Some Questions About God

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Re: Some Questions About God

Postby Мастер » Fri Jul 15, 2016 8:11 am

LLance has summoned me. The person who is really needed here is caveman1917 from CQ, but since he hasn't posted there in over a year, I'll attempt to carry the cave-torch for him.

Lance wrote:Who ever said it is a number? I think infinity exists outside the realm of numbers. It is something you can go toward but never reach. Kind of like that one 18 year old blonde girl.


It's a number if we decide it is a number. But more critically than whether we decide to call infinity a number or not, what are the properties of this infinity? Does it behave like a number, or is it somehow quite different than the other numbers? Therein lies the rub.

Lance wrote:
Dragon Star wrote:I can use an equation with infinity to prove that 1=0 which is false, so I'm skeptical that it's possible.

Can you show us that equation?


I think it is very important in such discussions to follow one rule, which is, do not believe anything, no matter how obvious it seems, unless you can prove it rigorously, step-by-step. In the world of internet mathematical proofs, if you want to know where the mistake is, look for the statement, "it is obvious that . . ." or "every child knows that . . ." - that's usually where you'll find it.

Dragon has presented an argument, but Llance questions the very first step, which does make an (unproven) assumption about how infinity behaves. Let's see whether we can take a different approach, but end up at the same destination.

Many of the familiar number systems are what are known as fields. A field is set of objects (often called "numbers"), together with two operations called "addition" (denoted by Image) and "multiplication" (denoted by Image), which follow eleven rules. These rules are not all that complicated, and should be mostly familiar from primary school arithmetic.

A1: If Image and Image are numbers, then Image is a number. (Additive completeness - you can add any two numbers, and the result is a number.)

A2: If Image and Image are numbers, then Image. (Commutativity of addition - it doesn't matter what order you add numbers in, you get the same answer either way.)

A3: If Image, Image, and Image are numbers, then Image. (Associativity of addition - it doesn't matter which order you do the addition operations in, you get the same answer either way.)

A4: There exists a number called Image such that, for any number Image, Image. (Existence of an additive identity - zero plus any number just gives you the same number right back.)

A5: For every number Image, there exists a number called Image, such that Image. (Existence of an additive inverse - every number, including zero, has a negative.)

M1: If Image and Image are numbers, then Image is a number. (Multiplicative completeness - you can multiply any two numbers, and the result is a number.)

M2: If Image and Image are numbers, then Image. (Commutativity of multiplication - it doesn't matter what order you multiply numbers in, you get the same answer either way.)

M3: If Image, Image, and Image are numbers, then Image. (Associativity of multiplication - it doesn't matter which order you do the multiplication operations in, you get the same answer either way.)

M4: There exists a number called Image such that, for any number Image, Image. (Existence of an multiplicative identity - one times any number just gives you the same number right back.)

M5: For every number Image, there exists a number called Image, such that Image. (Existence of a multiplicative inverse - every number, except zero, has a reciprocal.)

D1: For all numbers Image, Image, and Image, Image. (Distributivity of multiplication over addition.)

Note that the equality relation is reflexive (Image, every number is equal to itself), symmetric (if Image, then Image), and transitive (if Image and Image, then Image).

The rational numbers, the real numbers, and the complex numbers are all fields (with addition and multiplication defined in the standard way). The integers are not - the only integers with multiplicative inverses are Image and Image. So M5 does not hold for the integers.

Do these seem like good rules for numbers to follow? If so, we can have a look at what happens if we relax the annoying exception in M5 - every number except zero has a reciprocal. We can introduce a new number called "infinity" which is the reciprocal of zero, and see what happens if we insist that the new "infinity" number must satisfy the eleven rules above, along with all the other numbers. The conclusion that Monsieur Dragon reached is not far away.
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Re: Some Questions About God

Postby Heid the Ba » Fri Jul 15, 2016 10:31 am

With you so far Mactep.
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Re: Some Questions About God

Postby tubeswell » Fri Jul 15, 2016 10:52 am

which tends to further affirm the validity of my hypothesis
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Re: Some Questions About God

Postby Lance » Fri Jul 15, 2016 2:21 pm

:glp-s62:
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Re: Some Questions About God

Postby Lance » Fri Jul 15, 2016 9:33 pm

Мастер wrote:Do these seem like good rules for numbers to follow? If so, we can have a look at what happens if we relax the annoying exception in M5 - every number except zero has a reciprocal. We can introduce a new number called "infinity" which is the reciprocal of zero, and see what happens if we insist that the new "infinity" number must satisfy the eleven rules above, along with all the other numbers. The conclusion that Monsieur Dragon reached is not far away.

