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 Post subject: Big numbers D:
PostPosted: Fri Jul 29, 2011 11:14 am 
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http://prime.isthe.com/no.index/chongo/ ... ime-c.html This is pretty big number. D: What's your favorite large number?

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 11:21 am 
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LOL its prime too


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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 11:43 am 
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Largest known mersenne prime. (2ⁿ-1)

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 11:57 am 
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My favorite big number:


-d


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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 2:00 pm 
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I don't think I would have a favorite big number since there is always a number that is one bigger.
Maybe the number x such that x>1/d for any d>0? I think I like that one.

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 2:15 pm 
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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 2:38 pm 
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This one's also quite big:
Attachment:
Big number.png
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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 2:45 pm 
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Hello,

One of my favorite numbers had always been 10^100 also called a Googol. I learned about this number in 1980 watching Carl Sagan's Cosmos: A Personal Voyage. Or course, now everybody knows the misspelled version of the word that has become a certain search engine...

Here's the specific part of that specific episode: http://www.youtube.com/watch?v=gh4F5BQ8hgw#t=4m19s

Enjoy,

Skarabajo.

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 2:47 pm 
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Kelvin Stott wrote:
This one's also quite big
No, that's nothing. It's like saying 1/0 is a big number.

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 2:53 pm 
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I vote for Skewes number

Once called "The biggest number that ever came up in a mathematical proof"
(currently dwarfed by Grahams number though)

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 2:57 pm 
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2nd : 43252003274489855999 (and it's prime!)

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 2:58 pm 
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TomZ wrote:
Kelvin Stott wrote:
This one's also quite big
No, that's nothing. It's like saying 1/0 is a big number.
Prove it! :lol:

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 3:06 pm 
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Kelvin Stott wrote:
TomZ wrote:
Kelvin Stott wrote:
This one's also quite big
No, that's nothing. It's like saying 1/0 is a big number.
Prove it! :lol:

Technically 'infinite' is not a number but a mathematical concept. Any number is, once you name it, by definition finite.

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 3:16 pm 
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OK, then here's my favourite "big" number:

(10^78)! = number of permutations to arrange all the atoms of the universe in linear sequence.

This is roughly 10^(10^80), which is the number 1 with 10^80 zeros after it. Or just take the number 1 and multiply by 10 (add a zero) for each and every atom in the universe ... and then multiply that number by another 100! :D

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Last edited by KelvinS on Fri Jul 29, 2011 5:47 pm, edited 8 times in total.

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 3:19 pm 
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maarten wrote:
Technically 'infinite' is not a number but a mathematical concept. Any number is, once you name it, by definition finite.


Not directed at me I know but to clarify, my isn't infinity, it's a drunken 8.

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 3:20 pm 
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My favourite big number is 24,000,000. Or it will be if I win the Euromillions jackpot tonight! :lol:

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 3:31 pm 
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Tony Fisher wrote:
Not directed at me I know but to clarify, my isn't infinity, it's a drunken 8.


Then i is a just decapitated 1 :lol: ?

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 4:02 pm 
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Graham's number is considered by most people to be the largest known number with mathematical significance. It ends in ...95387. All digits are (in theory) possible to be calculated using an algorithm, although there isn't anywhere close to enough matter in the universe to find it in its entirety.

However, I've decided Grahams number has had it too good for too long. In my attempt to break its record as the largest mathematically significant number, I have posed the burning question of "What is Graham's number plus 1?"

Through incredibly complex mathematical processes, I have calculated that this number ends in ...95388. I have fittingly named it Kapusta's number.

...So, I guess that's my favorite big number.

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 5:54 pm 
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Kapusta wrote:
Graham's number is considered by most people to be the largest known number with mathematical significance. It ends in ...95387. All digits are (in theory) possible to be calculated using an algorithm, although there isn't anywhere close to enough matter in the universe to find it in its entirety.


Don't be too sure, I'm thinking that if all of the matter in the universe were somehow converted into rubiks cubes, the permutations of all of them(the total number of states of the group with each cube being considered) should at least come close to graham's number. If not, then holy crap that is a big number.

EDIT: forgot to mention that my favorite number is pi :)


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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 5:56 pm 
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Noreg89 wrote:
I'm thinking that if all of the matter in the universe were somehow converted into rubiks cubes, the permutations of all of them(the total number of states of the group with each cube being considered) should at least come close to graham's number. If not, then holy crap that is a big number.

That's kind of what I was getting at above, here:
Kelvin Stott wrote:
(10^78)! = number of permutations to arrange all the atoms of the universe in linear sequence.

This is roughly 10^(10^80), which is the number 1 with 10^80 zeros after it. Or just take the number 1 and multiply by 10 (add a zero) for each and every atom in the universe ... and then multiply that number by another 100! :D


EDIT: Mind you, if each of the 10^78 atoms in the universe was a Rubik's Cube with 4.3*10^19 permutations, then the total number of permutations of all atoms/cubes together would be:

(4.3*10^19)^(10^78)

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Last edited by KelvinS on Fri Jul 29, 2011 6:29 pm, edited 2 times in total.

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 6:00 pm 
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My bad lol I guess that's what I get for skimming over the topic :/


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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 6:12 pm 
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How about the number of elements in the set of all real numbers? Or along the lines of Hilbert's paradox of the Grand Hotel, the number of people needed such that they couldn't all fit into a hotel with countably infinitely many rooms, 1 guest to each room.

If you aren't familiar with the above question ask yourself which set has more elements, the set of all integers or the set of all rational numbers? Most would say there are more rational numbers then there are integers as the set of rationals includes the integers plus numbers not included in the integers but they would be wrong.

Carl

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 9:04 pm 
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Skarabajo wrote:
Hello,

One of my favorite numbers had always been 10^100 also called a Googol. I learned about this number in 1980 watching Carl Sagan's Cosmos: A Personal Voyage. Or course, now everybody knows the misspelled version of the word that has become a certain search engine...

Here's the specific part of that specific episode: http://www.youtube.com/watch?v=gh4F5BQ8hgw#t=4m19s

Enjoy,

Skarabajo.


