When/How does Infinity Become Infinite?

Devans99

As people have pointed out - and I acknowledge - maths is not my strong point. So I would be grateful to get input from those will more skill in the discipline on the following arguments:

1. $\infty=\sum_{n \to \infty} 1$
2. $= 1 + 1 + 1 + 1 + ...$
3. The first term in the series [2] is finite
4. If n is finite, then n+1 is finite
5. So by induction, the sequence [2] is finite for all natural numbers n
6. What about for n=∞? Well for no individual (n+1) step on the way to ∞, does a finite number ever become an infinite number. And infinite steps are just the sum individual steps, so it seems ∞ retains the property ‘is finite’ even at 'infinity'?

I think step [6] above is no doubt questionable, but it brings out the point: how exactly does a finite number ever become infinite? - We have no basic arithmetical operators to convert finite numbers into infinite numbers. To focus on this aspect, here is a similar argument that more graphically brings out the discontinuity between natural numbers and infinity:

A. $\infty=\sum_{n \to \infty} 1$
B. $= 1 + 1 + 1 + 1 + ...$
C. 1 is 0% of the way to counting all numbers / reaching infinity (because 1/∞=0%)
D. If n is 0% of the way to counting all numbers / reaching infinity, then so is n+1 (n+1/∞=0%)
E. (eg 100^100/∞=0% of the way to counting all numbers / reaching infinity)
F. So by induction on [C] and [D], for no natural n are we greater than 0% of the way to counting all numbers / reaching infinity
G. ∞/∞=UNDEFINED, so infinity is never in this sense 'completed'. In fact there is no number n such that n/∞=1%, n/∞=10%, n/∞=50%, so the sum of the series never gets anywhere near infinity / counting all numbers - there seems to be a massive discontinuous 'train wreck' at the end of the series associated with 'infinity' - it is unattainable?

Devans99

I think on step 6 above, saying 1*∞=∞ is a circular definition so is not too enlightening.

I think my doubts are around how infinity is defined. Limits seem to do a good job of defining potential infinity but how is actual infinity defined? Set theory defines 'countable actual infinity' as the the 'cardinality of the set of natural numbers' - which is about as meaningless a definition as could be imagined.

Actual infinity is not computable and not (it seems) definable, leading to a suspicion that it cannot be a real concept - it may exist only in our minds (along with talking trees and square circles) and is therefore not realisable.

Maybe @JohnDoe, @softwhere or @John Gill have opinions?

Amergin

By definition it would never become infinite, that's just what it'd be; there could be no point of becoming.
To tackle it from another angle, my cousin, who has a degree in mathematics, explained to me, there are infinities of different sizes. There is an infinity between the numbers one and two because numbers are infinitely divisible. So there's no need for conversion between finite and infinite: the two are part of the same system.

Devans99

There is an infinity between the numbers one and two because numbers are infinitely divisible. — Amergin

This point in a slightly different guise, is discussed at length on another thread:

https://thephilosophyforum.com/discussion/comment/362707

In short (and my opinion only) - a number has zero width, so the number of numbers between 1 and 2 is 1/0=UNDEFINED and not infinity.

If a number has non-zero width, that leads to a finite number of numbers between 1 and 2.

tim wood

If a number has non-zero width, that leads to a finite number of numbers between 1 and 2. — Devans99

How many might that be?

Devans99

↪tim wood

Numbers exist in our minds only so we can imagine a zero width number and an UNDEFINED number of numbers in a finite interval.

If you were to try to represent numbers in the reality external to your mind, you would find that any line used to represent the real number line would be composed of molecules so there would be a finite number of numbers on a real line segment.

jgill

It's interesting that the axiom of infinity and the axiom of choice can lead to such lengthy discussions. On the surface they seem so benign. The first simply says there is a set containing the natural numbers and the second says, roughly, that you can pick an item out of each set in a collection of sets. Most mathematicians go serenely about their investigations giving these two axioms little to no thought.

