Yep; Infinity is an unbounded quantity that is greater than every real number (Wolfram).

One would think it not too difficult. But it seems...

The first ordinal bigger than any real ordinal is called omega, and there are infinitely many others.

Yep; there is more than one infinity.

Infinity is a number greater than every integer.

And infinity + 1 = infinity is true only in the sense that if you take an infinite number and add one to it, you get an infinite number. Basically, only if infinity does not refer to a specific infinity (i.e. only if infinity - infinity =/= 0.) Otherwise, if you're working with specific infinity, infinity + 1 > infinity.

∞ and ∞+1 are the very same. That follows from the definition given above.

Not a lub but a gub

In computer programs infinity is used as a free-variable that might later become bound to a random finite number. So it is a logical concept rather than a concrete number.

OP.

Your argument about 1 divided by infinity is false.

I imagine this fits into your definition, and you'll force it.

I said in a different thread.

There has to be a great turning effort to keep anything infinite in continuum.

Infinity is always something's something, but this something dies with the user. How else are you going to conceptualize forever?

You project forever, but there must be turning forces and a specific something that is forever bound.

Otherwise it is just 'yep, forever[random, partial imagination of a great number]'.

There should be two definitions, literal infinity, and false numerical infinity.

I'm always annoyed at people who say infinity is not a number but a "concept." — Michael Lee

I've always been confused about this claim that infinity is not a number but is a concept. A number is a concept too isn't it?

I guess if we look at it from a quality-quantity perspective it'll begin to make sense. Firstly, infinity arises, even if only as a concept, in the world of numbers. Examples of infinity frequently used are numerical infinities e.g. the set of natural numbers {1, 2, 3,...}. These may be be soft evidence that the true home of infinity is the world of numbers,

I'm going out on a limb here and so forgive me if I don't make sense. I know there's a mathematical notion of countable and uncountable infinity which seems bit paradoxical since infinity is, by definition, uncountable. Kindly set aside the notion of countable vs uncountable infinities for the moment and consider only that infinity basically implies uncountable. Note that the term countable is formal and only describes the situation where a given infinite set can be put in 1-to-1 correspondence with the set of natural numbers.

That aside, compare this "fact" of infinity with other concepts that are considered uncountable. How about love, courage, joy? These concepts are categorized as uncountable i.e. unquantifiable and fall under the category of quality. So, it doesn't seem wrong to say that infinity is not actually a quantity, a number, but rather a quality like love or courage, etc.

I agree with you here. The idea is that infinity is not a quantitative concept (definable in relation to a numerical value), but a qualitative one. All qualitative and quantitative concepts are defined by their perceived potential.

infinity is, by definition, uncountable. — TheMadFool

But a trillion trillion is uncountable, too, in that sense.

It's like there is a wilful disregard for the mathematics here. See the slide to:

infinity is not a quantitative concept — Possibility

That's bullshit; in the technical sense - it's junk thinking indicating a lack of comprehension, wilful or otherwise.

But a trillion trillion is uncountable, too, in that sense.

It's like there is a wilful disregard for the mathematics here. See the slide to:

infinity is not a quantitative concept
— Possibility

That's bullshit; in the technical sense - it's junk thinking indicating a lack of comprehension, wilful or otherwise. — Banno

It's just an opinion. I'm not claiming this is a mainstream view on the issue. It just seemed right to look at infinity as a quality, being uncountable as it is.

Also, did you know that our ancestors could count only upto to 2? Look below:
Cardinal - Ordinal
1 - first
2 - second
3 - third
4 -fourth
.
.
.
n - nth

Notice that the names for ordinal numbers of the first two cardinals (1 & 2) are distinct viz. "first" and "second". All other ordinal numbers can be constructed from their respective numbers simply by adding "th". This is claimed to be be evidence of counting ability being limited to 2 and after that, 3, 4, 5,...,it was simply "many". So ancient counting looked like this: one (first), two (second), many. The many corresponds to the modern concept of infinity. As you can see, many and infinity represent a limit to quantification i.e. it spills over into the domain of quality - a concept and not a number.

Definition of 'number': an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations.

I don't see how inifinity is used in counting, nor is it a particular quantity. It is not finite, so not a number to my way of thinking. Personally I don't like all the maths based on larger and smaller infinities. Those definitions are like saying a colour is more black or less black.

Infinity is an equation of something and something. When does 1 = infinity? Infinity is it's own base number.

"It was 1. 1111,1111,1111,1111,1111r" - this would prove imagination is finite at most. (joke)

It is not created out of number, but mind of eternity through harmonious principles discovered in the world, or better yet, mind.