These seem like great rules!

My math skills end at high school algebra and trig. I even got lost on some high school geometry I was trying to help Fred (introduced elsewhere) with a couple of weeks ago. I can do complex arithmetic in my head quickly, but that's where my knowledge ends. I understand how orbits work and multiple body trajectories, but I've never had calculus and could never figure them out for myself.

So to naïve me, I don't buy that infinity is a "number" like any other. To me, it is the superset of all numbers great and small. I don't buy it as the reciprocal of zero either. I think zero is contained within the set that is infinity.
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Re: Some Questions About God

Postby Мастер » Fri Jul 15, 2016 9:38 pm

OK, if everyone is happy so far, I'd now like to prove that zero times any number is zero. Note that this is not one of the rules shown above, but we can prove it.

Step 1: By A4, there exists a number called Image.

Step 2: By D1, for any numbers Image, Image, and Image, we have Image. Note that this step rather implicitly assumes A1 and M1, i.e., we must assume that the sums and products referenced in the statement of D1 exist.

Step 3: In the result in Step 2, from D1, set Image, since in Step 1, we know zero exists from A4. Then Image. This result holds for any Image and any Image.

Step 4: By A4, we know that Image.

Step 5: Taking the result from Step 3, we can replace Image by Image, since from Step 4, we know they are equal. The result is Image.

Step 6: By A5, there exists a number Image such that Image. If the nested parentheses are getting cumbersome, we can write this as Image, where we have created a new operation, called subtraction; Image is defined as Image.

Step 7: From Step 6, we know Image. From Step 5, we know Image. Therefore, we can replace the first occurrence of Image in the first expression by Image, since they are equal. The result is Image.

Step 8: By A2, we have Image. From Step 7, we have Image, so by applying A2, we get Image. Recall that this can be written without the subtraction operation, as Image.

Step 9: By A3, we have Image. Since we already know that Image, it follows that Image.

Step 10: We already know that Image. Since Image, it follows that Image.

Step 11: By A4, we know that Image. Since Image from Step 10, it must be the case that Image.

So there it is, in step 11, any number times zero is equal to zero. I have proven this using A1, A2, A3, A4, A5, M1, and D1. If you believe there is some number (let's call it "infinity") that, when multiplied by zero, does not result in zero, you must believe that at least one of A1, A2, A3, A4, A5, M1, or D1 is wrong.

So zero times anything is zero, unless one of the eleven rules is wrong. In fact, it has to be one of seven rules (A1, A2, A3, A4, A5, M1, or D1), since M2, M3, M4 and M5 were not needed in the proof.

OK so far? Just a few more steps until the Dragon equality, Image.
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Re: Some Questions About God

Postby Мастер » Fri Jul 15, 2016 9:43 pm

Lance wrote:
Мастер wrote:Do these seem like good rules for numbers to follow? If so, we can have a look at what happens if we relax the annoying exception in M5 - every number except zero has a reciprocal. We can introduce a new number called "infinity" which is the reciprocal of zero, and see what happens if we insist that the new "infinity" number must satisfy the eleven rules above, along with all the other numbers. The conclusion that Monsieur Dragon reached is not far away.

These seem like great rules!

My math skills end at high school algebra and trig. I even got lost on some high school geometry I was trying to help Fred (introduced elsewhere) with a couple of weeks ago. I can do complex arithmetic in my head quickly, but that's where my knowledge ends. I understand how orbits work and multiple body trajectories, but I've never had calculus and could never figure them out for myself.

So to naïve me, I don't buy that infinity is a "number" like any other. To me, it is the superset of all numbers great and small. I don't buy it as the reciprocal of zero either. I think zero is contained within the set that is infinity.


This post came in after I began to work on my last one, so I didn't see it. But, where my posts are going is, if you do try to define infinity as the reciprocal of zero, you run into some problems. Either you have to accept the dragon equality, that zero is equal to one, or you have to add some exceptions to some of the other rules. So having gotten rid of the irritating exception to the rules by introducing infinity, you are forced to introduce other exceptions.