Here you go glad to help you out.

viewtopic.php?f=7&t=5438

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 Post subject: Re: Big numbers D:
PostPosted: Fri Jul 29, 2011 11:13 pm 
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wwwmwww wrote:
How about the number of elements in the set of all real numbers? Or along the lines of Hilbert's paradox of the Grand Hotel, the number of people needed such that they couldn't all fit into a hotel with countably infinitely many rooms, 1 guest to each room.

If you aren't familiar with the above question ask yourself which set has more elements, the set of all integers or the set of all rational numbers? Most would say there are more rational numbers then there are integers as the set of rationals includes the integers plus numbers not included in the integers but they would be wrong.

Carl


Actually, the number of rational numbers is a greater infinity than the number of integers...

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 Post subject: Re: Big numbers D:
PostPosted: Sat Jul 30, 2011 2:16 am 
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elijah wrote:
Actually, the number of rational numbers is a greater infinity than the number of integers...


Not true (typo by elijah?): there is a quite simple 1-1 correspondence matching each fraction with one unique integer. The sets of rational numbers and integers are then called having the same cardinality (in popular terms: are equally infinit).

please check http://www.jcu.edu/math/vignettes/infinity.htm

There are more real numbers than integers though.

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 Post subject: Re: Big numbers D:
PostPosted: Sat Jul 30, 2011 3:32 am 
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My favorite is what I've heard called the xkcd number. Take the Ackermann function and use Graham's number as both of the arguments.
http://www.xkcd.com/207/


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 Post subject: Re: Big numbers D:
PostPosted: Sat Jul 30, 2011 9:01 am 
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maarten wrote:
elijah wrote:
Actually, the number of rational numbers is a greater infinity than the number of integers...


Not true (typo by elijah?): there is a quite simple 1-1 correspondence matching each fraction with one unique integer. The sets of rational numbers and integers are then called having the same cardinality (in popular terms: are equally infinit).

please check http://www.jcu.edu/math/vignettes/infinity.htm

There are more real numbers than integers though.

Arg... beat me to the punch. I was going to post this last night but my youngest boy is on-line. Anyways here is how I usually prove to most that these sets are the same size. Its almost the same as the link maarten pointed you to.

All rationals can be expressed as A/B with B not equal 0. And A and B are integers. So create an infinite excel table like this.

Image

Each row contains one value of A and each column contains one allowed value for B. That is why there is no 0 column. Now you can start anywhere in a path that spirals outward that touches each and every cell. This proves there is a one-to-one maping between the set of integers and rationals. They are both countably infinite.

maarten is also correct that there are more reals then integers.

And its also possible to create sets that have more elements then the reals too. However I believe no one has yet found or created a set with more elements then the integers yet less elements then the reals. I believe I remember reading somewhere that such sets could exist and not lead to any contradictions (and might even be useful in some proofs) but I don't think anyone has yet made an example of such a set.

Carl

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 Post subject: Re: Big numbers D:
PostPosted: Sat Jul 30, 2011 10:47 am 
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wwwmwww wrote:
And it's also possible to create sets that have more elements then the reals too. However I believe no one has yet found or created a set with more elements then the integers yet less elements then the reals. I believe I remember reading somewhere that such sets could exist and not lead to any contradictions (and might even be useful in some proofs) but I don't think anyone has yet made an example of such a set.


It's not that no one has made an example, it's that there are consistent models of set theory with arbitrarily many sizes strictly between the naturals and the reals. The naturals have size aleph_0, as does any countable infinite set. The reals have size 2^(aleph_0), which is an uncountable cardinal (i.e. is strictly larger than any countable set). Now, for most practical purposes it is convenient to assume that 2^(aleph_0) is actually aleph_1, the first uncountable cardinal. This is called the continuum hypothesis (CH), and for a long time it was an open problem as to whether or not this was actually true. If you accept this as an axiom, then no, any set strictly larger than the naturals is at least as large as the reals.

But in modern times, it's a fairly routine exercise in set theory to show that this is actually independent of the commonly accepted set-theoretic model for mathematics (ZFC). In layman's terms, that means that [all of regular math] + [the reals have size aleph_1] is a completely sensible, non-contradictory system, but so is [all of regular math] + [the reals have size strictly larger than aleph_1]. (However, it is provable in ZFC that aleph_1 is the smallest the reals could be, since they're obviously uncountable). In fact, there are models of set theory where the reals have size aleph_n, where n is any natural number you want. That's a pretty cool result. So for example, you can build a model of set theory where the naturals have size aleph_0, the reals have size aleph_317, and then there obviously are many, many, many sets of size strictly larger than the naturals and strictly smaller than the reals.

To anyone interested in talking about infinity and knowing what you're talking about, read up on ordinals and cardinals. (Check wikipedia for a start).

To Kelvin Stott, I think you were being facetious by posting infinity^infinity^infinity^infinity^..., but you might be interested to know that while that doesn't mean anything, omega^omega^omega^omega^... does. (Omega being the first infinite ordinal, and the set of natural numbers. It's what most people are actually talking about when they write the sideways 8 symbol but don't know it). Specifically, omega^omega^omega^omega^... is called epsilon_0, and it's the first ordinal which cannot be expressed as finitely many arithmetical operations applied to omega. It is countable, however, so it's still the same size as the natural numbers. Surprising but true. And as far as ordinals go, epsilon_0 is still very small, since it's countable, but it's one of my favorites, since it is also the proof strength ordinal of Peano arithmetic.


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 Post subject: Re: Big numbers D:
PostPosted: Sat Jul 30, 2011 3:19 pm 
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So an infinite number of mathematicians walk into a bar. The first one says "I'd like a beer." The second one says "I'd like half a beer." The third one says "I'd like a quarter of a beer." The bartender says "Screw this" and pours two beers.


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 Post subject: Re: Big numbers D:
PostPosted: Sun Jul 31, 2011 1:44 pm 
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Ok... let me preface this by saying I'm a physicist... not a mathematician. I love trying to wrap my brain around some of these abstract concepts but I'm a bit out of my element when I do. So I many not also be 100% correct or even stating things in a correct way at times. So bare with me...

