# Thread: Different Types of Infinity

1. We generally call an infinitesimal number one that continues infinitely to the right side of the decimal place.

Infinity is generally regarded as a number that continues infinitely to the left side of the number line.

For example, .66545...., 3.13145352..., .555..., are all infinitesimals. However, it is also true that .777... is greater than .333..., therefore we should similarly suggest that 999... is greater than 777..., therefore there exists different types of infinity, some greater than others.

However, any infinite number is greater than any finite (i.e., real), number.

2.

3. Originally Posted by Ellatha
We generally call an infinitesimal number one that continues infinitely to the right side of the decimal place.

Infinity is generally regarded as a number that continues infinitely to the left side of the number line.

For example, .66545...., 3.13145352..., .555..., are all infinitesimals. However, it is also true that .777... is greater than .333..., therefore we should similarly suggest that 999... is greater than 777..., therefore there exists different types of infinity, some greater than others.

However, any infinite number is greater than any finite number (i.e., real), number.
Except for the last sentence this is WAY out in left field.

http://en.wikipedia.org/wiki/Non-standard_analysis

http://en.wikipedia.org/wiki/Hyperreal_number

http://en.wikipedia.org/wiki/Cardinal_number

http://en.wikipedia.org/wiki/Ordinal_number

4. You, and those like you, are the reason I am not pursuing an education beyond secondary school. Leave me alone.

5. Originally Posted by Ellatha
You, and those like you, are the reason I am not pursuing an education beyond secondary school. Leave me alone.
Then stop and think a bit before posting nonsense.

6. "The so-called 'intellectuals' still look down with infinite superciliousness on anyone who has not been through the prescribed schools and allowed them to pump the necessary knowledge into him. The question of what a man can do is never asked but rather, what has he learned? 'Educated' people look upon any imbecile who is plastered with a number of academic certificates as superior to the ablest young fellow who lacks these precious documents."

7. Originally Posted by Ellatha
"The so-called 'intellectuals' still look down with infinite superciliousness on anyone who has not been through the prescribed schools and allowed them to pump the necessary knowledge into him. The question of what a man can do is never asked but rather, what has he learned? 'Educated' people look upon any imbecile who is plastered with a number of academic certificates as superior to the ablest young fellow who lacks these precious documents."

An accurate quote of Adolph Hitler.

A completely inaccurate characterization of the attitude real mathematicians who are interested in ideas.

You might more profitably spend your time on good ideas than quotes from delusional tyrants, and erroneous or inane pseudo-mathematics.

8. A delusional tyrant with a genius level intellect that has convinced me of the aforementioned quote via discussion with you and other members.

9. is this something to do with Georg Cantor's set theory??

10. Originally Posted by Heinsbergrelatz
is this something to do with Georg Cantor's set theory??
Lol!

11. Originally Posted by Ellatha
You, and those like you, are the reason I am not pursuing an education beyond secondary school. Leave me alone.
Ellatha, all DrRocket did was inform you that your ideas are not correct in standard mathematics, and he knows from experience. It is clear that he knows mathematics and if you care the slightest bit about also accurately knowing and understanding mathematics, you would be wise to take what he says as the truth and use it constructively. And by not pursuing higher education you're simply letting 'those people' defeat you. If you care about mathematics at all, then learn from those who have been educated in the enormous base of knowledge that has been accumulated over hundreds (and more) of years. There is nothing wrong with that. No one is going to recreate every concept that others have come up with on their own, or improve on them without serious education in the matter, that's just the way it is.

12. Originally Posted by AlexP
Ellatha, all DrRocket did was inform you that your ideas are not correct in standard mathematics, and he knows from experience. It is clear that he knows mathematics and if you care the slightest bit about also accurately knowing and understanding mathematics, you would be wise to take what he says as the truth and use it constructively. And by not pursuing higher education you're simply letting 'those people' defeat you. If you care about mathematics at all, then learn from those who have been educated in the enormous base of knowledge that has been accumulated over hundreds (and more) of years. There is nothing wrong with that. No one is going to recreate every concept that others have come up with on their own, or improve on them without serious education in the matter, that's just the way it is.