You can conjure up a number called infinity, but it won't follow the rules above. It can't. Logically contradictory. The standard number systems don't have infinity in them for a reason - as soon as you introduce an "infinity" number, some of your rules get broken.
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Re: Some Questions About God

Postby Мастер » Fri Jul 15, 2016 9:54 pm

Thinking about this some more, the trajectory I'm on works by dropping the exception in rule M5. LLance, if you don't want to do that, there might be some other ways of showing that infinity "breaks" the rules. I'll have to think about it some more.
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Re: Some Questions About God

Postby tubeswell » Fri Jul 15, 2016 9:58 pm

:=D:
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Re: Some Questions About God

Postby Dragon Star » Fri Jul 15, 2016 10:23 pm

Howdy Mactep! Miss ya man!

Dragon Equality...nice ring to it.

As for the holy hell of a lot of math to prove or disprove my concept, much appreciated. I'm no math wizard by any means so I'm very interested to see the final proof. Thanks for that.

Tubeswell, I don't speak above IQ 140, lol, so you're on your own with your hypothesis. If I follow you're suggesting that based on this desconstuciting of maths that it would allow the continuum hypothesis to exist despite recent blabbing about by mathematician Katie Steckles using set to disprove it?
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Re: Some Questions About God

Postby Lance » Fri Jul 15, 2016 10:31 pm

Мастер wrote:Thinking about this some more, the trajectory I'm on works by dropping the exception in rule M5. LLance, if you don't want to do that, there might be some other ways of showing that infinity "breaks" the rules. I'll have to think about it some more.

I guess I don't think that infinity breaks the rules because the rules apply to numbers. And to me, infinity is not a number. You can't add 1 to infinity.
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Re: Some Questions About God

Postby Мастер » Sat Jul 16, 2016 2:53 am

Lance wrote:I guess I don't think that infinity breaks the rules because the rules apply to numbers. And to me, infinity is not a number. You can't add 1 to infinity.


So let's say, it would break the rules if it were a number.

So your way of dealing with infinity is to say that A1 (and I assume M1 as well) only applies to the set of numbers, which does not include infinity - so you can't perform maths with infinity. I think this is a very typical approach.

I have seen various attempts to turn infinity into a number, but always recognising that if you do that, you need to add some exceptions to these nice orderly rules I have written down. These attempts are often useful for various applications, but there doesn't seem to be a single, consistent way of doing it.

I have also seen (only on the internet) attempts to turn infinity into a number, and then apply all of the usual rules which apply to the other numbers, to infinity as well. When someone points out that this results in contradictions, the proponent always disproves this by calling the person bring forth the argument an idiot.
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Re: Some Questions About God

Postby Lance » Sat Jul 16, 2016 3:15 am

Мастер wrote:So let's say, it would break the rules if it were a number.

So your way of dealing with infinity is to say that A1 (and I assume M1 as well) only applies to the set of numbers, which does not include infinity - so you can't perform maths with infinity. I think this is a very typical approach.

This pleases me. I only know it as my approach. I'm glad I'm not alone.

Мастер wrote:I have seen various attempts to turn infinity into a number, but always recognising that if you do that, you need to add some exceptions to these nice orderly rules I have written down. These attempts are often useful for various applications, but there doesn't seem to be a single, consistent way of doing it.

Yeah. There was one a little ways back.

Мастер wrote:I have also seen (only on the internet) attempts to turn infinity into a number, and then apply all of the usual rules which apply to the other numbers, to infinity as well. When someone points out that this results in contradictions, the proponent always disproves this by calling the person bring forth the argument an idiot.

lol...

But I thought you were going to prove Dragon's conclusion?
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Re: Some Questions About God

Postby Мастер » Sat Jul 16, 2016 3:25 am

So if we're OK with what I've done so far, then I prove the Dragon Equality now. (This would appear not to apply to Llance's conception of infinity, as Llance does not treat infinity as being the value of 1/0.)

So, I have eleven rules for how numbers ought to behave. I have proven that zero times any number is zero; this was not one of the eleven rules, but it can be proven from them.

The usual thing that bothers everyone is this annoying exception in M5.

M5: For every number Image, there exists a number called Image, such that Image. (Existence of a multiplicative inverse - every number, except zero, has a reciprocal.)


It holds for every number, except zero. So what if we extend it to include zero, and call the reciprocal of zero "infinity"?

M5*: For every number Image, there exists a number called Image, such that Image. (Existence of a multiplicative inverse - every number, except zero, has a reciprocal.)