Iranon wrote:
It's not that no one has made an example, it's that there are consistent models of set theory with arbitrarily many sizes strictly between the naturals and the reals. The naturals have size aleph_0, as does any countable infinite set. The reals have size 2^(aleph_0), which is an uncountable cardinal (i.e. is strictly larger than any countable set).


aleph_0 is a "size". When you get right down to it... it isn't a number. I assume 2^(aleph_0) means to raised to that power. What does the power operation mean in this context? How do you raise a number to a power that isn't a number?

Iranon wrote:
Now, for most practical purposes it is convenient to assume that 2^(aleph_0) is actually aleph_1, the first uncountable cardinal.


Why is it necessary to assume there is a first? Look at the real numbers? If you assume x is the first positive real number then what is x/2? I can see an assumption like that causing problems down the road. Not saying I disagree as I'm certainly not aware of an uncoutable set with a cardinal less then the reals but this is a part I don't fully understand.

Iranon wrote:
This is called the continuum hypothesis (CH), and for a long time it was an open problem as to whether or not this was actually true. If you accept this as an axiom, then no, any set strictly larger than the naturals is at least as large as the reals.

But in modern times, it's a fairly routine exercise in set theory to show that this is actually independent of the commonly accepted set-theoretic model for mathematics (ZFC). In layman's terms, that means that [all of regular math] + [the reals have size aleph_1] is a completely sensible, non-contradictory system, but so is [all of regular math] + [the reals have size strictly larger than aleph_1]. (However, it is provable in ZFC that aleph_1 is the smallest the reals could be, since they're obviously uncountable). In fact, there are models of set theory where the reals have size aleph_n, where n is any natural number you want. That's a pretty cool result. So for example, you can build a model of set theory where the naturals have size aleph_0, the reals have size aleph_317, and then there obviously are many, many, many sets of size strictly larger than the naturals and strictly smaller than the reals.


So if the reals have size aleph_317, how do you produce a set with size aleph_316? Let's assume I can produce such a set and we call it George. This means George exists in the completely sensible and non-contradictory system [all of regular math] + [the reals have size strictly larger than aleph_1]. Correct? So what happens to George in the completely sensible and non-contradictory system [all of regular math] + [the reals have size aleph_1]. Does George still exist and if so is George now the same size as the reals? I just don't know. Isn't this now a contradiction? George can't both be smaller then the reals AND the same size as the reals. And once George has been created I don't see how it can stop to exist when I go from one completely sensible and non-contradictory system to another.

If you can't create a set with size aleph_316 that is smaller then the reals then what is the advantage of talking about the size of the reals as aleph_317 in the first place? You are over my head at this point.

Iranon wrote:
To Kelvin Stott, I think you were being facetious by posting infinity^infinity^infinity^infinity^..., but you might be interested to know that while that doesn't mean anything, omega^omega^omega^omega^... does. (Omega being the first infinite ordinal, and the set of natural numbers. It's what most people are actually talking about when they write the sideways 8 symbol but don't know it). Specifically, omega^omega^omega^omega^... is called epsilon_0, and it's the first ordinal which cannot be expressed as finitely many arithmetical operations applied to omega.


Ok... lost me again. Arithmetical operations are defined for numbers. Ordinals aren't numbers. Would you agree with that statement? So help me here. What do expressions like these mean?

omega+omega
omega-omega
omega*omega
omega/omega
omega^omega
omega!
sqrt(omega)
ln(omega)

I have no idea what these mean or even if they all mean anything.

Iranon wrote:
It is countable, however, so it's still the same size as the natural numbers.


Define "it". I'm now totally lost. Are we talking about the set of all ordinals? aleph_0 and omega are the same thing... correct? And if 2^(aleph_0) is an uncountable cardinal how is aleph_0^aleph_0^aleph_0^aleph_0^... countable. Maybe I'm missing something... what is the difference between ordinals and cardinals?

Iranon wrote:
Surprising but true. And as far as ordinals go, epsilon_0 is still very small, since it's countable, but it's one of my favorites, since it is also the proof strength ordinal of Peano arithmetic.


Ok... you are so far over my head at this point I don't even know what to ask anymore... maybe this one physicist went to the bar and drank as much as an infinite number of mathematicians once too often.

Carl

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 Post subject: Re: Big numbers D:
PostPosted: Mon Aug 01, 2011 1:18 pm 
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wwwmwww wrote:
Ok... let me preface this by saying I'm a physicist... not a mathematician. I love trying to wrap my brain around some of these abstract concepts but I'm a bit out of my element when I do. So I many not also be 100% correct or even stating things in a correct way at times. So bear with me...

Iranon wrote:
It's not that no one has made an example, it's that there are consistent models of set theory with arbitrarily many sizes strictly between the naturals and the reals. The naturals have size aleph_0, as does any countable infinite set. The reals have size 2^(aleph_0), which is an uncountable cardinal (i.e. is strictly larger than any countable set).


aleph_0 is a "size". When you get right down to it... it isn't a number. I assume 2^(aleph_0) means to raised to that power. What does the power operation mean in this context? How do you raise a number to a power that isn't a number?

This is going to be a lot easier since you already know the difference between ordinals (well-ordered sets) and cardinals (sizes of sets). For cardinals, 2^kappa is defined to be the size of the power set of kappa. You should be able to convince yourself that the reals are the same size as the power set of the naturals.
wwwmwww wrote:

Iranon wrote:
Now, for most practical purposes it is convenient to assume that 2^(aleph_0) is actually aleph_1, the first uncountable cardinal.


Why is it necessary to assume there is a first? Look at the real numbers? If you assume x is the first positive real number then what is x/2? I can see an assumption like that causing problems down the road. Not saying I disagree as I'm certainly not aware of an uncoutable set with a cardinal less then the reals but this is a part I don't fully understand.


You're right to be wary, but what I said is okay. We need to go to the distinction between ordinals and cardinals again. omega_1 is defined to be the set union of all countable ordinals. That means that by definition, it is (a) an ordinal, (b) larger than any countable ordinal, and (c) the smallest thing with those two properties. Note that (b) means "uncountable". aleph_1 is defined to be the smallest cardinality strictly larger than aleph_0.
wwwmwww wrote:

Iranon wrote:
This is called the continuum hypothesis (CH), and for a long time it was an open problem as to whether or not this was actually true. If you accept this as an axiom, then no, any set strictly larger than the naturals is at least as large as the reals.

