No its not an odd or even number; its infinity.
Is infinity an odd or even number? (2011)
371–380 of 398 posts
Re: Is infinity an odd or even number? (2011)
#372Here's one that gets me: what's the sign of infinity? What do I mean? 1/d retains the sign of d across all finite and infinitesimal values. 1/ε = +∞, 1/-ε = -∞ So what about 1/0? Neutral infinity.
1/0 is not defined. There is only lim(1/ε) when ε->0+ = +∞ and lim(1/ε) when ε->0- = -∞
Re: Is infinity an odd or even number? (2011)
#373Earlier quoted context omitted.
> And then infinitely later There can be no 'and then' after infinitely later. I don't see why stepwise is important but that must be the key to Cantor's proof. If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned. The ordering could be entirely random, I don't see how it makes a difference. There will always be enough integers to m…
> If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned. Exactly correct! Any bijection between the naturals and the reals would suffice to show that they're the same cardinality; the order does not matter. I think where you're getting confused is just in who's trying to do what; who's the "protagonist" and "antagonist" in the proof.…
I feel the number he generates via any procedure would be on the list since all of them are on the list. What am I missing about the plausibility of a procedire that must be possible which must generate a real number not in the series?
Re: Is infinity an odd or even number? (2011)
#374The short answer to this question is "No".
Re: Is infinity an odd or even number? (2011)
#375Earlier quoted context omitted.
Dividing by 0 or -0 is a valid floating-point operation because there's an infinity in the number system, and JS uses double precision floating point for all numbers. Python has an integer type and a double type, and division by 0 is disallowed for integers, but okay for doubles.
your explanation makes sense, however python doesn't allow division of floating point number by 0 either: >>> type(1.0) >>> 1.0/0 Traceback (most recent call last): File " ", line 1, in ZeroDivisionError: float division by zero However numpy lets you do it - it is only a warning >>> import numpy as np >>> >>> np.divide(1.0,0) __main__:1: RuntimeWarning: divide by zero encountered in true_divide inf
Wow, Python really does hold your hand.
>>> 1/0.0
Traceback (most recent call last):
File "", line 1, in
>>> 1.0/0.0
Traceback (most recent call last):
File "", line 1, in
Re: Is infinity an odd or even number? (2011)
#376Earlier quoted context omitted.
> But "even" and "odd" are all about whether you can partition something into an equal number of pairs or not. Sez you. I can just as easily define even and odd in terms of whether or not I can arrive at a given position in a (potentially infinite) sequence taking by taking two steps at a time.
Of course you cannot actually end up at ω by taking two steps at a time, but your point is still well taken.
Yes, I know. ω is odd :-)
Re: Is infinity an odd or even number? (2011)
#377Earlier quoted context omitted.
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.
No problem at all with your analogy for a child, but the dirty truth of the universe is that hugs are finite and infinity can be counted (sometimes)
Re: Is infinity an odd or even number? (2011)
#378Earlier quoted context omitted.
> If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned. Exactly correct! Any bijection between the naturals and the reals would suffice to show that they're the same cardinality; the order does not matter. I think where you're getting confused is just in who's trying to do what; who's the "protagonist" and "antagonist" in the proof.…
I feel the number he generates via any procedure would be on the list since all of them are on the list. What am I missing about the plausibility of a procedire that must be possible which must generate a real number not in the series?
> I feel the number he generates via any procedure would be on the list since all of them are on the list
The claim is that all of them are on the list. The constructed number proves that claim false.
It's a proof by contradiction. If you assume there is any way to write an infinite numbered list of all reals, then Cantor shows it's possible to come up with a number not on your list. The construction uses your list as input, and given any list, can always produce a real number not on that list. Therefore there is no way to write an infinite numbered list of all reals.
It relies on the fact that real numbers have (countably) infinite digits, and therefore infinite "degrees of freedom" to be different. This may be one reason it's hard to accept. A "true" real number can contain infinite information in a single number. For instance, we can jam all of the naturals into a single real by just concatenating their decimal representations: 0.1234567891011121314151617181920212223...
This one single real number encodes the full infinite natural number line. That hopefully gives you a sense of why the "infinite digits" definitions of reals makes them qualitatively "bigger" than any number that has finite representation.
Re: Is infinity an odd or even number? (2011)
#379Earlier quoted context omitted.
I feel the number he generates via any procedure would be on the list since all of them are on the list. What am I missing about the plausibility of a procedire that must be possible which must generate a real number not in the series?
> I feel the number he generates via any procedure would be on the list since all of them are on the list The claim is that all of them are on the list. The constructed number proves that claim false. It's a proof by contradiction. If you assume there is any way to write an infinite numbered list of all reals, then Cantor shows it's possible to come up with a number not on your list. The construction uses your list a…
No I still don't get it, it's like saying that infinity^2 is larger than infinity. If 0^2 is no larger than 0, then it must be the same for infinity.
I see how a list of reals is like 2D list of infinities, so one grows from the middle and the other grows from the end, but they're both still infinite. I guess I'm still stuck in a 'mechanical' approach and not a mathematical one. I'm not sure I want to leave ;) This has been fascinating to think about anyway.
Re: Is infinity an odd or even number? (2011)
#380Earlier quoted context omitted.
> And then infinitely later There can be no 'and then' after infinitely later. I don't see why stepwise is important but that must be the key to Cantor's proof. If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned. The ordering could be entirely random, I don't see how it makes a difference. There will always be enough integers to m…
> If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned. Exactly correct! Any bijection between the naturals and the reals would suffice to show that they're the same cardinality; the order does not matter. I think where you're getting confused is just in who's trying to do what; who's the "protagonist" and "antagonist" in the proof.…
> Write your pairing down on an infinitely long sheet of paper.
If I do that then Cantor will never have a chance to give me his 'gotcha' number, because I'll always be writing on my infinite paper ;)