Does infinity exist?

So to recap:

The definition that the infinite is a set that can be bijected with a subset of itself cannot be understood without already knowing what an infinite set is, so the definition is not a definition.

You’re welcome to submit a new one.

Haha, no, I am a very serious student of phenomenology and Russell is just facile in that respect. Heidegger is my orientation. Its about what we can expect semantics to amount to, before even syntax is arrived at. For example, the basic difference of verbs and nouns is of house entirely artificial but syntax relies on it 100%. Thats a true philosophic problem as see it and one Ive resolved but that is not for here. I will publish it eventually.

Russell and Wittgenstein both jumped directly to syntax. Wittgenstein returned from there, seeing a mistake. But Russell is just a specialist in syntax and I now respect him for that.

Greatly, basically from someone I wasn’t interested in anymore, to someone I will be looking into.

Thank you, this is very interesting indeed!

Perfect.
this moves in the direction I wished.

By the way in the meantime a remark for your discussion with Serendipper -

in f(x) x is indeterminate. That means there is no limit to it that is prescribed. But thats the basic idea of variables. That doesn’t make the possibilities necessarily infinite, but still without prescribed finitude.

See this is all semantics and not syntax. Pre-syntactic considerations for definition.

This is why James S Saint so much valued Definitional Logic.

I love things that enable us to compute reality without imposing excess weight of ideas and conditions on it.
When we don’t decide top down “this is this because we just called it that” - but when we look at what propositions do more or less automatically with themselves.
Like, what a proposition implies about assumptions is much more interesting to me than what it results in. Thats why I think philosophy is detective work, and why I love it.

I hope you and Serendipper find out the value in your conflict I can see both sides and this enabled me to mine it for some of my own substance.
Conflict is really good. Like a battery, two chemicals at war. Usually a third dog walks away with the bone, but sometimes the conflicting ones can also walk away with booty.
Which just means the power to think even deeper.

The timing of this math disclosure for me is perfect. Im going on a job, havent had an assignment in quite some while, why I was going so much into philosophy. Someone suggested it to me. He is always pretty smart about what I can do with free time. Last time he suggested I build a boat I didn’t do that but I should have. Maybe after this job.

I honourably greet you, men of science.
Ha dato frutti!

[tab]serpens iam procul est[/tab]

I don’t see that at all. Let’s say I’m sitting in a room. Sets come down a chute onto my desk. My job is to examine them and determine whether or not there’s a bijection from the set to a proper subset of itself. If NO, I toss the set into the a bin marked Finite. If YES, I toss the set into the bin marked Infinite. Note that Finite and Infinite are only meaningless words that label the two bins. The bins could say “Foozle” and “Gibbous” for all it matters.

So the set {1, 2, 3, 4, 5} comes down the chute. I carefully examine it, and I determine that there is NO BIJECTION from that set to any of its proper subsets. So I toss it in the Finite bin.

Now the set {0, 1, 2, 3, 4, 5, …} comes down the chute. I notice that it can be placed into bijection with the set {0, 2, 4, 6, …}, a proper subset of itself. So I toss it in the Infinite bin.

At no point do I have any prior knowledge of what an infinite set is. All I can do is determine whether or not there is a bijection between a set and one of its proper subsets. Galileo made this same observation in 1638. en.wikipedia.org/wiki/Galileo%27s_paradox. The same thing was noted by Indian and Arab mathematicians far earlier. Were they all part of the same mathematical conspiracy?

I don’t follow your objection.

Well, I’m using the exact same definition that YOU GAVE IN YOUR INITIAL POST:

Do you happen to have a cat? The reason I ask is that my cat often types on my keyboard and writes things on discussion forums without my knowing. I wonder if this is what happened to your initial post, since somebody using your handle gave this exact definition.

Peace, brother.

How did you make the determination that the set can be placed into bijection with the subset without having prior knowledge that the set was unending/unlimited/unbounded?

There is no way. You must know the set is unending before you can show the bijection.

I have no idea. I do know Galileo did not like the church since they censored him and placed him on house arrest for heresy where he remained until he died. Anyway this appeal to authority is beside the point and more groping in vain attempt to substantiate dogma rather than admitting you’re wrong and moving on.