Does 1.1R imply 1.1r^i?(last joke)

Also, did you know that our ancestors could count only upto to 2? Look below:
Cardinal - Ordinal
1 - first
2 - second
3 - third
4 -fourth
.
.
.
n - nth

Notice that the names for ordinal numbers of the first two cardinals (1 & 2) are distinct viz. "first" and "second". All other ordinal numbers can be constructed from their respective numbers simply by adding "th". This is claimed to be be evidence of counting ability being limited to 2 and after that, 3, 4, 5,...,it was simply "many". So ancient counting looked like this: one (first), two (second), many. The many corresponds to the modern concept of infinity. As you can see, many and infinity represent a limit to quantification i.e. it spills over into the domain of quality - a concept and not a number. — TheMadFool

That is interesting...and I have often wondered about that kind of thing. But I suspect it is a lot more complicated than that.

If an ancient had 5 hens...and one went missing, I suspect he would not just say..."I had many yesterday and I have many today, so no problem."

I'm always annoyed at people who say infinity is not a number but a "concept." For one thing, that is way too broad and says absolutely nothing; I cannot think of anything in mathematics or logic that is not a concept. I say it is a number for if it is not then one divided by infinity would not equal zero anymore than one divided by a cat is a number. Here's my definition of infinity, and for simplicity I'm only referring to positive infinity: infinity is a number, but it has a characteristic that all real numbers do not possess. Namely, it is a number that is greater than any particular real number. All the rules of arithmetic applicable to real numbers do not carry over to use of infinity. Examples: infinity plus a real number is infinity: infinity divided by infinity is not equal to one: infinity subtracted from infinity is not equal to zero. — Michael Lee

The problem that arises for me is...IF "infinity"...why do we only look at things for "here" out infinitely (whatever that is). IF truly infinity...then the concept should begin at "nothing"...perhaps not even the thought or the notion/concept.

It's more a complex shape; you have come out of a womb, it is a complex time.

Understanding the evidence about forever is like understanding how you exist out of a womb.

You get to grips with the shape, over time channeling thought, you'll project some sort of infinity.

Depends on whether we're talking about infinity in the general sense of the word (as in, any number greater than every integer) or in the specific sense of the word (as in, specific number greater than every integer.) It's not like there is only one number greater than every integer. There are many such numbers. And infinity in the general sense of the word does not refer to a specific number of that sort (hence why infinity + 1 = infinity holds true.)

I'd say anything that can be said to be greater than or less than something else is a quantity (and therefore a number.) Infinity > every integer. Therefore, infinity is a number.

That aside, compare this "fact" of infinity with other concepts that are considered uncountable. How about love, courage, joy? These concepts are categorized as uncountable i.e. unquantifiable and fall under the category of quality. So, it doesn't seem wrong to say that infinity is not actually a quantity, a number, but rather a quality like love or courage, etc. — TheMadFool

But love, courage and joy are neither greater than nor less than any sort of number.

I'd agree it is qualititive and quantitive.

It is a number but also a concept.

You can have infinite numbers, but that doesn't make infinity a number, it makes the number infinity.

That's bullshit; in the technical sense - it's junk thinking indicating a lack of comprehension, wilful or otherwise. — Banno

In what technical sense? Infinity is historically a philosophical concept, representing an unbounded limit in relation to value/potential - whether quantitative OR qualitative.

I will concede that it’s more likely both or neither - that it refers beyond the outer limit of value/potential - although in mathematics it’s more representative of this boundless outer limit itself. But to define infinity as a supposedly quantifiable concept would be inaccurate: this is usually accepted by mathematicians, at least in a conceptual sense.

If an ancient had 5 hens...and one went missing, I suspect he would not just say..."I had many yesterday and I have many today, so no problem." — Frank Apisa

Piraha is a documented and currently spoken language that has relative terms roughly translated as ‘one’, ‘two’ and ‘many’ - but no terms for more exact numbers. From the studies conducted, it’s fair to say that some would not have noticed the missing hen, while others would.

I cannot think of anything in mathematics or logic that is not a concept. — Michael Lee

Totally agreed.

Mathematics is only about abstractions expressed in language.

infinity is a number, but it has a characteristic that all real numbers do not possess. Namely, it is a number that is greater than any particular real number. — Michael Lee

Cantor's work is really interesting in this regard.

Countable infinity and uncountable infinity cannot possibly be the same. Cantor's diagonal proof is simple but certainly surprising too. You may want to check. It is amazing.

If $\aleph_0$ represents countable infinity and $\aleph_1$ uncountable infinity, then the continuum hypothesis says that:

$\aleph_1 = 2^{\aleph_0}$

There cannot possibly be one infinite cardinal. According to Cantor's proof, there are at least two. Furthermore, there is this assumption that infinity is actually an infinite sequence of infinities:

$\aleph_0, \aleph_1, \aleph_2, \aleph_3, \aleph_4, ...$

And that the next infinite cardinal is two to the power of the previous one (generalized continuum hypothesis):

$\aleph_{i+1}=2^{\aleph_i}$

Through the Löwenheim-Skolem theorem, this explains why first-order arithmetic (PA=Peano's Arithmetic) has more than one model (=more than one Platonic world that satisfies the theory) -- one for each infinite cardinality -- and therefore why Gödel's incompleteness theorem ends up proving that PA is "inconsistent or incomplete".