Dr. Rocket stated that he would not provide me with any aid in mathematics anymore, to which I said good. He has clearly not followed through with his statement--your statements are not convincing or impressive, and they seem to follow the routine outlined by many other members on this forum that cannot think for themselves.

Heinsbergrelatz,
I don't believe so.

13. Hello Ellatha,

I am sorry to cut in (being an inexperienced intellect myself).

Reading from your messages, may I ask what exactly do you expect us to do then? Given that you have posted an idea of yours, what do you expect from us? To accept it and develop it further? Or to critique and improve?

I would like to know out of curiosity.

14. Originally Posted by Heinsbergrelatz
is this something to do with Georg Cantor's set theory??
In fact it nothing much to do with anything, really.

What the OPer seems to be groping toward (but failing in spectacularly) is the following: Yes there (at least) two kinds of infinity as far this term refers to the cardinality of sets.

However, the following might be worth pointing out: if I say there are infinitely many numbers (natural, real or complex), I actually mean the sets are of infinite cardinality. This slight linguist abuse rarely confuses. However, the expression "x is an infinite (or finite) number" always does, and in fact makes little sense unless it refers to the cardinality of a set.

Cantor showed that, whereas a set that can be put into one-to-one correspondence with a subset of the natural (i.e. counting) numbers is by definition countable (not due to him, I believe), even though it may be infinite (hence the term "countably infinite"), the set of real numbers cannot be put in such a correspondence with the set of counting numbers, and is thereby said to be "uncountably infinite".

Unfortunately, all this seems to have been too much trouble for the OPer to check up on before posting.

15. Originally Posted by Guitarist
In fact it nothing much to do with anything, really.
You mean nothing other than mathematics. You say that my post relates to nothing, and yet you write several paragraphs discussing what it relates to. After typing up arbitrary information the subject, you bring up the dogmatic opinion that I should have "checked up" on it before posting, and yet I'm not familar with nonstandard analysis in the least. Your statements are not as insulting as you would like them to be, rather they simply make you appear like an idiot for throwing in a bunch of words that are generally taken as insulting.

Furthermore, I'm interested in particular why you are a moderator on this forum: for an elongated period of time you posted nothing on this forum except for a promise to clear up some misconceptions in a particular thread I wrote, to which you never followed through with, and when you did finally decide to post it was because you were offended by a word that I wrote, "no."

See my signature anid this thread (http://www.thescienceforum.com/Hospi...eem-28561t.php) for my remaining opinion.

Tikai,
Discuss further, bring up issues with specific points, etc...

16. Originally Posted by Ellatha
... and yet I'm not familar with nonstandard analysis in the least.
That is painfully obvious. Non-standard analysis is the branch of mathematics in which infinitesimals are made rigorous. Your version is, not surprisingly given your self-admitted ignorance of the subject, complete nonsense.

Unfortunately non-standard analysis has not been able to provide proofs of any theorems that cannot be proved using conventional methods.

You would be barking up the wrong tree, if you could locate the forest.

Rather than posting trash and expecting admiration, perhaps you ought to actually learn some real mathematics.

If you want to understand infinite numbers, then the theory of cardinal and ordinal numbers would be a good place to start. See the links in my first post in this thread.

On the other hand, if you still feel put upon when your errors are corrected, and consider yoursellf a mathematical genius as you have stated elsewhere in this forum, then you might consider some profesional counseling.

17. Originally Posted by DrRocket
That is painfully obvious. Non-standard analysis is the branch of mathematics in which infinitesimals are made rigorous. Your version is, not surprisingly given your self-admitted ignorance of the subject, complete nonsense.
If I did not say that this thread related to non-standard analysis, and you suggest that it doesn't, than the error lies in trying to compare the two. I did not do so; you did.

Originally Posted by DrRocket
Rather than posting trash and expecting admiration, perhaps you ought to actually learn some real mathematics.
When did I say I expected admiration?

Originally Posted by DrRocket
On the other hand, if you still feel put upon when your errors are corrected, and consider yoursellf a mathematical genius as you have stated elsewhere in this forum, then you might consider some profesional counseling.
I never said I was a mathematical genius, nor do I want any counseling. In fact, I said that I do not think like a mathematician or a scientist.

http://en.wikipedia.org/wiki/Cardinal_number

http://en.wikipedia.org/wiki/Ordinal_number

And I'm still not sure about something.