I just copied the dictionary as a starting point, but it turned-out that particular definition cannot be a definition which is something I didn’t know at the time. That is the purpose of discussions: to discover truths and errors.

@Serendipper

I never agreed to that.
There are two kinds of infinities, a general infinity, which means the One has no limits whatsoever, and a particular infinity, which means a thing is unlimited in some ways, and limited in others.

It’s not nonsense, it’s conceptually possible, and in actuality it may very well be the case.

Me: It doesn’t matter if you could add or subtract road, the fact of the matter is, at this moment, the road ends this way, and it doesn’t end that way.

Matter, energy, space and time.

It doesn’t matter if you can add or subtract apples, there’s still an infinite amount of total apple.

We don’t know that for sure, as of yet our instruments may be too blunt to detect things smaller than quarks.

And for all we know, everything is essentially made up of waves, much to the chagrin and dismay of scientific materialists, who prefer to carve everything up into neat little bits and pieces.

Or maybe Aristotle was right when he uttered: “The whole is more than the sum of its parts”.

It’s not logically or grammatically incorrect to say a wall is infinitely tall, or infinitely wide, a thing can be infinite in some quality without having to be infinite in all qualities.

Whether it’s empirically incorrect is another matter.

I take this objection seriously. I’ve asked it of myself from time to time. I’ll outline my thoughts.

  • First, if we accept the axiom of infinity, which says that there’s an infinite set, we can establish the bijection with no trouble. But the axiom of infinity bakes infinity into mathematics, so it’s too strong for our purposes.

The next step back from the axiom of infinity is the Peano axioms. Here we have each of the natural numbers 0, 1, 2, 3, …, but we do not have a completed “set” of them; where a set is defined as something that obeys the particular axioms you’re assuming.

It was in this context that Galileo made his famous observation that the counting numbers may be placed into bijection with a proper subset of themselves, in particular the squares. That is not a “appeal to authority” as you put it, but rather a statement of historical record. 1638, in his final work, Two New Sciences. You can find a copy online these days.

More generally if I assert that the planets revolve around the sun, I am not making an “appeal to authority” when I accept this universally agreed upon conclusion of the greatest scientific minds in history. “Oh yeah, Newton and Einstein, you’re just appealing to authority.” Is that the argument YOU are making here? That Galileo was some jerk and YOU know better?

If you take a step back and view your position with objectivity you will agree that the burden of proof is on you.

Before analyzing the case of Peano, let me first cover the third alternative: You deny that all of 0, 1, 2, 3, … exist. You claim that at some point, there aren’t any more. You deny not only infinite sets, but mathematical induction too.

That position is called Ultrafinitism.

I completely agree with you that from an ultrafinitist position that it’s meaningless to talk about a bijection. There’s no map that inputs n and outputs 2n. At some point you put in a big n and it says, “Sorry Dave, I can’t do that.”

Ultrafinitism is a really interesting idea. There have been a couple of serious ultrafinitists, though many more adherents are cranks. That said, ultrafinitism is useless. Even if it’s true it’s useless. You can’t do math with it and if you can’t do math you can’t do science and then we’re back to living in caves and throwing rocks. If you reject mathematical induction you lose all of finite math, combinatorics, everything. First you threw out calculus, and now basic probability theory?

So I reject ultrafinitism for the same reason I reject solipsism. Even if it’s right it’s worthless. It’s an essentially nihilistic philosophy.

  • Ok. Back to Peano. We have 0, 1, 2, 3, … and each one of them “eventually” exists. We have the law of induction; which says that if 0 exists; and if whenever n exists, n+1 exists; then all natural numbers exist. Of course “exist” just means mathematical existence.

Now what sense can we make of a notion such as the function, or mapping, denoted as f(n) = 2n, which inputs a natural number n, and outputs an even number 2n?

If we are in the realm of ZF set theory, this is no problem. Given the SET of natural numbers we can form another set that serves the role as a function; namely, the set of ordered pairs (n, 2n).

But without the axiom of infinity, we can’t get that off the ground. What, then, is a function?