But in modern times, it's a fairly routine exercise in set theory to show that this is actually independent of the commonly accepted set-theoretic model for mathematics (ZFC). In layman's terms, that means that [all of regular math] + [the reals have size aleph_1] is a completely sensible, non-contradictory system, but so is [all of regular math] + [the reals have size strictly larger than aleph_1]. (However, it is provable in ZFC that aleph_1 is the smallest the reals could be, since they're obviously uncountable). In fact, there are models of set theory where the reals have size aleph_n, where n is any natural number you want. That's a pretty cool result. So for example, you can build a model of set theory where the naturals have size aleph_0, the reals have size aleph_317, and then there obviously are many, many, many sets of size strictly larger than the naturals and strictly smaller than the reals.


So if the reals have size aleph_317, how do you produce a set with size aleph_316? Let's assume I can produce such a set and we call it George. This means George exists in the completely sensible and non-contradictory system [all of regular math] + [the reals have size strictly larger than aleph_1]. Correct? So what happens to George in the completely sensible and non-contradictory system [all of regular math] + [the reals have size aleph_1]. Does George still exist and if so is George now the same size as the reals? I just don't know. Isn't this now a contradiction? George can't both be smaller then the reals AND the same size as the reals. And once George has been created I don't see how it can stop to exist when I go from one completely sensible and non-contradictory system to another.

If you can't create a set with size aleph_316 that is smaller then the reals then what is the advantage of talking about the size of the reals as aleph_317 in the first place? You are over my head at this point.


Here's where we need to be very careful to maintain a difference between ordinals and cardinals. Cardinals measure size, as you know. Ordinals are just well-ordered sets of various lengths (called order-types), which is not quite the same. For the first question you asked, the answer is simple. aleph_316 is already defined, it is literally the 316th largest cardinality an infinite set can have, in the same way that aleph_0 is the smallest an infinite set can be, aleph_1 is the next size up, then aleph_2, and so on. So that's a perfectly well-defined cardinality for a set to have, and it doesn't matter what the elements are -- take any well-ordering of cardinality aleph_316 and there's George. It's just a set, albeit a very big one.

I'm glad you asked your next question, what the heck happens to George in ZFC+CH... Your misconception is that you're asking "what size is this set" when you need to be asking "what size does this model say this set is?". When you vary the axioms you're working with (for example by accepting/rejecting CH), you're working in a different system, with different provable truths/falsehoods. Our old model, without CH, could prove the statement "reals have size 2^aleph_0, which is aleph_317, and george has size aleph_316." With CH, those sizes actually just collapse. Our new model proves "reals have size 2^aleph_0, which is aleph_1, and george has size aleph_0." Imagine all the ordinals laid out in a line, starting with 0 and extending infinitely far to the right. Notice that no matter what model we're working in, huge swaths of those ordinals are clumped together as having the same cardinality. (More on this in the answer to your next question). There are all the finite numbers, followed by all the countable ordinals (which all have size aleph_0), followed by omega_1, and then all the uncountable ordinals forever. Our first model essentially broke up the countable ordinals into lots of different cardinalities (316 of them), and then said okay, that's all the countable guys, omega_1 is the next size up, at aleph_317. Our second model said no, all those countable guys are the same size, they're all just aleph_0, but as soon as I hit uncountable guys they're clearly a larger cardinality, so omega_1 has size aleph_1.

The problem is that we like to think of math as "these are true statements" and "these are false statements", but once you start messing around with different axioms, you really need to be saying "these are true statements in this particular model" and so on. Some statements are true in all (consistent) models, some statements are false in all models, but most statements depend on what your axioms say. To go a little further into this particular case (so it makes a little more sense), remember that "cardinality" is really tied up with "what bijections exist between which sets". If there's an injection from one set to another, the range set has cardinality at least as large as the domain set. If there's an injection the other way too, then they have the same cardinality. That's just what cardinality means. But the key detail is that the existence of functions is tied to particular models. For example, suppose you have some countable model of set theory (ZFC). That means there are only countably many elements in its universe. Now, that model proves all the things that ZFC normally does, for example "there is a set called the real numbers which is uncountable". How can this be? There are only countably many elements in the model! Well, what's really going on is the model says "okay, here's my set that I think is the reals. What size is it? Hmm, let me look at all the bijections I know about. I don't see any bijections between that set and the natural numbers, therefore it must be uncountable". Of course, we as outside observers know that there are such functions, since that particular set is countable, but those functions aren't part of the model, so it "knows" they don't exist. Anyway, we've gotten pretty fair afield of your question, but the point I was making is that asking what cardinality a set has is something you do *in the confines of a particular model*.
wwwmwww wrote:

Iranon wrote:
To Kelvin Stott, I think you were being facetious by posting infinity^infinity^infinity^infinity^..., but you might be interested to know that while that doesn't mean anything, omega^omega^omega^omega^... does. (Omega being the first infinite ordinal, and the set of natural numbers. It's what most people are actually talking about when they write the sideways 8 symbol but don't know it). Specifically, omega^omega^omega^omega^... is called epsilon_0, and it's the first ordinal which cannot be expressed as finitely many arithmetical operations applied to omega.


Ok... lost me again. Arithmetical operations are defined for numbers. Ordinals aren't numbers. Would you agree with that statement? So help me here. What do expressions like these mean?

omega+omega
omega-omega
omega*omega
omega/omega
omega^omega
omega!
sqrt(omega)
ln(omega)

I have no idea what these mean or even if they all mean anything.


I do agree with those statements. However, addition, multiplication, and exponentiation *are* defined for ordinals as well. They are similar, in that ordinal +/*/^ agree with finite +/*/^ on finite inputs, but obviously they're different operations. Subtraction, division, roots, etc are not defined. Factorial is only defined on the naturals (though the Gamma function "extends" ! to the complex plane), and so on. The actual definitions are very technical, so I'll spare you the tedium and spare me the myriad opportunities to mis-remember them, but remember that ordinals are really just well-orderings. So for example, omega+omega says "take a list of length omega, and append another list of length omega to the end of it". Doing that process again would give you omega+omega+omega, and you can see that we're on our way to multiplication, that's omega*3. Warning: ordinal multiplication is *not* commutative. 3*omega and omega*3 are different objects. omega*omega says "take omega-many lists of length omega, and string them all together end-to-end". Doing that process again would give you omega*omega*omega, and so on until we get to omega^omega. For cardinal exponentiation we define a^b in terms of the number of functions from a to b, but this is close enough to get a general intuition. So for example when I said earlier that 2^aleph_0 was the size of the power set of aleph_0, that's because you can represent any subset of aleph_0 by a function from aleph_0 to {0,1}, taking value 1 if the element is in the set and 0 otherwise...

wwwmwww wrote:

Iranon wrote:
It is countable, however, so it's still the same size as the natural numbers.