My teacher said the following (in non-mathy speak):

Is this correct? If so is it relative to:
http://en.wikipedia.org/wiki/Cardina...er#Subtraction
If the axiom of choice holds and given an infinite cardinal σ and a cardinal μ, there will be a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.

19. Originally Posted by m84uily
My teacher said the following (in non-mathy speak):

Is this correct? If so is it relative to:
http://en.wikipedia.org/wiki/Cardina...er#Subtraction
If the axiom of choice holds and given an infinite cardinal σ and a cardinal μ, there will be a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.
Cardinal numbers play no role in integration.

You can say very little about the integral of a function absent any information about the function.

So, no, what you wrote is not "correct".

20. Originally Posted by Guitarist
Originally Posted by Heinsbergrelatz
is this something to do with Georg Cantor's set theory??
In fact it nothing much to do with anything, really.

What the OPer seems to be groping toward (but failing in spectacularly) is the following: Yes there (at least) two kinds of infinity as far this term refers to the cardinality of sets.

However, the following might be worth pointing out: if I say there are infinitely many numbers (natural, real or complex), I actually mean the sets are of infinite cardinality. This slight linguist abuse rarely confuses. However, the expression "x is an infinite (or finite) number" always does, and in fact makes little sense unless it refers to the cardinality of a set.

Cantor showed that, whereas a set that can be put into one-to-one correspondence with a subset of the natural (i.e. counting) numbers is by definition countable (not due to him, I believe), even though it may be infinite (hence the term "countably infinite"), the set of real numbers cannot be put in such a correspondence with the set of counting numbers, and is thereby said to be "uncountably infinite".

Unfortunately, all this seems to have been too much trouble for the OPer to check up on before posting.
Thanks for that brief description regarding set theory. Though most of the terms like cardinal numbers and ordinal numbers are pretty new to me. I just got to know about them 2 weeks ago when i was reading about Georg Cantor's work in the book "God created the integers"- stephen hawking, and so far im not doing a great job understanding the whole thing . Nevertheless i find it very interesting indeed.
Just a question, when i was reading about Cantor's work, this number called Aleph number came out. How do you write the "Aleph" the hebrew letter in terms of TeX???

21. Originally Posted by DrRocket
Originally Posted by m84uily
My teacher said the following (in non-mathy speak):

Is this correct? If so is it relative to:
http://en.wikipedia.org/wiki/Cardina...er#Subtraction
If the axiom of choice holds and given an infinite cardinal σ and a cardinal μ, there will be a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.
Cardinal numbers play no role in integration.

You can say very little about the integral of a function absent any information about the function.

So, no, what you wrote is not "correct".
What if I were to clarify and say:

22. Originally Posted by Heinsbergrelatz
Thanks for that brief description regarding set theory. Though most of the terms like cardinal numbers and ordinal numbers are pretty new to me. I just got to know about them 2 weeks ago when i was reading about Georg Cantor's work in the book "God created the integers"- stephen hawking, and so far im not doing a great job understanding the whole thing . Nevertheless i find it very interesting indeed.
Just a question, when i was reading about Cantor's work, this number called Aleph number came out. How do you write the "Aleph" the hebrew letter in terms of TeX???

To learn about cardinals and ordinals a good source is Naive Set Theory by Paul Halmos.

23. Originally Posted by DrRocket

To learn about cardinals and ordinals a good source is Naive Set Theory by Paul Halmos.
Thanks for the source. Oo and how did you write that using TeX??
For the undergraduate mathematics courses, is set theory usually taught if you are going for mathematics as a major??

24. Originally Posted by Heinsbergrelatz
Originally Posted by DrRocket

To learn about cardinals and ordinals a good source is Naive Set Theory by Paul Halmos.
Thanks for the source. Oo and how did you write that using TeX??
For the undergraduate mathematics courses, is set theory usually taught if you are going for mathematics as a major??

You shoild see the code in the window when you reply.

