to check into room 1, the guest in room 1 moves to room 3, the guest in room 3 moves into room 5, etc. Only the odd numbered rooms have guests to be relocated. But infinity divided by 2 is still infinity, it is still an infinite inconvenience, so there is no solution that will minimize the room moving.
Anyway, White Light was recently republished after about 15 years out of print. Grab it if you can, it is a cyberpunk classic.
I mentioned this in a previous thread, but it’s definitely appropriate to bring it up here. I’ve heard that this year a new axiom has been proposed (Woodin’s axiom) which, adjoined to FZC set theory, implies the continuum hypothesis is false. Not necessarily remarkable, but for the fact that I’ve heard that it does have an intuitive appeal, and there is some speculation as to whether it will be accepted “at large” among the mathematical community. Unfortunately, I don’t know what the axiom states, but I’ve been wondering if this could ultimately turn into an “axiom of choice” type debate among mathematicians.
There is a field of mathematics called “complex analysis”. The “complex” part refers to complex numbers that involve both real numbers and the square root of -1, “i”, but it also gets complex in the sense of being complicated Obviously the set of complex numbers includes the set of integers.
In complex analysis, the infinity that you reach by counfing forward from zero is the same “number” that you reach by counting backwards from zero. You can think of mapping the complex numbers one-to-one to points on a sphere where (0 + 0i) is one pole and the antipodal pole is the one point to which infinity maps. No matter what path you follow from one pole to the other, there’s only one infinity. You can even do path integrals that include the infinite point on the path (that is, there are continuous curves that go to infinity and “come back out the other side” fo infinity without leaving the set of complex numbers).
This is one of the less weird attributes of complex analysis. Don’t get me started on the Schwarz-Christoffel transform, or residue analysis! {grin}
It’s been a long time since I took complex analysis, and I’d be hard-pressed to go into much more detail.
Earlier I said that they are equivalent. I was talking about cardinality and I stick by what I said. But I cannot argue with Jinx saying negative infinity is less than infinity. There is more than one way to define infinity. Cardinal numbers represent “how many” elements there are in a set and they are all non-negative. There are ways to define negative infinities that are distinct from positive infinities. So, the answer to:
“Are both these infinities equivalent?” depends upon what system of infinities are we talking about and exactly what do you mean by equivalent.
Cabbage
Years ago I read an article that said since the CH was independent of the axioms, CH or (not CH) could be added as an axiom without a contradiction. It proposed some system called non-Cantorian set theory where there was an uncountable set of reals that was not equivalent to the entire set of reals. This article sure didn’t make it intuitive, but the point was just to show that if you assume CH, you can build a model where CH is false. I haven’t heard of Woodin’s axiom. (I haven’t been keeping up.) I’d be interested to find out what it proposes.
Yeah, I’ve been wanting to find out more about Woodin’s axiom as well. Here is where I first heard of it, and, really, that’s about the extent of my knowledge of it. I know that Woodin is a professor of mathematics at the University of California at Berkeley, specializing in large cardinals. If nothing else, I suppose you could try emailing him.
I suppose you’re probably familiar with the debate that surrounded the axiom of choice earlier this century. The axiom of choice implies some very non-intuitive things (that any set can be well ordered, or that a basketball can be taken apart into a finite number of pieces, and put back together into a ball the size of the sun). Ultimately, though, there’s so much mathematics that can’t be done without the axiom of choice that it was eventually commonly accepted. Like I said, it will be interesting to see if anything similar happens with Woodin’s axiom.
I’ve seen systems where the Axiom of Choice was assumed to be false. I always thought they seemed unworkable. I like the Axiom of Choice; it just seems right. I think Cantor assumed GCH. I would be a little disappointed if it is decided to make GCH false. I already did a search for “Woodin’s axiom” and the only link that turned up was the one you supplied. That is a good link though.
I just found the following joke by Jerry Bona “The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn’s Lemma?”
Roppy Bork said "Infinity is not a number. Don’t try arithmetic with it - it doesn’t work. … 0 is not a natural number. "
There are systems where Infinities are numbers and arithmetic works just fine. I think that zero is the most natural number. Systems that include zero are much more elegant. But if you want to exclude zero from the natural numbers Well go ahead, but it is not right or wrong.
