There are a few different conceptions of infinity in mathematics.

There's what the usual infinity symbol represents: $\infty$, which usually denotes a limiting process: $\lim_{x\rightarrow\infty}f(x)$, IE what value $f(x)$ tends to when $x$ becomes arbitrarily large. Formally this corresponds to a definition of a limit and can be considered shorthand for it.

Then you've got cardinal numbers, which count how many of something there are. The smallest infinite cardinal is called $\aleph_0$, which is the size of the set of natural numbers $\mathbb{N}=\{0,1,2,3,...\}$. Then there are ordinal numbers, which agree with cardinal numbers up to $\aleph_0$ and can disagree beyond that - they correspond to different ways of ordering infinite sets of things. For example, the standard ordering of $\mathbb{N}=\{0,1,2,3,...\}$ is given the symbol $\omega$, which denotes its order type. If you removed 42 from $\mathbb{N}$ and stuck it on the end (after the infinity of numbers), you'd have the same set of elements but it would look like $\{0,1,2,...,41,43,44,...,42\}$, and this is given the order type $\omega+1$. You can separate out the odds and evens similarly and end up with $\omega+\omega=2\omega$. This operations allow you to define standard arithmetic operations on infinities relating to orders, and similarly for cardinals.

In the first case, infinity is a shorthand for a limiting process (the infinity is hidden in the quantifier 'for all epsilon'), in the second case infinite objects are referred to explicitly.

in the second case infinite objects are referred to explicitly. — fdrake

Does that mean that in the second case "infinite" is being used as a quantity?

In the first case it's easier to think of as a direction. In the second case - for cardinals - they give the size of infinite sets, so yes they are probably quantities since they represent the magnitude of something.

Following the "default definition", quantity stands for the magnitude of countable and reducible things. I mean, in a geometric view, would be like distance, the space between the initial and the final point. When you're counting something, you're presuming that there's a limit and when you reach the limit you'll be known the quantity.

Also, a x quantity of filler stuff fill in a x quantity of fillable things.
For example, two shoes fits in a box, if you increase the quantity of shoes to 3 shoes, it won't fit any more because it crosses the limit, it can also happen in the negative way. The thing is that with you have an infinitely large box, with an infinite amount of shoes in, no matter how many shoes you take off from the box, it won't change nothing. So talking about quantity doesn't make sense any more.

"Limiting processes" tend to have a somewhat uneasy relationship with the axioms of Set Theory and Peano Arithmetic which underlie damn near everything about number theory. If you are talking about aleph-null infinities then, of course, every aleph-null infinity has a precise numeric value (though this value is impossible to identify).

Is this a criticism of the epsilon-delta and epsilon-N convergence/continuity criteria?

Not at all; I am trying to tread a delicate line between "mathematics" and "mathematical philosophy". Most non-mathematicians, and even many mathematicians, conceive of mathematics as the quintessential, monolithic embodiment of perfect rationalism, the ultimate logical system; and of course, it isn't. There are many logical grey areas, even at basic levels. For example, is 0.9... equal to 1? Or is it the largest real number which is less than 1? There are persuasive mathematical arguments on both sides.

"The wise man doubts often, and his views are changeable; the fool is constant in his opinions, and doubts nothing, because he knows everything, except his own ignorance" (Pharaoh Akhenaton).

Ok. Well epsilon-N implies 0.9 recurring = 1 anyway. AFAIK it's even true in non-standard analysis. 1 - infinitesimal isn't the same thing as 0.9 recurring.

No, infinity is not a quantity it is a direction on any scale in which it is listed as a measurement.

East is not a location, destination, or even an obtainable goal. It is a direction relative to the current position and might more properly be stated as "east of here," wherever here may be, in the same sense that x + 1 is not an absolute quantity but instead something greater than x.

From a Hegelian perspective, I would rationally perceive infinity not as a quantity.

1. A quantity is a specified amount of something. It has a limit. The infinite is that which has no limits and so cannot be quantified. Therefore, not a quantity as not quantifiable.

This is just... no. Look, even if I take your definition of quantity, I can easily show infinity is a quantity. Take the set of Natural Numbers (o, 1, 2, 3...). In set theory, the concept of "size" is formalized as what is known as "cardinality". The cardinality (size) of the set of Natural Numbers is infinity, specifically aleph-null. QED. You can say the Natural Numbers have "no limit" in the sense that it can always get bigger, but that doesn't mean it's impossible to quantify.