I think that this is the core question you are asking. I have a strong answer.

  • The map n → 2n is a computable function on the natural numbers. That is: There is a Turing machine that, when given the number n, eventually outputs the number 2n, in a finite number of clearly defined steps.

Therefore even if someone rejects the axiom of infinity; if they are a constructivist who demands that all mathematical concepts and objects should be computable; then the bijection between the naturals and the evens exists.

Do you see my point? In order to define the map that sends n to 2n, I do not need any “a priori” unbounded or infinite sets or collections. All I need is a FINITE string of symbols that represents the operation of a Turing machine (or a Python or Java or Javascript program, same thing) that inputs the number n, and outputs the number 2n. And such a thing exists.

A computer program is a purely finitistic object. It’s a finite string of symbols. A finite string of bits if you like. And there’s a computer program that doubles any natural number.

So: I say to you:

If you are a constructivist, you must accept the bijection.

If you are an ultrafinitist, you are a nihilist. I can’t defeat your argument with logic. Only with practicality. So in the end my position is based on pragmatics. Usefulness.

Ok thanks for reading I think there’s a good anwer there, I hope you’ll give some thought to what I said.

Haha. I’ll resist the temptation to point out that not only don’t you seem to read what I write; you don’t even seem to read what YOU write. I’ll keep that in mind. No just kidding. I think you asked a very good question and I gave you my best answer.

** tl;dr: If you are an ultrafinitist, you are right. I can’t defeat your idea with logic. Only with a claim to practicality. And by practicality I mean ALL of known math and science. My gosh, you even reject induction? That’s a lot worse than just rejecting the axiom of infinity.

On the other hand if you understand that the map that sends n to 2n is expressible in a finite number of symbols; and is a computable function in the sense of Turing; then that mapping exists; without any need to invoke boundless collections. Unless you are ready to throw out all of computer science too?

To me your argument is irrefutable; yet nihilistic. It denies everything.

My ego lured me back. The weekend seems legit.

first I thought Serendipper had hit it on the head and nailed it but then I was convinced again by the argument about it being computable.

I thought it was pretty much standard approach though, that he started out with a definition that he just wanted to test.

The definition is pretty damn resilient in its ability to be validated, and also to clarify what we are talking about when we mean mathematical infinity.
As a way of nuancing you could suggest that infinity is a mathematical presupposition required to biject sets with their own subsets. Lol. I mean math is a circle, right?

As we know not in all logics the lack of yes means no and the lack of no means yes, some have for example null when the machine can’t allocate the question to an answer.

like if we apply Peano to a trivalue logic, for the proposition that the set of natural numbers is finite, we get not false but null.

Thats probably not irrelevant if you want to do quantum computing, which maybe for that reason hasn’t emerged yet even though they said they were about to crack it 20 years ago.

I think maybe you sometimes need to roll up infinities or wrap them up. I know thats what they do with whole dimensions in string theory.

Come think, every infinity is a dimension.
So thats why its a problem I see now that things can be present in different sets.

Yeah all this points to type theory, because it doesn’t predicate infinity but just allows for it, or basically allows for finitude to be endlessly postponed but still dominate whats actually there.

I think these are actually philosophical problems, like nitpicking, or being impossible, its really just showing the weak links in the theory, the points where it can be challenged. Its not an attack to destroy but to mine the rock for its minerals.

Also its a loss of pleasure. Infinity if it can’t be defined with some empirical tasty arguments, then the whole concept of it kind of diminishes. If you could show infinity to be as elegant as a galactic cylinder or matrix of primes - but well there you have it. Infinity is really well disclosed when you look at the logic behind the succession of primes.
There is absolutely no logic to be found. Thats a fucking infinity for the mind, a total mind bender, which is what infinity requires to be if it can be at all computed philosophically.
Because the phenomena that math describes are important, and it is not cool to have math deviate too lightly from these. Not out of conservatism, but a miners instinct… more knowledge might be scattered in the rocks nearby than far off.

Then how do you define infinity?