We have now gotten rid of that annoying exception. Zero is no longer special; it works just like every other number. You can take its reciprocal, just like every number; we will call the reciprocal of zero "infinity", and write it as Image. But having gotten rid of one exception to the rules, we need to introduce another exception, or we get some rather alarming results.

Using the eleven rules, but with M5 replaced by M5*, here's what we get.

Step 1: By A4, there exists a number called Image.

Step 2: By M5*, there exists a number Image, which is the reciprocal of Image. For convenience, we give this number Image a name, which is Image. However, the name isn't important; what is important is that it exists. This number has the property Image, as required by M5*.

Step 3: We have proved above that Image. This applies to any number, including Image. So Image.

Step 4: From Step 2, Image. From Step 3, Image. By the transitivity of equality, we have Image.

So we have now established the Dragon Equality, by replacing M5 with M5* (and thereby removing the exception, you can't divide by zero). If you allow division by zero, and don't add some exceptions to some of the other rules, then zero is equal to one. But it gets better.

Step 5: For any number Image, by M4, we have Image.

Step 6: But since we have proved that Image, this means Image.

Step 7: We have already proved that for any number Image, Image.

Step 8: From Step 6, Image. From Step 7, Image. By the transitivity of equality, it follows that Image. Note that this result applies for any choice of Image.

So if we try to introduce infinity by saying its what you get when you divide by zero, without relaxing any of the other rules, we find that every number is equal to zero. Oddly enough, infinity is also equal to zero. So if that's what you do, the only possible number system is the one that only has one number in it. Probably not the most useful number system ever invented.

So if you take the approach of saying infinity is what you get when you divide by zero (that is, eliminate the exception in M5), then you have two choices - you need to introduce some exceptions to some of the other eleven rules, or you end up with a number system that only has one number.

There may be some other ways to try to turn infinity into a number, but I'll take a break for now.
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Re: Some Questions About God

Postby Мастер » Sat Jul 16, 2016 3:26 am

Lance wrote:But I thought you were going to prove Dragon's conclusion?


Just did - you posted while I was working on it :) Using the eleven rules, but modifying M5 to eliminate the "no divide by zero" exception, I get that every number (not just one) is equal to zero.
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Re: Some Questions About God

Postby Lance » Sat Jul 16, 2016 4:22 am

Мастер wrote:
Lance wrote:But I thought you were going to prove Dragon's conclusion?

Just did - you posted while I was working on it :) Using the eleven rules, but modifying M5 to eliminate the "no divide by zero" exception, I get that every number (not just one) is equal to zero.

Okay, I think I'm with you here. Did you actually prove infinity is not a number?
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Re: Some Questions About God

Postby Мастер » Sat Jul 16, 2016 6:35 am

Lance wrote:Okay, I think I'm with you here. Did you actually prove infinity is not a number?


I don't think anyone can prove that :) But, what I did prove is, if you try to treat infinity as a number, defining it as the reciprocal of zero, it can't possibly follow all the rules we expect the other numbers to follow - if you do, the conclusion is that all numbers are same. A number system with only one number in it :)

I believe there are number systems that do include "infinity" as a number (in fact, I think most of these systems include not just one, but infinitely many "infinity" numbers). However, the behaviour of the various infinities in those number systems probably does not correspond to people's usual intuition about infinity.
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Re: Some Questions About God

Postby Lance » Sat Jul 16, 2016 6:51 am

Okay, I follow. Thank you.

I've thought a bit more about zero too. And I think I will retract what I said earlier about it being just another of the numbers, great and small, that are within infinity. In the respect of a number line that goes from the infinitely small to infinitely large it is still true. Zero probably falls right in the middle. But there is another way to look at it; where zero is nothing and there is no such thing as "less than nothing". In that respect, I could say zero falls outside the set of numbers too. So that one exception that is made for it no longer applies.

Basically zero is at one end and infinity is at the other, with all the numbers coming between them. Zero and infinity are the bounds of the set of numbers but not included within it. Like a glass can contain water but is not part of the water. The container shapes the contents but is separate from it.

Any number times zero is zero.
Any number times infinity is infinity.

So there are even 2 rules that they follow that none of the numbers do.

:mrgreen:
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Re: Some Questions About God

Postby Мастер » Sat Jul 16, 2016 7:27 am

Thing is, if you're doing arithmetic with zero and infinity, what exactly does it mean if we say they are not numbers? What is the definition of a "number"?