\aleph

25. Originally Posted by Ellatha
Furthermore, I'm interested in particular why you {i.e. me}are a moderator on this forum:
It is considered Very Bad Form on this forum to openly challenge a moderator's credentials, though not strictly "illegal".

If you (or any one else for that matter) truly believe I am not "up to the job" then please, by all means, complain to the admins, currently In(Sanity) and SkinWalker.

Though I was very active, possibly over-active in the past, the reason for my recent lack of posting here is well known to my favourite members. Does that disqualify me from deleting viagra, shoe and iPhone adverts? Believe me, that is most of what I do here

26. I just find it annoying that you would think that using your posting time to attack my character is more important than increasing discussion on mathematical matters. I do not care about this (or any other) forum enough to complain.

27. Originally Posted by Ellatha
I just find it annoying that you would think that using your posting time to attack my character is more important than increasing discussion on mathematical matters. I do not care about this (or any other) forum enough to complain.
Guitarist did not, ever, attack your character. He is far to circumspect, and even-handed to do that. Guitarist suffers fools gladly.

I do not, in public forums where they pose a threat to the naive but genuinely curious.

You have made the transition from the genuinely curious to someone posing as knowledgeable but posting only the false or trivial. You are doing an excellent job of making a fool of yourself via the objective content, or lack thereof, in your posts. Guitarist's comments were very clearly directed toward that content, and therefore toward "increasing discussion of mathematical matters".

28. Originally Posted by DrRocket
Guitarist did not, ever, attack your character.
Interesting than that he admitted to insulting me. Stop scanning, and start reading.

I'm interested in the results of the following experiment, for the sake of my argument:

Measure your heart rate at its normal level. Read the following statement:

"I am several times smarter than Isaac Newton."

Measure the heart rate at its apex, and subtract what it was at its normal level.

I think the results will demonstrate that your emotions blind your intellect to a tremendous extent; you would do yourself well to fix this problem.

29. Originally Posted by Ellatha
Originally Posted by DrRocket
Guitarist did not, ever, attack your character.
Interesting than that he admitted to insulting me. Stop scanning, and start reading.
Guitarist has made precisely two posts thus far in this thread :

Originally Posted by Guitarist
In fact it nothing much to do with anything, really.

What the OPer seems to be groping toward (but failing in spectacularly) is the following: Yes there (at least) two kinds of infinity as far this term refers to the cardinality of sets.

However, the following might be worth pointing out: if I say there are infinitely many numbers (natural, real or complex), I actually mean the sets are of infinite cardinality. This slight linguist abuse rarely confuses. However, the expression "x is an infinite (or finite) number" always does, and in fact makes little sense unless it refers to the cardinality of a set.

Cantor showed that, whereas a set that can be put into one-to-one correspondence with a subset of the natural (i.e. counting) numbers is by definition countable (not due to him, I believe), even though it may be infinite (hence the term "countably infinite"), the set of real numbers cannot be put in such a correspondence with the set of counting numbers, and is thereby said to be "uncountably infinite".

Unfortunately, all this seems to have been too much trouble for the OPer to check up on before posting.
Originally Posted by Guitarist
It is considered Very Bad Form on this forum to openly challenge a moderator's credentials, though not strictly "illegal".

If you (or any one else for that matter) truly believe I am not "up to the job" then please, by all means, complain to the admins, currently In(Sanity) and SkinWalker.

Though I was very active, possibly over-active in the past, the reason for my recent lack of posting here is well known to my favourite members. Does that disqualify me from deleting viagra, shoe and iPhone adverts? Believe me, that is most of what I do here
As anyone can clearly see the content of your posts, not your character, is the subject addressed by Guitarist. Now, you may personally take criticism of the content of your post as an insult or an attack on your character, but an objective reader would not.

If there was any insulting I would opine that it was in your reply to Guitarist, though a "rapier-like wit" seems to be absent:

Originally Posted by Ellatha
You mean nothing other than mathematics. You say that my post relates to nothing, and yet you write several paragraphs discussing what it relates to. After typing up arbitrary information the subject, you bring up the dogmatic opinion that I should have "checked up" on it before posting, and yet I'm not familar with nonstandard analysis in the least. Your statements are not as insulting as you would like them to be, rather they simply make you appear like an idiot for throwing in a bunch of words that are generally taken as insulting.

