These seems to be the accepted (mathematical) rules regarding infinity, the finite and zero.

Does anyone know why they are valid or able to provide a link to a demonstration of their substance. E.g how do you prove infinity x infinity = infinity without employing finite principles.


i + f = i
i - f = i
i × f = i
i / f = i

i + i = i
i - i = indeterminate
i × i = i
i / i = indeterminate


f × 0 = 0
f / 0 = i
0 / 0 = undefined
i / 0 = undefined

regards

leeaus

My thoughts
The finite exists with-in infinity and not separate to infinity.so one can not divide or add to infinity as this would be a contradiction to what infinity is

Go to the library and get a book on Set Theory that includes a chapter or so about Transfinite numbers. A man named Cantor did the first work on this topic prior to 1900.

He starts with the notion that you can decide which of two sets has more or less elements by pairing the elements. If each element of one set can be paired with an element of another set, the two sets have the small number of elements (or members). If the pairing process results in one set having some unpaired members, it is the larger set.

The above is not necessary for small finite sets, since you can merely count the members in each set. For large finite sets and for infinite sets, counting takes too much time. For such sets you try to devise a pairing algorithm.

Cantor defines an infinite set as one which can be put into one-to-one correspondence with itself. He coined the term transfinite number for the cardinal number of memebers of an infinite set. For example, the set of all integers is infinite because the following pairing works. 1 --- 2
2 --- 4
3 --- 6
. . .
n --- 2nEvery even integer can be paired with a member of the set of all integers, with none left over from either the entire set or the subset. This shows that the set and the subset have the same number of members.

He called the set of all integers Aleph₀ (usually pronounced Aleph null). The above pairing indicates a proof that 2*Aleph₀ is the same as Aleph₀. Aleph_n is sometimes used as the name of a set, and sometimes used to name the transfinite number of members in the set. You have to decide from context which meaning is intended.

He proved that the set of all subsets of a set has more members than the set itself. The set of all subsets of Aleph₀ is called Aleph₁. You can keep this up and get Aleph₂, Aleph₃, et cetera.

I think he proved that Aleph₀ is the smallest transfinite number.

I think he proved that there is no transfinite number between Aleph_n and Aleph_n+1.

He proved that the set of all 2D points, the set of all 3D points, and the set of all points in any finite interval have the same number of members as the set of all real numbers. I think he called this transfinite number the power of the continuum.

Cantor proved that the set of all real numbers has more than Aleph₀ members, but I do not think that anybody has proven what Aleph_n has the same number of members. Everytbody is pretty sure that is is Aleph₁. There is a simple proof that it is Aleph₁. I came up with this proof a long time ago, and was told that many others had also invented it, and that it is invalid. It looked good to me, but better mathematicians than I do not accept it.

The finite exists with-in infinity and not separate to infinity.so one can not divide or add to infinity as this would be a contradiction to what infinity is

I'll try not to come down on your quackery too hard, but doesn't 2 "exist within" 5, then? You can add, multiply and divide those two numbers without "contradiction."

Since 0/0 or infinity/infinity do not have practical use in the form of purely numerical division, I think pondering crazily about them doesn't really make sense. If those 0 and infinity is a function of something (that is normally the case) we can then apply L'Hopital's rule and get their meaningful result.

Perhaps the question wasn’t asked very well Dinosaur.

Infinity + 1 = infinity applies a finite principle (addition) to infinity. Counting is the finite principle.

To me what Cantor does is acknowledges both odd and even numbers, for example, extend indefinitely. Then concludes from this the existence of several infinities. And then applies finite principles to his several infinities.

With the limitation of a unique set as QQ implies, how would one establish that infinity can be subject to addition is probably the way the question should have originally been framed.

Regards

leeaus

[Cantor] called the set of all integers Aleph₀ (usually pronounced Aleph null).Aleph, by the way, is simply the first letter of the Hebrew alphabet.

Cantor realized that it was getting pretty confusing to use Roman letters like e and i for universal constants. And it's not much better to use Greek letters like pi, since some of them already have other uses such as mu as an abbreviation for the prefix "micro-', capital sigma as the symbol for summation of a series, and "gamma" as a type of electromagnetic radiation.