2. Infinity is not limited to numbers (because it has no limit). if you say infinity is only a number you have broken the law of none contradiction as you have put a limit on something defined as having no limits. Therefore, infinity contains numbers but numbers do not contain infinity as numbers are limited to number.

A better way to think about it is there are different kinds of infinite numbers, some larger or smaller than others. The set of Real numbers, for instance, is a larger infinity than the infinity of the Natural numbers. Cantor proved this with a proof by contradiction. No one is contradicting themselves saying there are infinite quantities.

You can add 1 to any real number, so infinity isn't a real number. Infinity is a concept.

Colloquially, infinite is a quantity that's not a number, $|\mathbb{R}| \sim \infty \notin \mathbb{R}$.
But it's ambiguous (hence the $\sim$).
As it turns out there's more than one infinite, there are infinitely many different infinites, no less (Cantor).
Anyway, Dedekind and Tarski came up with different (general) definitions that can be shown equivalent.

You can separate out the odds and evens similarly and end up with ω+ω = 2ω. — fdrake

Very nice post.

A quibble. I just happen to be brushing up on ordinal arithmetic this week. Ordinal multiplication is defined backwards from our intuition in my opinion. $\alpha \times \beta$ is defined to be $\beta$ copies of $\alpha$ concatenated.

So for example $2 \omega$ means $\omega$ copies of 2 strung together. The ordinal 2 represents the order 0, 1. If you line up $\omega$ of those, you get ... drum roll ... $\omega$.

On the other hand, $\omega 2$ is two copies of $\omega$ side-by-side. You can visualize this as 0, 2, 4,6 , 8,...,1, 3, 5, 7, ... the evens-before-odds order. That's $\omega + \omega$.

Other than that quibble, great post.

https://en.wikipedia.org/wiki/Ordinal_arithmetic#Multiplication

You can add 1 to any real number, so infinity isn't a real number. Infinity is a concept. — GreenPhilosophy

Nothing to prevent you from adding 1 to infinity.

Yes there is. If it is Infinity then it should already contain the 1 you’re attempting to add to it. If it doesn’t contain that 1 being added then it’s not infinity, as it is limited to not containing the 1 you are adding. This means what you are calling ‘infinity’ is not limitless at all and so not worthy of the title. — Mr Phil O'Sophy

Seriously, you can even add infinity to infinity. Plenty of cases where that happens in mathematics.

I understand that mathematics uses the concept of multiple infinities. I’ve been exposed to the idea before. — Mr Phil O'Sophy

Do you understand though?

I’m saying that I fundementally disagree with it. What ever they are adding is more worthy of the title ‘indefinite’ than infinity. — Mr Phil O'Sophy

So, we have established that you DON'T understand it.

As I said before. If you try to have more than one infinity then you create a problem. — Mr Phil O'Sophy

Repeating an error ad infinitum does not correct it.

Infinity is boundless, without limit, Etc. — Mr Phil O'Sophy

And some of those are bigger, infinitely bigger, than the others.

If you have two infinity’s, A & B, then you are saying that in order to add infinity A to infinity B that A does not contain B. Which is to say that both A and B are limited or bounded to A and only A or B and only B — Mr Phil O'Sophy

You have never studied mathematics.

This making two infinity’s then leads to the logical conclusion that it is an indefinite number; an undisclosed amount that is limited to not containing that which you wish to add to it; not an infinite quantity as the mathematitions like to insist. — Mr Phil O'Sophy

Indefinite in number, you say.

You haven’t actually confronted my rebuttal, only used an appeal to authority fallacy a kin to ‘the mathematitions disagree with you so you’re wrong’. — Mr Phil O'Sophy

You have no rebuttal short of "I don't understand this".

So it would appear that I understand the problem more than you do, unless of course you can demonstrate why i’m wrong, which so far you haven’t. — Mr Phil O'Sophy

So it's you versus Cantor?

Simply agreeing with authority without actually confronting the argument being made against it ad infinitum is not itself an argument. — Mr Phil O'Sophy

Demonstrating your lack of comprehension does not constitute an argument.

Yes I have. — Mr Phil O'Sophy

Primary school doesn't count.

Please feel free to actually deal with the argument. I’m genuinely interested to hear a counter argument, which you have failed to offer so far. — Mr Phil O'Sophy

You don't have an argument.