I have already given my opinions of infinities as being limit concepts, with virtually no mention of "the point at infinity". Infinitesimals, on the other hand, have been placed in a proper mathematical model and can be used to generate calculus. I would think these tiny little objects might generate more controversy than infinity, but apparently not. As part of the hyperreal number system they are intimately connected with infinities.

I never went beyond naive set theory, so these mathematical notions are merely mildly amusing. Keep in mind what I said before: the foundations of mathematics is a relatively recent subject that attempts to place all the math that has worked so well over millennia in "proper" logical frameworks. I suspect there are a few analytic philosophers on this forum who know far more than me.

tim wood

we can imagine... — Devans99

...whatever we want, and in our imaginations we can dispense with any need for consistency - or anything else at all; after all, it's just our imagination. But dreamland is not a reasonable ground for argument, so let's leave it.

If a number has non-zero width, that leads to a finite number of numbers between 1 and 2. — Devans99

This is what you wrote, not me or anyone else. Since it's a finite number, likely it's calculable. Or at least describable. So, take your least width - to get the most - and tell us how many.

Devans99

↪tim wood

There are only three possibilities. A number of the real number line must have:

1. A zero width
2. A non-zero width
3. A undefined width

So how many reals between 0 and 1?

1. 1/0=UNDEFINED
2. 1/(some finite small number)=(some finite large number)
3. 1/UNDEFINED = UNDEFINED

I believe that a number is a purely imaginary concept so we can imagine it to have zero width so definition 1 above is what we do in our minds. This is also consistent with the definition of a point in maths as having zero extent - so we can imagine a point corresponding to a number on the real number line.

Now you could claim that a number has an infinitesimal width. But that is 1/∞ and in my opinion ∞ and its inverse are not well defined - the point of the whole OP.

tim wood

purely imaginary concept so we can imagine — Devans99

Maybe this is it. Two very different meanings of "imagine."

1) Ok
2) Finite means definite. But it's only definite for a particular value. But no particular value can be assigned, therefore not finite.
3) Ok.

Infinitesimal, sure. 1/∞ is not well-defined in terms of finite numbers. But that's the wrong criteria.

jgill

When you get down to the nitty gritty and use numbers, as in a computer program, they are all rational and thus countable. So how many rational numbers (fractions) do you think lie in the interval [0,1]? :chin:

Devans99

2) Finite means definite. But it's only definite for a particular value. But no particular value can be assigned, therefore not finite. — tim wood

As you can imagine I am not a believer in the possibility of continua occurring in nature and I am not alone in this belief (eg loop quantum gravity). If spacetime is actually discrete, then case 2 above will have a particular, assignable, finite width. Time will tell. I'd point out that we will never, ever be able to empirically prove spacetime is continuous, but we may be able to prove it is discrete.

Devans99

When you get down to the nitty gritty and use numbers, as in a computer program, they are all rational and thus countable. So how many rational numbers (fractions) do you think lie in the interval [0,1]? :chin: — John Gill

Depends on the hardware/software you are using I guess. Computers use discrete binary representations of numbers. Computers are real. Reality is real. Continua maybe just a figment of our imagination as far as reality goes. Time will tell.

tim wood

As you can imagine I am not a believer — Devans99

Near as I can tell, all of your arguments are conditioned by an IF, whether belief, opinion, imagination, even small-i if. You use that IF as a fence behind which you indulge whatever your IF and your "imagination" will give you. Behind this redoubt you're unassailable, but also useless and a waste of time.

NOS4A2

↪Devans99

There would be no “when” because there is no beginning or end to infinity. I’m prone to think a symbolic representation of infinity suggests finitude, but then again I am a layman in math.

Devans99

↪tim wood

All your counter arguments have been countered so you resort to mud slinging. Just typical of you.

tim wood

All your counter arguments have been countered so you resort to mud slinging. Just typical of you. — Devans99

There's [il'esprit d'escalier][/i], but darned if I can find spirit of the playpen.

Devans99

↪tim wood

Thats not a counter argument.

tim wood

↪Devans99

That's ok, yours aren't arguments. If only they were.

A Seagull

↪Devans99

Yes you have a good point. It seems that most people do not understand infinity.
Infinity just means 'without end'. It is not a number and cannot be used in any calculation.