Unfortunately, aleph poses a big problem because a lot of fonts don't have the Hebrew alphabet. This one, for instance.

a question about infinity.
If you could see for an infinte distance and do it infinitly fast how much time does it take to see a million miles?
or

How far can you see if you look for 10 minutes?

AD1
How is the number five regarded as infinite?

or the number two for that matter? I fail to see the context of your comment.
Are not the numbers 2 and 5 a part of infinity.

Infinity equals everything and 2 and 5 are just a part of it.

Infinity is a mind divided by zero.... it is just a notion of no substance.

Wouldn't waste brain thoughts on something that has no meaning.

If you need to manipulate infinity, you know you are not in reality anymore...

Bit like the madness that comes from dividing GR by zero to gain a singularity, or trying to define God.

It is incorrect to say that infinity has no meaning. At least one meaning has been clearly explained above.

I think that Zarkov was only suggesting a value for trying to define infinity.... or should I say worthwhileness.....

So it is to be concluded, as far as this forum is concerned, the application of finite arithmetical rules to infinity cannot be proven from a simple unadulterated continuation of numbers.

Unless there is some far flung misunderstanding, infinity + 1 = infinity has no proven or demonstratable mathematical foundation without the breaking of 1,2,3,4... into various sub series is the conclusion of this forum it is taken?

thankyou leeaus

So it is to be concluded, as far as this forum is concerned, the application of finite arithmetical rules to infinity cannot be proven from a simple unadulterated continuation of numbers.

Correct. Different rules apply to infinity.

Unless there is some far flung misunderstanding, infinity + 1 = infinity has no proven or demonstratable mathematical foundation without the breaking of 1,2,3,4... into various sub series is the conclusion of this forum it is taken?

That is incorrect. The mathematical foundation was laid by Georg Cantor, among others.

James can I ask you whether you fully agree with those rules and the foundations laid down by Georg Cantor?

Sorry JR Cantor uses sub series. The task put to the undoubted wisdom of this forum is to establish infinity + 1 = infinity from the unique series 1,2,3,4,5…………………….

You may be tight with delight going down the hierarchy of infinite sets path. I’m not and have this academic pretty much non argumentative question. Not saying Cantor is right or wrong. Just wanting to see if someone can supply a mathematical foundation to infinity + 1 = infinity without duplicating the unique.

That is all. Cantor is irrelevant to this task so I guess with the further understanding that this post supplies you with, your answer is there is no mathematical foundation to infinity + 1 = infinity without a duplication of the unique as Cantor spent his life doing. Doesn’t seem that you applied your self to the question. Its an academic one. Take your time this time.



Regards

leeaus

QQ:

From what I know of them, they are self-consistent and very helpful, so I agree with them within their own framework.



leeaus:

Not saying Cantor is right or wrong. Just wanting to see if someone can supply a mathematical foundation to infinity + 1 = infinity without duplicating the unique.

What does "duplicating the unique" mean?

That is all. Cantor is irrelevant to this task so I guess with the further understanding that this post supplies you with, your answer is there is no mathematical foundation to infinity + 1 = infinity without a duplication of the unique as Cantor spent his life doing.

You guess wrong. Your post hasn't provided me with further understanding. Instead, it has confused me by introducing the new, unexplained concept of "duplication of the unique". Within your framework, I cannot give you an answer at all until you explain your terminology. Up to now, I have been talking about mathematics.

Doesn’t seem that you applied your self to the question.

You didn't ask your question clearly enough, it seems.

I await your response with baited breath.

Gee JR is pretty straight forward.

1 2 3 4 5……..

1 3 5 7 9……

Once you start doing this sort of thing, as Cantor does, can you see all that you are really doing is manipulating symbols with respect of a projection to infinity and getting several projections.
In the lower line the 3 is the 2 from the above line, the 5 is the 3, the 7 is the 4, the 9 is the 5 etc. That is the duplication of the unique, subtle and manipulative as it may be.