Mathematicians who work in mathematical analysis - particularly what is called classical real or complex analysis - don't often say such-and-such is infinite, rather they say a process tends to infinity, provided, for every bound one might conjecture, the process eventually moves beyond that bound. For example, suppose we state

$\underset{x\to a}{\mathop{Lim}}\,F(x)=\infty$

We may say, "F(x) goes to infinity as x goes to a", but what we mean is that for each
$M\gt0$ there exists a positive number $\varepsilon =\varepsilon (M)$
such that

$\left| x-a \right|\lt\varepsilon \Rightarrow F(x)\gt M$


$-\infty$ can be used in a similar manner. And there are comparable definitions for complex valued functions in the complex plane.

It doesn't go much beyond this sort of (Cauchy-Weierstrass) definition in analysis. Orders of infinity and the like normally don't appear in the literature. However, soft or modern analysis does move in the general direction of set theory and algebra. And set theory is a different story; most of the posts in these kinds of threads pertain to that subject. :cool:

That is interesting...and I have often wondered about that kind of thing. But I suspect it is a lot more complicated than that.

If an ancient had 5 hens...and one went missing, I suspect he would not just say..."I had many yesterday and I have many today, so no problem." — Frank Apisa

:lol: If you find it difficult to believe try physicist George Gamow's book, One, Two, Three,...,Infinity which was published in 1947.

How's this, then?
Loosely, ∞ is a quantity that's not a number, and one ∞ is the quantity of numbers.

There are more reals than naturals though, so which kind of number do you mean?

But love, courage and joy are neither greater than nor less than any sort of number — Magnus Anderson

I may be guilty of reading too much into it but there is a sense in which infinity defies quantification, at least one aspect of it and that's the difficulty or impossibility of fixing its numerical value. All I did was look at other areas where we face a similar situation and the world of quality seemed an obvious one.

It bears mentioning that the existence of the qualitative concept doesn't mean that what we view as qualities are unquantifiable per se. By that I mean quantification that will appear quite crude and unsophisticated to scientists and mathematicians but still qualify as a quantification. For instance, taking the qualities love and courage it is common practice to make statements like, "he loves me more than you" or that "x is less courageous than y". Such statements, as indicated by the words "more" and "less", directly involve quantification, the crude and unsophisticated kind I mentioned a few lines before. The whole point of quantifying is to aid us in decision-making when two or more options are available; the choice we make based on the quantitative differences between the option. For instance Mary may choose to marry John than Tom because she feels John loves her more.

Now, it seems rather easy when I put it the way I did but actually making a decision requires accurate values for what is being compared. On what basis can one ever infer, say, that x loves y more than z? Only when we can actually fix the exact numerical value of how much x loves y and how much z loves y i.e. we need actual numbers to work with: If I could determine that on the love scale, x loves y 9 and and z loves y 7 then I could rightly infer that x loves y more than z loves y. The difficulty is that this isn't possible and the word "more" or "less", in the case of love and also for other qualities, suffer from the absence of a fixed numerical value. Isn't this exactly the same problem with infinity for it too has no fixed numerical value?

don't often say such-and-such is infinite, rather they say a process tends to infinity — jgill

The affinely extended real number system simply adds infinity as two real numbers in a Cantor-like approach:

Extended real number line. In mathematics, the affinely extended real number system is obtained from the real number system ℝ by adding two elements: + ∞ and − ∞ (read as positive infinity and negative infinity respectively), where the infinities are treated as actual numbers.[1] It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration.[2] The affinely extended real number system is denoted R ¯ or [−∞, +∞] or ℝ ∪ {−∞, +∞}. — Wikipedia on the affinely extended real number system

According to the explanations this extension approach is entirely consistent with existing results in mathematical analysis. This extension approach is actually what the Löwenheim–Skolem theorem does for Peano's arithmetic (PA) and other theories with infinite models:

It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models. — Wikipedia on the Löwenheim–Skolem theorem

If you can extend the real number system with uncountable infinity then you can extend it with any other infinite cardinality (upwards), since the theory is unable to control the cardinality of its infinite model. However, this also depends on whether this real number system is still a first-order theory.

Still, the fact that an affinely extended real number system is possible, suggests that mathematical analysis may have exactly the same interpretation problem as Peano's arithmetic (PA), i.e. if one infinite cardinality satisfies the model, then all other upward infinite cardinalities also do.

Hence, mathematical analysis could suffer from the same fundamental interpretation problem surrounding infinity.

You can't add what takes from you, sometimes; number isn't everything but what gave the idea of number.

Respectively number steals the idea of infinity or you naturally think 1.

Reformatting is something that should happen often in thought.

The answer to life is a number, but it's likely not base 4.

Therefore infinity is a something, and a number of it. We make use of the infinite. Pulsation, for example, where we love doing things repetitively. Addiction, etc.