Define "it". I'm now totally lost. Are we talking about the set of all ordinals? aleph_0 and omega are the same thing... correct? And if 2^(aleph_0) is an uncountable cardinal how is aleph_0^aleph_0^aleph_0^aleph_0^... countable. Maybe I'm missing something... what is the difference between ordinals and cardinals?


Nope. "It" here is omega^omega^omega^.... That object is probably best viewed as the smallest ordinal which is at least as large as omega, and omega^omega, and omega^omega^omega, and so on for every finite tower of exponents. The thing you're missing is very subtle... Ordinal exponentiation and cardinal exponentiation are different (see the last sentence above, cardinal exponentiation is *not* repeated multiplication). aleph_0^aleph_0^... is indeed uncountable, just like 2^aleph_0 is, but omega^omega is not, and neither is omega^omega^omega^... Remember how a countable union of countable sets is countable? Well, 2^aleph_0 (or aleph_0^aleph_0) ask "how many functions are there from 2 (or aleph_0) to aleph_0? Clearly uncountably many. But 2^omega and omega^omega say "take omega-many sets of size 2 and stick them together end-to-end" and "take omega-many sets of size omega and stick them together end-to-end". Both of those are clearly countable. From there's it's not a far jump to see that omega^omega^omega^... is also a countable union of countable sets.
wwwmwww wrote:

Iranon wrote:
Surprising but true. And as far as ordinals go, epsilon_0 is still very small, since it's countable, but it's one of my favorites, since it is also the proof strength ordinal of Peano arithmetic.


Ok... you are so far over my head at this point I don't even know what to ask anymore... maybe this one physicist went to the bar and drank as much as an infinite number of mathematicians once too often.

Carl


Ha! Just in case your brain isn't completely turned to jelly by now, what I meant by "proof strength ordinal of PA" is this. PA is basically the framework for arithmetic, it's just a bunch of axioms that say "the natural numbers exist and mathematical induction is a valid thing to do", more or less. Now, remember when we were saying that sometimes a model will fail to contain certain functions, and as a result will think all sorts of crazy things about the sizes of some sets? It can get worse. Ordinals are just long well-ordered lists, and you should be able to check if something is a well-ordered list or not, right? Weeeellllll, sometimes. It turns out that any given model will correctly identify ordinals up to a point, but eventually the ordinals will get so big that the model just fails to identify them as a well-ordering. However, just like our countable model of ZFC knew that the reals were uncountable, so it had a (countable) set that it thought (erroneously) was the set of real numbers, these models will still have sets that it thinks are these huge ordinals, and it will think those sets have the properties that those huge ordinals are supposed to have, but it will be wrong about what sets they actually are. For PA, that breakpoint is at omega^omega^omega^... For all the ordinals smaller than that, PA says "yep, that's an ordinal", but once you hit that one PA says "I can't tell if that's even an ordinal anymore".

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 Post subject: Re: Big numbers D:
PostPosted: Mon Aug 01, 2011 2:46 pm 
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Thank you Iranon for your extensive post!

I didn't want to do the entire lecture on Cantors "Mengenlehre" in a topic 'just' about big interesting numbers. But here you go! you did really well explaning it all, thanks again.

Just got a tweet from "mathematicsprof" on the latest new on the continuuum hypothesis, just to let you know.

Please check To Infinity and Beyond

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 Post subject: Re: Big numbers D:
PostPosted: Mon Aug 01, 2011 3:35 pm 
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maarten wrote:
Thank you Iranon for your extensive post!
Yes, thank you Iranon. I'll have to fry a few brain cells before I'm sure I understand all of that but I think I followed most of that. The difference between ordinals and cardinals I think is what had me confused the most.
maarten wrote:
Thank you maarten. That was a very interesting article.

Ok... another question that I think makes sense.

(1) Is the set of all cardinals countable?

The article talks about "a step in a staircase leading to ever-higher levels of infinities stretching up as far as, well, infinity." We start with aleph_0 and aleph_1 appears to be defined as the first uncountable cardinal. How do we know there IS a first? Just as with the reals there isn't a first positive number couldn't it also be the case that the set of all cardinals is a continuum and therefore be uncountable?

Thanks,
Carl

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 Post subject: Re: Big numbers D:
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wwwmwww wrote:
Ok... another question that I think makes sense.

(1) Is the set of all cardinals countable?

The article talks about "a step in a staircase leading to ever-higher levels of infinities stretching up as far as, well, infinity." We start with aleph_0 and aleph_1 appears to be defined as the first uncountable cardinal. How do we know there IS a first? Just as with the reals there isn't a first positive number couldn't it also be the case that the set of all cardinals is a continuum and therefore be uncountable?

Thanks,
Carl


First, not only are there uncountably many cardinals (or ordinals), there are so many cardinals that they don't even form a set. The collection of all cardinals (or ordinals) is something called a proper class. Remember when I said described aleph_1, aleph_2, and so on, and each aleph_n was the smallest cardinality above aleph_(n-1)? Well, the natural thing to do would be then ask if there's an aleph_omega, which is bigger than all the aleph_n... Yes, there is. In fact, the aleph numbers are indexed by any *ordinal*, not by any natural number, and if you assume CH then every cardinal is of the form aleph_a for some ordinal a. I think you were imagining aleph_omega and thinking that would be the biggest cardinal. But no, you can keep going, just like the ordinals keep going past omega. aleph_(omega+1) is the next one, for example. You're confusing the the word "continuum" being used on the one hand for a dense linear order without endpoints (like the reals) and on the other hand being used for the size of the reals (just uncountable).