Infinity isn't a real number, but it is an extended real number. Infinity can be used to describe infinite things, such as an infinitely sized universe.

By the way, I'm pretty bad at math, so don't take my word for it. I should just stop before I spread false information.

The thing you missed here is the unspoken inference you make. The cardinality of the set of Natural Numbers is not infinity (which is defined as having no limits) as by referring to Natural Numbers you are limiting it to Natural Numbers alone. You are not including anything which is not a Natural Number, it does not include different colours, shapes, texture etc. It is a concept limited to that which is considered a natural number.

How was it unspoken if I literally said the assumption (the the natural numbers are infinite)? That aside, you aren't making sense. That the natural numbers do no, for instance, include the Real Numbers does not entail that the set of Natural Numbers is not infinity. In mathematics, infinity is not (as you claimed) defined as "having no limits". In this case that's especially obvious, because by "limit" you're already sneaking in the assumption of finitude (e.g. the natural numbers are finite, somehow, because the set doesn't include other types of numbers). This argument makes no sense.

You can say that the numbers have no end.. or could go on forever.. or go on indefinitely.. but you cannot refer to them as infinite as you contradict yourself by describing them as such. As they are limited... to Natural Numbers. I am aware that mathematicians are fond of using the word infinite, but I would argue that its an illogical thing to do. As I think I have sufficiently shown.

No because then you're not talking about the infinite any more.

Consider the following:

1. There are two infinite numbers, A and B
2. A is not B, and B is not A.
3. A is larger than B.

this isn't a description of something without limits. You are specifically saying that A is limited to A and does not include B. And that B is limited to B and does not include A. These are limits.

You can say it has no limits in one specific sense but has limits in others, but then you are not referring to the infinite or to a limitless thing anymore.

You are simply ignoring the definition of infinity that mathematicians use and thereby conclude that it's incoherent because of we assumed your definition we'd get a contradiction. QED, your definition is wrong because it leads to a contradiction. That's ridiculous.

Your argument makes an obvious assumption, namely that all infinite sets are of the same size That's quite literally rejected in mathematics. Infinite sets which are countable, like the natural numbers, have the ability to be put into a one-to-one correspondence with a proper subset of themselves, e.g. we can map all the even numbers onto the set of natural numbers. Uncountably infinite sets (e.g. the reals) cannot do this mapping with the natural numbers, entailing that such sets are larger. Your definition leaves no real ability to use infinity in mathematically useful ways, e.g. Calculus.

You are if you are saying this thing has no limits when it defined within the specific limits of Real or Natural numbers as in the examples you gave. You are therefore saying that this thing is both limited and not limited simultaneously. Which is a contradiction. It cannot be A and ~A.

Incorrect. The natural numbers are the counting numbers, so they do no include the reals. That does not entail the Natural Numbers have a finite *cardinality*, it simply means the set of natural numbers leaves out particular types of numbers. This simply means the set of natural numbers has a particular size of infinity.

It comes down to semantics. Infinity can be considered a quantity in terms of transfinite math - so there are actually many "infinities" (aleph-0 is less than aleph-1; there are "more" real numbers than integers). But it's not a quantity in a sense that it corresponds to anything that exists in the material world.

I would say that space and time exist, and both are generally thought to be infinite.

The existence of an actual infinity (vs a potential infinity) is controversial among philosophers. I'm of the opinion an actual infinity cannot exist. I feel strongest about the impossibility of an infinite past, because that would entail a completed infinity: how could infinitely many days have passed?

Physicists accept the possibility of infinity in space and time simply because there is no known law of nature that rules it out. That doesn't imply the philosophical analysis is wrong, it just means that we don't know of any particular limits.

My opinions are consistent with the dominant opinion among philosophers prior to Cantor's set theory, but that doesn't seem like a very good reason to believe an actual infinity exists in the world.

A lot of this, in my estimation, doesn't make sense under scrutiny.

I'm of the opinion an actual infinity cannot exist. I feel strongest about the impossibility of an infinite past, because that would entail a completed infinity: how could infinitely many days have passed?

Um, before every day there is another day. QED. Or to put it more directly, the cardinality of the set of days prior to day "n" can be put into a one-to-one correspondence with the members of the set of natural numbers. Ergo, the number of past days are infinite. I don't know if this is actually true, but there is no logical argument against the *possibility* of it.