You can count the integers as far as you like and you will never reach 'infinity'. So to say that the integers are infinite is just to say that you can count them as far as you like without reaching the end.

Similarly no physical measurement can ever be 'infinite'. So to claim that time or space are 'infinite' is just to hypothesize that the end of time or space can never be reached.
To hypothesize that some things are infinite, such as hotel rooms, is to enter a fantasy world, where nothing is real.

Devans99

↪A Seagull

Thanks @A Seagull!

This forum has many folk like @tim wood who accept the received 'wisdom' of Cantor and co without question - it is refreshing to talk to someone with an open mind!

BTW Did you know the reason actual infinity is enshrined within maths as a number is that Cantor was a devout Lutheran, believed that God was infinite and believed that God was talking to him telling him to put infinity into maths!

Fast forward to today and ironically, it is mainly the atheists who believe in infinity - theists and deists often question its existence and rightly so.

quickly

I think step [6] above is no doubt questionable, but it brings out the point: how exactly does a finite number ever become infinite? - We have no basic arithmetical operators to convert finite numbers into infinite numbers. To focus on this aspect, here is a similar argument that more graphically brings out the discontinuity between natural numbers and infinity: — Devans99

The correct terminology is that the series $\sum_{i = 0}^\infty 1$ diverges. In other words, the limit of the partial sums $\sum_{i = 0}^n 1$ does not converge to a number as $n$ tends to infinity. The notation $\sum_{i = 0}^\infty 1 = \infty$ does not mean that the series equals some number called 'infinity' and denoted ' $\infty$ '.

softwhere

.

Devans99

↪quickly

I agree with you, $\sum_{n \to \infty} 1$ , tends towards but never attains a purely imaginary concept/state called infinity. (Actual) Infinity is not a number. The arguments given in the OP lend weight to this claim.

Accepting the non-numeric / purely imaginary / unrealisable status of infinity implies:

- The commonly given definition of infinity is wrong: 'a number greater than any assignable quantity or countable number'
- Transfinite arithmetic is a work of pure fiction
- Ideas about space and time that assume the existence of actual infinity are not mathematically sound

quickly

↪Devans99

You are misinterpreting my response. I was responding to your argument by clarifying what people mean when they say things like "the series equals infinity." This is simply a shorthand for "the series diverges" or "the limit of the partial sums does not converge to a real (complex) number." It doesn't follows that cardinal and ordinal arithmetic is "false" (I'm not even sure what that means) or that space and time cannot be unbounded.

Devans99

↪quickly

My argument in the OP is that alpeh-zero must be finite. What problem(s) do you have with that argument?

quickly

↪Devans99

By definition, $\aleph_0$ is the cardinality of the natural numbers. Your argument does not establish that the natural numbers have finite cardinality. I thought that clarifying what people mean when they use the series notation might disabuse you of the notion that there must be some "transition" from "finite numbers" to "infinite numbers" happening.

Devans99

By definition, ℵ0 is the cardinality of the natural numbers. — quickly

Just a definition. Sort of like defining ∞=∞. Meaningless IMO.

Your argument does not establish that the natural numbers have finite cardinality. — quickly

Why?

jgill

Cardinality has to do with sets being in a 1:1 correspondence with each other. $\aleph_0$ is not a number in the common sense of the word. It denotes an equivalence class. It is not infinity. The terms "bijection", "injection", and "surjection" apply to functions and replace words such as "onto" or "into", which were easier to comprehend BITD, IMHO. They were introduced by a mysterious Frenchman, whose identity is baffling, in the 1930s. (Please don't try to explain "who" this was!) :nerd:

quickly

The problem with the traditional terminology is that "one-to-one" is ambiguous between injection and bijection. If you think about the etymology of each word, they are very descriptive: iacere means to throw or cast, and the prefixes (from French/Latin) describe the ways elements of the domain are "thrown" or "cast" into the range.

jgill

One-to-one and onto was not so difficult to say. But your generation learned differently. And it's the coin of the realm, now. :nerd:

When/How does Infinity Become Infinite?

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