What you have to do is show the mathematical foundation of infinity + 1 = infinity only using the upper line of numbers shown here. Can or can’t do. Unbait that breathe. Don’t think it can be done is why I have been stating the conclusion of this forum etc.

Regards
leeaus

Leeaus: Dealing with transfinite arithmetic requires a shift in the way you count, the way you do arithmetic, and how you decide if two sets have the same number of members. I am not an expert in this, but believe that the following is valid. The set containing all integers expect one is an infinite set.
The set containing all of the integers, including one is also an infinite set. These two sets contain the same number of members, because they can be put into a one to one correspondence with each other (see below). Hence adding one element to an infinite set does not change the cardinality of the set. In lay terms, infinity + 1 = infinity.The following shows how to match members of the above two infinite sets.1 ---- 2
2 ---- 3
3 ---- 4
. . .
n ---- n + 1
. . .In the above pairing, every member of each set is paired with a member of the other set. No member of either set is left unpaired. Hence they have the same number of members.

Note that the same pairing could be done with the set excluding the first n integers and the set of all integers.n + 1 ---- 1
n + 2 ---- 2
n + 3 ---- 3
. . .
n + k ---- k
. . .Adding a finite number of members to an infinite set does not change the cardinality (Id est: Does not change the number of members).

When somebody objected to proofs such as the above, the reply by Cantor (or some other mathematician was something like the following.If each arithmetic property of a transfinite set was equivalent to the corresponding arithmetic property of a finite set, the so called transfinite set would be a finite set.

If you wish to deny the existence of transfinite sets, you are being logically consistent, and need not concern yourself with their properties. In this case, do not talk about their properties with those who accept the existence of transfinite sets.

If you accept the existence of transfinite sets, you must accept the counterintuitive consequences of their properties as proven by Cantor and others.

A third alternative is to accept the existence of transfinite sets, define them in a fashion different from the Cantor definition, and prove whatever you can about your special transfinite sets.The above is a paraphrase, not a quote. The final paragraph is my own idea, which seemed pertinent in the context of this thread.

Thanks Dinosaur

I wish to deny the existence of transfinite set as you allow that I am free to do. Also I do not wish to concern myself with the properties of transfinite sets. That should have plainly been seen to be my position already. Like this is nonsense to me <FONT COLOR=blue> The set containing all integers expect one is an infinite set</FONT>

The infinity + 1 = infinity etc ideas are dependent on the existence of transfinite sets. That is the simple point I am trying to make concrete by asking if someone can provide a foundation for infinity + 1 = infinity without using transfinite sets.

regards

leeaus

Like this is nonsense to me <FONT COLOR=blue> The set containing all integers expect one is an infinite set</FONT>

So you are saying that you can count all integers except the integer number "1" in a finite time ? (I take "1" as an example, but you are free to choose another number). Let's say you can count one number each second, how many years would it take ?

Well, why not! Let's start the counting right here: 0, we left out 1 so that doesn't count, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 ...

Let me know when you finish :).

Like this is nonsense to me: The set containing all integers expect one is an infinite set

Why?
I'm no mathemetician, but it seems straightforward to me that there are infinity integers greater than 1?

Leeaus: You cannot be serious!! Deal with infinity + 1 without using concepts relating to infinity? This is a joke. Correct?

What field of mathematics or logic would you like to see used to deal with infinity + 1? Elementary school arithmetic? Perhaps some handy dandy Aristotlean syllogisms? Theory of equations from algebra? Perhaps complex analysis, excluding functions which grow without bound? Perhaps finite difference theory or topology?

How about another topic? Tell me about algebra, but I do not want to hear about any symbolic representation of numbers. Or perhaps tell me how to find the roots of a polynomial, but do not use any algebraic concepts.

Do you think there is some subtle flaw in the methods Cantor used? do you understand his concepts?

Crisp and Pete you miss the point. If you leave a number out of an infinite series it is a finite series or contained within infinity as QQ would probably put it.
Crisp you more or less uncover the charade surrounding infinity. A human inability to count beyond a certain magnitude gives rise to the theory of infinity. If you were immortal and began counting now when would you arrive at infinity. Each extra number you counted would just like going 1,2 etc no how big in magnitude it seems to us mortals.