Second, you're right to be suspicious of the existence of a "first" uncountable cardinal, but think of it this way. The first uncountable cardinal is the size of the first uncountable ordinal. Okay, great, so now we have to show that there is such an ordinal. First we need a few facts about what ordinals really are. They are just sets which are well-ordered (linearly ordered with no infinite descending chains) by inclusion. The standard way of representing them (due to von Neumann) is to start by identifying the ordinal 0 with the empty set. From there, there are two kinds of ordinals, successor ordinals (which are the kinds that come immediately after another one, like 1, 2, 3, and so on), and limit ordinals (which are the ones that don't have an immediate predecessor, like omega).

For the first kind, the successor of an ordinal a is defined to be the set a union {a}. So writing 0 for \emptyset, the ordinal 1 is {0}, the ordinal 2 is 1 union {1}, which is the same as {0,1}, the ordinal 3 is 2 union {2}, which is the same as {0,1,2}, and so on. Limit ordinals are defined to be the set of all ordinals below them, so for example omega is the set {0,1,2,3,...} of natural numbers. Notice that every ordinal is literally the set of ordinals below it. This makes for many nice properties, the most important of which is that if you have two ordinals, one of them must be smaller, and that one is literally a subset of the other. For that reason, think about what happens when you take the set union of a bunch of ordinals. If any one of them is the largest, then clearly the others are all subsets, so you just get the largest one. But if there is no largest one, then you get a set which is still well-ordered and contains all of them -- and that's exactly what a limit ordinal is. You should be able to convince yourself that taking unions of ordinals is the same idea as taking the supremum of them. Also because of the well-ordering, no ordinal can be a subset of itself, since no ordinal is larger than itself. Among other things, that means there's no largest ordinal, since if there was one, it would be an ordinal which contains every ordinal, including itself. On another side note, the ordinals are not values, they're sets, so while you can visualize them as all lying on a number line, they are discrete points with gaps between the successor ordinals and a dense cloud of points to the left of all the limit ordinals.

Now, what happens if we take the union of all countable ordinals? We know we get an ordinal, since any union of ordinals is an ordinal. We know that ordinal is bigger than all the countable ordinals, since it contains them all as subsets, and hence can't be countable. And any ordinal which is smaller than that one we just constructed must be a subset of it, but by definition all its subsets are countable. So we've got an uncountable ordinal such that every ordinal below it is countable. That's the first uncountable ordinal. Notice that this situation is exactly analogous to how there's no last finite number, but if you union them all up you get omega, the first infinite ordinal. There's no last countable ordinal, but if you union them all up you get omega_1, the first uncountable ordinal.


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 Post subject: Re: Big numbers D:
PostPosted: Tue Aug 02, 2011 10:10 am 
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An interesting discussion, but in the main incomprehensible.

My favourite number is e^(pi*i) :)

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 Post subject: Re: Big numbers D:
PostPosted: Tue Aug 02, 2011 10:35 am 
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Gus wrote:
My favourite number is e^(pi*i) :)

That's neither big nor real, nor rational, but interesting how often it pops up everywhere... :D

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 Post subject: Re: Big numbers D:
PostPosted: Tue Aug 02, 2011 10:43 am 
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Er e^(pi*i) = -1 which is real and rational!
One of the most beautiful equations out there that also brings together most of the important things in maths (yes I am English) e^(pi*i) + 1 = 0.

Iranon - one thing to say to you... Ultimate-L ! discuss :P :shock:

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 Post subject: Re: Big numbers D:
PostPosted: Tue Aug 02, 2011 10:58 am 
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oxymoronicuber wrote:
Er e^(pi*i) = -1 which is real and rational!
One of the most beautiful equations out there that also brings together most of the important things in maths (yes I am English) e^(pi*i) + 1 = 0.

Whoa, you're right. I typed it into mathematica and it gave -1. Completely unexpected, I've never seen this before, it's amazing, like the E=mc^2 of mathematics! Who discovered this, where does it come from??

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 Post subject: Re: Big numbers D:
PostPosted: Tue Aug 02, 2011 11:22 am 
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Iranon are you a mathematician? If not you certainly could be.

Anyways... what I don't get is how the cardinals could be discrete (not continous) and still not be countable.

You talk about limit ordinals and successor ordinals. Can I then also assume there are limit and successor cardinals?

If so it appears you could present them in a table like this

0, 1, 2, 3, 4, 5, ...
aleph_0, aleph_1, aleph_2, aleph_3, aleph_4, aleph_5,...
aleph_omega, aleph_(omega+1), aleph_(omega+2), aleph_(omega+3), aleph_(omega+4), ...
aleph_(omega+omega), would the next one be aleph_(omega+omega+1)? I'm guessing at this point.
.
.
.


The first column is the limit cardinals and the remaining columns are all the successor cardinals which follow it.

Then once you have this tabel you could use an argument just as presented here:

http://www.jcu.edu/math/vignettes/infinity.htm

to map them to the positive integers. If they aren't countable, where does this argument fail?

Let me guess... going down the first column it takes forever to get to aleph_(omega*omega) so you don't have aleph_(omega^omega) in this table?

Its just that my mind wants to equate countable with discrete and uncountable with continous.

Carl

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 Post subject: Re: Big numbers D:
PostPosted: Tue Aug 02, 2011 11:54 am 
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Kelvin Stott wrote:
oxymoronicuber wrote:
Er e^(pi*i) = -1 which is real and rational!
One of the most beautiful equations out there that also brings together most of the important things in maths (yes I am English) e^(pi*i) + 1 = 0.

Whoa, you're right. I typed it into mathematica and it gave -1. Completely unexpected, I've never seen this before, it's amazing, like the E=mc^2 of mathematics! Who discovered this, where does it come from??

Euler. See:

http://en.wikipedia.org/wiki/Euler%27s_identity

I seem to recall reading somewhere that either Euler (or Gauss?) said that if it isn't immediately obvious why e^pi*i + 1 = 0 you won't ever be a great mathematician.

I suppose I won't ever be great because e^ix = i * sin x + cos x still requires a ton of thought for me.

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 Post subject: Re: Big numbers D:
PostPosted: Tue Aug 02, 2011 12:03 pm 
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bmenrigh wrote:
Thanks!

bmenrigh wrote:
I seem to recall reading somewhere that either Euler (or Gauss?) said that if it isn't immediately obvious why e^pi*i + 1 = 0 you won't ever be a great mathematician.