However, this wasn't even really what I was suggesting. Between any two moments of time there's another moment. That's what I had in mind. And it's even clearer with the divisibility of space. It's nearly always taken to be a continuum, meaning it would be infinitely divisible.

That doesn't imply the philosophical analysis is wrong, it just means that we don't know of any particular limits

What philosophical analysis? If we are adopting perfectly standard mathematics (or even most non-standard math systems) there is no contradiction whatsoever in supposing the past days are infinite. This will play into a bigger point I make at the end.

My opinions are consistent with the dominant opinion among philosophers prior to Cantor's set theory, but that doesn't seem like a very good reason to believe an actual infinity exists in the world.

I hope it doesn't come across rude, but that just reads as "If you ignore the last 150 years of mathematics most philosophers would agree with me". Well that's... a defense anyone can make to defend their belief in whatever.

Look, my broader issue/point is this. The interplay between our beliefs about the world and the formal tools (maths, logics) is more complex than often made out (i.e. the influence goes both ways). However, generally the idea is that our physics needs math to guide it's conjectures, and our beliefs about the world ought to be in line with the dominant physical theories. If maths has explicated infinity as a coherent, precise concept - and it has - then presumably it becomes irrational to say (as I understand you to be saying) that "Yea yea, there's infinity in mathematics and in physics, but if you try to apply it to real things it entails a contradiction." I just don't get it.

Infinity is not a contradictory concept, so how is it supposed to produce a contradiction if applied to real things? Or is it supposed to be a category mistake? But how does that work? We talk about infinite collections in mathematics all the time, it's central to set theory. That doesn't mean infinite collections (or other infinite whatevers) can exist in our universe, just that you cannot rule them out as incoherent and thus fail to obtain in every possible universe.

I'm of the opinion an actual infinity cannot exist.

I believe that beyond time and space, that infinity is only a mathematical construct.
We can not deal with infinite time or space, and nothing else in our world becomes infinite.

We measure the totality of energy by the rate of electrical flow times the amount of time, giving us Kilowatt hours.
If we multiply an infinite universe times the infinitely small, moving point in time in which it exists, perhaps the infinities cancel. This seems like a comfortable, perhaps pathological workaround to the notion that nothing can be infinite.

1. A quantity is a specified amount of something. It has a limit. The infinite is that which has no limits and so cannot be quantified. Therefore, not a quantity as not quantifiable. — Mr Phil O'Sophy

I would say that it simply isn't countable or computable. Yet it does in my view quite clearly define a quantity.

The problem is that math starts from counting. And hence we have all the problems with infinity.

If different infinities have different values then is it really a definite quantity?

Ammount of real numbers = infinite
Ammount of even real numbers = infinite
Does this mean that infinity divided by 2 equals itself?

Does this not violate the definition of a real number? Seems like there was something on this in "analytical geometry"...

Ammount of real numbers = infinite
Ammount of even real numbers = infinite

I meant whole, or natural numbers, not real numbers.

If different infinities have different values then is it really a definite quantity? — 3rdClassCitizen

Different levels infinities have different sizes. They're size is definite. "Infinity" is not one value.

Amount of real numbers = infinite
Amount of even real numbers = infinite
Does this mean that infinity divided by 2 equals itself?

The real numbers have a cardinality (size) larger than that of the even numbers. The even numbers have the same cardinality as the natural numbers, aleph-null, and so are "countably" infinite. The real numbers cannot be put into a one-to-one correspondence with the natural numbers, being that the reals are a larger infinity. Hence the reals are "uncountably" infinite.

Infinity is a Mathematical fiction and should be applied carefully to the World of Physical Things. For example we can say that there are an Infinite number of Natural Numbers. Natural Numbers are Mathematical concepts. But there can not be an infinitely large Pencil in the Universe. A good old fashioned Pencil is made out of a core of Lead or graphite (lets just say Lead). surrounded by a tube of Wood and then a coat of Paint. Take a point exactly in the center of the Lead and then let the Pencil grow in size to Infinity. You will have a Universe that is completely filled with Lead. You can never get to the Wood no matter how far you travel away from the center point (assuming we can travel through Lead). There will be no Wood or Paint in this Universe. The Pencil will become something less than it was when it becomes Infinite. You can not really have an Infinite Pencil.