Dinosaur that is the point. You can't deal with infinity without unproven assumptions about the nature of infinity. The Cantor concepts are about a hierarchy of infinities. Hierarchies are finite impositions. The concepts that you employ that relate to infinity are simple finite principles in vain hope. They have no laid out foundation that treats infinity purely as infinity.
At this stage it would have to be the conclusion of this thread that infinity + 1 = infinity is dependent on the assumption of varying levels of infinity. Happy to leave it at that.

Regards

leeaus

If you leave a number out of an infinite series it is a finite series
??? Tell me you're joking?

Here's an easy question: How many positive integers are there?

Infinity is a valid concept, its just that your mortal brain leeus is unable to comprehend the formulations proposed by mathematical immortals such as Cantor, Haussdorf etc.

leeaus, are you familiar with Hilbert's Hotel?

Like this is nonsense to me: The set containing all integers expect one is an infinite set

You seem to think the set {...-3,-2,-1,0,2,3...} is not infinite. Is it finite then? or something else? (by the way, when you say "integers" do you really mean "natural numbers"?)


If you leave a number out of an infinite series it is a finite series or contained within infinity as QQ would probably put it.


So do you think 1, 2, 3, ... is an infinite sequence? What about 0, 1, 2, 3, ... ? If the second one is infinite, surely the first one is finite by your strange logic (or somehow "contained within infinity", whatever that means).

For that matter, what do you mean by "finite series"? You seem to be using a different definition than any mathematician I've met if you think removing one term from an "infinite series" many can get you something finite, but I'd like to be clear on this. Also, what the hell do you mean by "contained within infinity"?

Shmoe Not sure how this has got argumentative. Why does infinity +1 = infinity. Because you can start counting at 1 or 100 and the counting never ends in either case is the apparent answer. But that doesn’t make a total sense. There are always 99 more numbers in the count that begins at 1.
This is getting a bit silly with respect of the original intent of the thread but what if each number had its own symbol instead of being made of combinations as most are.
0,1,2,3,4,5,6,7,8,9 then another symbol unrelated to the first 9 + 1 and so on. Isn’t that all concepts of infinity amount to. Instead of keeping on putting breath taking combinations together at the high end, you say to hell with it, this is infinity at what works out to be an arbitrary or humanly determined point. It is not a mathematical point. Can you or anyone say infinity is more than an inability to continue counting.

(An inability to continue counting + 1 = an inability to continue counting. To me that is not is a mathematical foundation for anything much at all.)

Pete the amount of positive integers is countless. Hilberts hotel burnt forever didn’t it.

Ryans the gents you refer died. (neither immortal)

Regards

leeaus

Leeaus

Say your concept is correct, and you deal with infinities in the usual way that you do, i.e. inf - inf =0

O.K. Say I take the sum of all the positive real numbers = inf. That's cool, if we had enough time we could count them, so we roughly no how "big" it is going to be (i.e. cardinality)

This infinity is called countable, we know each element and thus count it.

Now what about the set of all irrational numbers? We know the sum of this set is infinite, yet we do not have a suitable method to determine all of its elements.

So how big is the set of the positive reals - the irrationals.

You can't say that is zero, because I know that for example pi and sqrt(2) are elements of this set.

And this is a big problem and is not trivial.

Take as an example again This sum

lim (n->inf) ( 1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n) ) =0.5772156649 0153286061 ...

Now the summation diverges(i.e. is infinite) and ln(inf)=inf but subtraction of the latter from the former yields a finite number!

How would you propose to do this?

Just to state the obvious; The concept of Infinity can only be used as an absolute, anything less is not describing infinity.

An "Infinte quantity of random numbers stated sequentially" for example is both difficult and easy to "conceptualise" but impossible to "realise".

An infinite quantity of random numbers stated in one instant of time or as I would call it A-linea is described simply as a single symbol.

An Infinite "bit" of infinity is almost an oxymoron.