I suppose I won't ever be great because e^ix = i * sin x + cos x still requires a ton of thought for me.
I wouldn't say I'm bad at maths as I came top at uni (many years ago), but I'd have to agree - I'll never be great! Just as well I didn't stick to it as my day job. :lol:

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 Post subject: Re: Big numbers D:
PostPosted: Tue Aug 02, 2011 12:14 pm 
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wwwmwww wrote:
Iranon are you a mathematician? If not you certainly could be.

Anyways... what I don't get is how the cardinals could be discrete (not continous) and still not be countable.

You talk about limit ordinals and successor ordinals. Can I then also assume there are limit and successor cardinals?

If so it appears you could present them in a table like this

0, 1, 2, 3, 4, 5, ...
aleph_0, aleph_1, aleph_2, aleph_3, aleph_4, aleph_5,...
aleph_omega, aleph_(omega+1), aleph_(omega+2), aleph_(omega+3), aleph_(omega+4), ...
aleph_(omega+omega), would the next one be aleph_(omega+omega+1)? I'm guessing at this point.
.
.
.


The first column is the limit cardinals and the remaining columns are all the successor cardinals which follow it.

Then once you have this tabel you could use an argument just as presented here:

http://www.jcu.edu/math/vignettes/infinity.htm

to map them to the positive integers. If they aren't countable, where does this argument fail?

Let me guess... going down the first column it takes forever to get to aleph_(omega*omega) so you don't have aleph_(omega^omega) in this table?

Its just that my mind wants to equate countable with discrete and uncountable with continous.

Carl


I'm a math grad student at the moment. So no, I'm not a mathematician, but yes, that's my goal for the future.

Are there successor and limit cardinals? Absolutely. Cardinals are just special types of ordinals. (Specifically, they are ordinals which have cardinality equal to themselves. omega has size omega, so that's a cardinal. omega+1 has size omega, so that's not a cardinal. And so on. The successor of a cardinal k, written k+, is basically the size of the least ordinal bigger than k. What's the successor of the cardinal 5? Well, 5 is the set {0,1,2,3,4} as an ordinal, and the smallest ordinal which contains that as a strict subset is {0,1,2,3,4,5}, which is the ordinal 6. So 5+=6, as we expect. Limit cardinals are just defined to be all the other ones -- cardinals which are not the successor of any smaller cardinal.

Anyway, the crux of what you're doing is trying to use a diagonalization argument to conclude that there are only countably many ordinals. You are completely correct that there are only countably many ordinals which you can write down in terms of addition and multiplication of finite numbers and powers of omega. Remember that omega^omega^omega^... that we talked about before? (It's called epsilon_0). It turns out that every ordinal below epsilon_0 is indeed equal to some expression that looks more or less like a "polynomial in omega", like (omega*omega*omega)*4+omega*19+omega+9384, or something like that. Of course they aren't really polynomials in any meaningful sense, but you get the idea. Past epsilon_0, they can't be written down like that. Now, epsilon_0 is still countable, so if we came up with some kind of crazy notation that worked perhaps we could still continue our list, but there's no way we'll ever get past omega_1 and beyond. omega_1 is an uncountable ordinal, great. So try and list off the ordinals that are smaller than it -- that's all the countable ordinals. In the back of your mind we're trying to list these off all as subscripts for alephs, but it's the same thing. How many countable ordinals are there? Well, as a set, remember ordinals are literally the set of ordinals below them. So there are exactly omega_1-many ordinals below omega_1. In other words, there are uncountably many countable ordinals. And that's what you were trying to avoid. So any list or grid or whatever you try to do to list off all the aleph_a cardinals will fail, and more specifically it will fail somewhere before aleph_(omega_1).


oxymoronicuber wrote:
Iranon - one thing to say to you... Ultimate-L ! discuss :P :shock:


Eh. I tend to agree with Hamkins. It's a nice article, and a rare thing for not only model theory but large cardinal axioms (which are very obscure, and something my eventual career very well may go into) makes it to the public sphere. But my general feel is that no, there is no one correct model of set theory and mathematics which is the "correct" one, just infinitely many consistent models, some with slightly nicer or slightly worse properties. I'd have to read Woodin's actual paper to see what it actually was, as the article was necessarily vague about it, but I would be very surprised if the larger mathematical community accepted it as anything other than a beautiful accident.


Kelvin Stott wrote:
Whoa, you're right. I typed it into mathematica and it gave -1. Completely unexpected, I've never seen this before, it's amazing, like the E=mc^2 of mathematics! Who discovered this, where does it come from??


That identity carries Euler's name, but it's doubtful that he was the first to actually discover it. It's fundamental to complex analysis. For a quick justification of why it's true, the complex exponential function is defined as e^a=cos(a)+isin(a), and then the result is just a simple calculation, but that should feel like cheating. That's the definition of the complex exponential function because sin and cos work like that in the complex plane, not the other way around. There are lots of ways to prove it, but I think looking at power series is the best way. If you know what Taylor series are, this should make perfect sense.

Taylor series for sin(x):
x-x^3/3!+x^5/5!-x^7/7!+...
Taylor series for cos(x):
1-x^2/2!+x^4/4!-x^6/6!+...
Taylor series for e^x:
1+x+x^2/2!+x^3/3!+x^4/4!+...

Now plug in iz for x in e^x. You get
1+iz+(iz)^2/2!+(iz)^3/3!+(iz)^4/4!+(iz)^5/5!+(iz)^6/6!+...
=1+iz-z^2/2!-iz^3/3!+z^4/4!+iz^5/5!-z^6/6!+...
=(1-z^2/2!+z^4/4!-z^6/6!+...)+i(z-z^3/3!+z^5/5!-...)
=cos(z) + i sin(z).

Which is pretty damned neat. Of course I'm sweeping all the technical details under the rug about why real Taylor series are still valid in the complex plane, and the convergence of those series, and so on, but that's the idea. Trust me that the details check out okay, and are not very illuminating.


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 Post subject: Re: Big numbers D:
PostPosted: Tue Aug 02, 2011 10:53 pm 
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wwwmwww wrote:
Its just that my mind wants to equate countable with discrete and uncountable with continous.