Furthermore, I'm interested in particular why you are a moderator on this forum: for an elongated period of time you posted nothing on this forum except for a promise to clear up some misconceptions in a particular thread I wrote, to which you never followed through with, and when you did finally decide to post it was because you were offended by a word that I wrote, "no."

See my signature anid this thread (http://www.thescienceforum.com/Hospi...eem-28561t.php) for my remaining opinion.

30. Do you ever know what's going on? The points I made related to content outside of this thread; your continually demonstrate a clear lack of understanding of what's being discussed.

31. Originally Posted by Ellatha
Do you ever know what's going on? The points I made related to content outside of this thread; your continually demonstrate a clear lack of understanding of what's being discussed.
So, one is supposed to understand that your post, in this thread, has nothing to do with the content of this thread.

You tell me that you have some enormous IQ. Then why do you say such incredibly stupid things ?

BTW, my characterization of the gentlemanly deportment of Guitarist and his willingness to suffer fools gladly extends well beyound the bounds of this thread. So, despite your widening of the basis for your criticism of Guitarist, you are still WAY off base.

Put a sock in it.

32. Originally Posted by DrRocket
So, one is supposed to understand that your post, in this thread, has nothing to do with the content of this thread.
If you do not understand what is being discussed, than stop pretending that you do: you are only doing the public a disservice in pretending that you do.

Originally Posted by DrRocket
You tell me that you have some enormous IQ. Then why do you say such incredibly stupid things ?
I refrained from doing so in the public forum in order to not appear arrogant to the public while still addressing a previous point. You have childishly chosen to simply reveal the content that I attempted to conceal. Grow up.

Originally Posted by DrRocket
BTW, my characterization of the gentlemanly deportment of Guitarist and his willingness to suffer fools gladly extends well beyound the bounds of this thread. So, despite your widening of the basis for your criticism of Guitarist, you are still WAY off base.

Put a sock in it.
God forbid anyone offend your precious ego; maybe I should have gone so far as to lie in order to make you happy. Needless to say the word childish exemplifies your behavior.

33. Originally Posted by DrRocket
To learn about cardinals and ordinals a good source is Naive Set Theory by Paul Halmos.
I fully endorse this recommendation. I found it as a used hard cover the back end of last year for fractionally over 10 euros on Amazon (or was it Alibris?). It is a really nice tour around what it says on the cover.

Of course, set theory in all its glory, is more than Halmos offers (by his own admission) and is, speaking for myself, a rather dry - not to say arid - subject. But each to their own.

Whether or not set theory in any form is a mandatory course in undergrad math courses, I am not (of course) qualified to say. I will simply throw out this.....

,,,,,in so-called "advanced" mathematics you will have to go quite a distance to find a mathematical object (ring, field, group, vector space,....) that cannot be said to be a SET with some additional algebraic structure. If this is true (I think it is, though I can think of at least one exception), then an understanding of naive set theory is, at the very least, helpful.

The exception I am thinking of are topological spaces, which, just being sort of "jazzed up" sets, require an even deeper understanding of set theory, though I cannot convince myself they are algebraic in the loose sense that I used above. Likewise, manifolds are sort of "jazzed up" top. spaces, so the same applies.

PS I apologize for my imprecise language here, I hope it was not too confusing

34. Originally Posted by Guitarist
Originally Posted by DrRocket
To learn about cardinals and ordinals a good source is Naive Set Theory by Paul Halmos.
Of course, set theory in all its glory, is more than Halmos offers (by his own admission) and is, speaking for myself, a rather dry - not to say arid - subject. But each to their own.
There are three very appealing characteristics of Halmos's book

1. As you note, it is inexpensive.

2. It is not nearly as dry and arid as axiomatic set theory (a sure cure for insomnia)

3. It is quite short (see item 2).

35. Originally Posted by Heinsbergrelatz
is this something to do with Georg Cantor's set theory??
Yes, Georg Cantor created a whole hierarchy of transfinite (i.e. beyond finite) numbers. The great mathematician David Hilbert called this vast domain, rich for