Warning to others:

As far as I am aware, leeaus does not believe that the set of all positive integers (for example) is infinite. He may or may not believe that the set of all integers, positive and negative, is infinite. On the other hand, it may be that he doesn't even consider that set to be infinite, since it does not include all real numbers.

This has been argued with him extensively in previous threads, but I don't think he has the ability to grasp the relevant concepts.

Ryans, plausibly I am the worst communicator ever. The point is I do not accept that something like infinity – infinity = zero is at all sensible. Or a countable infinity as you intone. Thought that would have been clear. I do respect your post and know what you are getting. But all I see is an inability to continue with finite steps being referred to as infinity.

JR is fairly accurate about what I believe not to be infinity. He is right as well about an inordinately long thread about infinity started by God of Course took place on this forum last year and personally couldn’t be bothered with such again.
Now not trying to say that cardinal numbers and the like are an errant philosophy. Just trying to establish if the finite principles like addition and multiplication can be rationally applied to a concept of infinity without a philosophy about cardinal numbers and the like. Is that so hard to understand? For anyone wishing to reply it is pretty much a yes or no situation. It is not argument about what you believe to be infinity or what I believe not to be infinity. Plausibly JR will provide a link to the aforementioned thread if you want to know what I believe not to be infinity although in the main the thread was about a quasi proof of infinite distance that JR was juggling around.

Regards
leeaus

Possibly infinity is more of the nature of a logical operator then of a number (the same as 0 (zero)). Infinity would then mean "all" (eg. elements) and 0 none (element). Then the statement infinity + 1 is a nonsence because 1 is already included in "all" and cannot be added. What ever you do with the "all" you destroy its nature (of being "all")by adding to or substracting from it. If you add 1 to "all" and insist on it then "all" will have to be deprived of its basically infinite nature, it will become "not-all" and therefore countable/finite. Funny thing is that "all" when deprived of its infinite nature by adding "all" + 1 will not become infinity again (finite "all" + 1 do not form infinity).

Just trying to establish if the finite principles like addition and multiplication can be rationally applied to a concept of infinity without a philosophy about cardinal numbers and the like. Is that so hard to understand? For anyone wishing to reply it is pretty much a yes or no situation. It is not argument about what you believe to be infinity or what I believe not to be infinity.

What you do or don't think infinity is is rather critical when you are trying to discuss what properties it has. You can believe infinity is whatever you like. Personally, I think it's an orange dancing rhinoceros named Hugo. It's no suprise that when I try to discuss how Hugo performs the tango that no one understands since I haven't bothered to tell people about Hugo's charming little traits.

You have your own concept of infinity that doesn't seem come close to matching anyone else's. So you shouldn't be suprised when people question you to try to interpret what the heck you're saying. Eventually they'll give up and go try to pet the rhino.

cheers

Shmoe....a good point....Maybe leeaus will enlighten us all as to what he believes Infinity to be.????

Leeaus:The obvious answer is no!!! The concepts you learned in grammar school do not apply to infinity or related concepts.

I would have answered your question sooner, but did not realize what you were asking.

Thankyou dinosaur.

(Schmoe and QQ, thought I had made it plain that I don’t believe in the existence of infinity. It’s a human word that tries to convey that a never ending series of finite steps actually is something more than a never ending series of finite steps. All the best with Hugo, Schmoe. We he rolls in the mud after his dance he will lose his dreadful colour scheme thankfully. )

Thanks again, dinosaur. If I reach old age, intend to write a book about an interesting life I’ve had which would have to include something about the non existence of infinity. A no to the posed question was the presumed academic position. Just wanted to make sure.

Regards

leeaus

Do you believe in the concept of infinity?
Is it possible to imagine a hotel with infinity rooms and infinity guests?

Just trying to establish if the finite principles like addition and multiplication can be rationally applied to a concept of infinity without a philosophy about cardinal numbers and the like.

Yes, if we're talking about the same concept.

Hello Pete
Your proviso doesn't refer to what concept. Dinosaur says no. I think no is the right answer unless you can provide other information.

regards

leeaus (not real name)

Incidentally, pete, the way you make imagination a condition of your hotel implies a lack of checked out principle.

regards again

leeaus