Ok... I think I can see past this notion now. Isn't the set of all possible subsets of the integers uncountable? That would be another discrete set if so.

I'm now wondering what it really means to be continous. I don't think the rationals are considered continous but the gap between then can become arbitrarily small. So what exactly is it that makes the reals continous? I remember reading of other number systems which are super sets of the reals and include numbers called infinitesimals such that zero + an infinitesimal is smaller then any real positive number. Doesn't this imply that there are then gaps in the real numbers or how else do you make room for such numbers? And how would you quantify the gaps in the rationals? Is this another topic that touches on the particular model you are working in? In one model the reals are continous and in other (one that includes the infinitesimals) the aren't? In the normal model I assume all the infinitesimals would be indistinguishable from zero.

And while we are talking about odd number systems has anyone ever done any work with the hyper complex numbers? My high-school senior math text had one page on the topic (this is ~25 years ago... give or take a few) and I've never really seen them mentioned since. As I recall it had three different square roots of -1. In addition to i it also had j and k. It had properties like:

i*i = j*j = k*k = -1
i*j = k
j*i = -k
etc.

Which makes me think of a few questions I don't know the answers to.

(1) Is the set of complex numbers bigger then the reals?
(2) Are the hyper complex numbers a bigger set then the complex numbers?
(3) Aside from being very interesting things to twist one's mind around... are there practical applications of these different levels of infinity? Not trying to knock abstract math, I enjoy it myself and I also know many apparently totally abstract ideas have found there way into physics. A good example would be the Dirac delta function. It just seems the visable universe is finite and space and time are quantitized so in reality we never have to deal with a true infinite... but countable set. We'll never have ALL the digits of pi for example so I'm curious what knowing the set of reals is larger then the set of integers really gets us, let alone under what conditions would something like (omega*omega*omega)*4+omega*19+omega+9384 come up?

I guess its questions like that last one that got me into physics and not math... but I've had friends that went on to get Ph.D's in math and it was my last year in grad school that Fermat's Last Theorem was proven. That's something I'd love to understand but I doubt I ever will.

Carl

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 Post subject: Re: Big numbers D:
PostPosted: Wed Aug 03, 2011 12:59 am 
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To take a break from the infinite to discuss the ridiculously(understatement of all time) large yet finite:

TBTTyler wrote:
My favorite is what I've heard called the xkcd number. Take the Ackermann function and use Graham's number as both of the arguments.
http://www.xkcd.com/207/


Here is a question:

Which is larger: A(G64, G64) or G(G64)?

Also, which sequence grows more quickly?
A(G64, G64), A(A(G64, G64), A(G64, G64)), A(A(A(G64, G64), A(G64, G64)), A(A(G64, G64), A(G64, G64))) ...
or
G(G64), G(G(G64)), G(G(G(G64))), G(G(G(G(G64))))...

Also, could a power tower containing a googol levels of googol come close to any of the above numbers?

Also, with Euler's Equation, I kind of prefer the form e^(tau*i) = 1. Now if someone could find an elegant way to relate e, tau, i, and 1 with phi.

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 Post subject: Re: Big numbers D:
PostPosted: Wed Aug 03, 2011 3:26 am 
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I think e^(pi*i)+1=0 is the most beautiful equation in maths because it is simple and contains five of the most important constants in mathematics. I was blown away when I first saw it.

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 Post subject: Re: Big numbers D:
PostPosted: Wed Aug 03, 2011 4:37 am 
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wwwmwww wrote:
(1) Is the set of complex numbers bigger than the reals?
(2) Are the hyper complex numbers a bigger set then the complex numbers?
(3) ...are there practical applications...


(1) No, you can write x+iy as one single real by writing down alternating digits of x and y. This maps every real to uniquely to a complex number (and the other way around)

(2) No, same reasoning

btw: during my math master (well, it was called drs at the time in the Netherlands) we used to call them Quaternions

(3) I can only think of indirect applications (as part of a proof).

@Gus, @Kelvin Stott: The formula actually won a beauty contest for formula's (2nd was F=ma) and a math-beauty contest (2nd was Euclid's proof the number of primes is (countable!) infinite). You can even buy shirts with the formula on it!

@Iranon: Euler had a simpler (and in my opinion nicer) proof for e^iz = cos(z) + i sin(z) using only a formula for cos(a+b) and lim(1+1/n)^n =e, n>inf (the original limit that led from interest calculations to the 'discovery' of e).

(does anyone consider this thread being derailed? I still like it!)

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 Post subject: Re: Big numbers D:
PostPosted: Wed Aug 03, 2011 10:10 am 
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wwwmwww wrote:
And while we are talking about odd number systems has anyone ever done any work with the hyper complex numbers? My high-school senior math text had one page on the topic (this is ~25 years ago... give or take a few) and I've never really seen them mentioned since. As I recall it had three different square roots of -1. In addition to i it also had j and k. It had properties like:

i*i = j*j = k*k = -1
i*j = k
j*i = -k
etc.
The numbers you're mentioning are called Quaterions. The iconic definition of them is
i^2 = j^2 = k^2 = i j k = -1
And they are absolutely useful. They can be used to represent rotations in graphics/modeling. The great advantage of quaternions is that they do not suffer from gimbal lock like you would have with a roll/pitch/yaw system. I also think they are just plain easier to use.

There are other hyper complex systems such as Octonion(8 parts), and Sedenion(16 parts). You actually create any 2^n sized hyper complex system by constructing it from the 2^(n-1) system. Though it seems that you seem to lose certain mathematical properties along the way. Quaterions lose the communicative property of multiplication. Octonions lose the associative property of multiplication. Sedenions lose something called "alternative" which I don't know much about.

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 Post subject: Re: Big numbers D:
PostPosted: Wed Aug 03, 2011 10:48 am 
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GuiltyBystander wrote:
The numbers you're mentioning are called Quaterions. The iconic definition of them is
i^2 = j^2 = k^2 = i j k = -1
And they are absolutely useful.
Interesting... I'll have to look more into these. By the way I mentioned pi yesterday and that caused my mind to go off on another tangent last night. Unsure if I should put that here I started a new thread.

Carl

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