This strikes me as something analogous to a thing that was mentioned at CQ - the number "one" used to be considered a prime number. Today, it isn't. Is this really meaningful, or is it just an arbitrary definition? Nobody changed their opinion about the properties of the number "one" - they just changed their definition of "prime number".

I feel we have something similar here. Is infinity a number or not - well, that's a matter of the definition of "number". But whether we call it a number or not, how does it behave? In my list of the eleven properties of a field, there are two operations - addition and multiplication. Can you add to infinity? Can you multiply by infinity? (You multiplied by infinity in your last post, but didn't add.)

Also, when you say any number times infinity is infinity. Let's pick two - is two times infinity the same infinity, or is it some kind of larger infinity? If it's the same, we need to outlaw the idea of an additive inverse (per A5) for infinity in my list of properties. If two times infinity is infinity, and infinity has an additive inverse, then we have a bit of a problem.

If infinity has an additive inverse, then

Image

But if

Image,

then we can make the substitution,

Image

But using the multiplicative identify, M4, we have

Image

So we can make the substitution,

Image

Now if you accept that Image (I haven't proved this), then the above result can be rewritten as

Image

But then by the distributive law (D1), we have

Image

Then by associativity of addition (A3), we have

Image

By the distributive law (D1) again, we get

Image

OK, I'm getting tired of typing all these things, so a few steps all in one - infinity and negative infinity are additive inverses, one times infinity is infinity by LLance's rule, and one times zero is zero (by either M4, or by the result I proved above, that zero times anything is zero). We get

Image

Whether we choose to call "infinity" a number or not, an infinity with the properties you described (e.g., any number times infinity is infinity) cannot follow the eleven rules. There must be some "infinity exceptions" somewhere.

If you say that two times infinity is not the same infinity, but a somehow larger infinity, that might be a different story. Then there is not just one infinity, there are many infinities. I'll have to think about that one some more.
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Re: Some Questions About God

Postby Lance » Sat Jul 16, 2016 7:42 am

I guess to me, 1*infinity and 2*infinity are equivalent, but more realistlly, more nonsense than anything. If infinity is not a number that you can't multiply it and get a meaningful result.
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Re: Some Questions About God

Postby Мастер » Sat Jul 16, 2016 8:53 am

Lance wrote:I guess to me, 1*infinity and 2*infinity are equivalent, but more realistlly, more nonsense than anything. If infinity is not a number that you can't multiply it and get a meaningful result.


That's pretty much what CQ might call the "mainstream" view. The most commonly used number systems have no infinity. Although conventional mathematics is full of statements like, what happens to x2 as x goes to infinity, it is very careful to avoid applying arithmetic operations to it - infinity plus three is not meaningful, consistent with what you say above.

Sometimes people do treat it like a number and perform arithmetic operations on it, but they have to recognise that it is a deviant number that doesn't work like the others. Zero is also a bit of a deviant, but this is unavoidable - if you eliminate the zero exception in rule M5, the only number system that satisfies all eleven rules is, the system that only has one number :)
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Re: Some Questions About God

Postby Lance » Sat Jul 16, 2016 2:45 pm

These are all human constructs though anyway, right? I mean, there is no "infinity" out there somewhere defying definition and laughing at us for even trying...
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Re: Some Questions About God

Postby Arneb » Sat Jul 16, 2016 3:57 pm

Some (e.g., Max Tegmark) think there is infinity out there. Lots of it.
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Re: Some Questions About God

Postby Мастер » Sat Jul 16, 2016 4:27 pm

Lance wrote:These are all human constructs though anyway, right?


I agree with that. Numbers were not discovered in a cave somewhere, we invented them. We can invent different number systems. Some systems of numbers may be more or less useful as models for real, physical things out there.

Lance wrote:I mean, there is no "infinity" out there somewhere defying definition and laughing at us for even trying...


It should laugh at us if we assign to it properties that are logically inconsistent. (E.g., there is an infinity number that can satisfy my eleven rules.) As long as we are logically consistent, the worst thing any infinity can say is, "that's not me they're talking about, it's some other infinity".
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Re: Some Questions About God

Postby Мастер » Sat Jul 16, 2016 4:29 pm

Arneb wrote:Some (e.g., Max Tegmark) think there is infinity out there. Lots of it.


I'm quite well qualified to talk about the infinity that is in there, in our minds. I am much less well qualified to talk about any infinity out there, in the real world. The most I can say is, some number systems with an "infinity" number in them have proven to be useful in modelling real world phenomena.
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