Unlimited in some ways and limited in others is unlimited in quantity and limited in identity/category. One thing is sure: we cannot have a quantity which is both limited and unlimited, so if there are limits, they are not imposed on the quantity that is said to be infinite, but rather they are imposed on the identity of the uncountable things.

I can’t conceptualize it and you can’t either. In actuality, where in the universe do you suspect that it may be possible to draw an infinite line in one direction, but not in the other?

Then, at this moment, it is not infinite because there exists a place for more road. How can you propose having an infinite road when clearly we could make it longer? A road that is truly infinite would extend around and around the universe many many times until it occupied every planck cube in the universe, completely displacing all matter, and until it eventually connected with itself for lack of having anywhere else to go. To say that isn’t so is to say the road has a boundary which would make it not infinite.

Ok, now I forgot why I needed to know that lol

Yes it does matter because if there is a place for another apple, but no apple is there, then we have found a boundary and therefore the number of apples is not infinite.

Size must have a zero like temperature and speed or else it couldn’t exist. We can’t get infinitely colder, infinitely slower, infinitely smaller and if we could, then temperature, speed, size would have no significance/meaning.

I think so.

I don’t know… maybe.

Infinitely wide is ok because it extends in both directions, but infinitely tall would be a wall that extends around the universe until it connected with the other side of the earth.

Huh? Infinite doesn’t mean maximized or maximal.

Pretend you’re talking to a 5-year old kid who doesn’t know what infinity means. If you say “we have an axiom of infinity”, the kid will look at you stupid. If you say, “we can align the set with a subset of itself”, the kid will look at you stupid. If you say, “the infinite is the unbounded, unlimited, unending” then the kid will say “Ooooh!” You cannot do any of these acrobatics (axioms, bijections) until you make it clear what infinity means.

There is no way I could perform a bijection of a set with a subset without already understanding the set is unending. So I would need to know what infinity means before I could understand the definition.

I accept that planets revolve around the sun not because of authority, but because it’s the most sensible scenario.

The infinite is something that cannot be observed (much like god) and there is no evidence for (much like god), yet I am required to prove that it doesn’t exist? There is also no evidence of a teapot orbiting the earth, so do I have to prove that doesn’t exist too?

What’s the biggest number you can think of? Now make it bigger: square it, factorial, define new symbols to reduce the size and continue on and on until you run out of room on the forum to write that number down. Regardless what you devise, you will find a biggest number, but you won’t be able to do anything with it other than bask in its glory.

The issue isn’t whether the bijection is possible, but whether it defines infinity.

I can integrate an area over a height to yield a volume without using infinity. I can add my grocery bill without infinity. What do you need infinity for?

What’s the limit of 1/x where x → 23+10^10^10^10^10^10^10^10^10^10? Zero right? If it’s not zero, then how far am I off? No machine that could ever exist will be capable of discerning the difference, so from the perspective of practicality, we don’t need infinity.

Induction isn’t the same as deduction and if n exists, you cannot say with certainty that n+1 exists. However, if n+1 exists, you can say with certainty that n exists (deduction).

Yes I understand your point, but we still must have a priori understanding that the turing machine never ends. Once again, infinity must be understood before it can be defined with the turing machine.

Yes it does because there are no boundaries. If there are boundaries, then it’s not infinite.

If you claim there are infinite apples and I find a place with no apples, then you don’t have sufficient apples to go around in order to fill every place and therefore the quantity of apples is limited or else there would be apples in every place that apples could exist.

If there could be an apple here, but there is no apple here, then the only reason to explain that is insufficient apples.

If I would claim there are infinite apples available, then yes that would apply. Also with one apple. But to have infinite apples in an infinite universe would not necessarily guarantee any apples within reach.

Like if time is infinite, which it is since beginnings are part of the concept time, then even if you are born now, some of your actions will still have effects for an infinite duration, unless the universe collapses into a singularity which I think is fantasy.

So your deeds would (do) resound in the infinite future, but not in the also infinite past.

Of course it would be easy to negate infinity if you start out with the assumption of a finite universe.

You can just say “the universe is finite, thus it isn’t infinite, thus nothing in it is infinite”.
But who will think this makes sense? Only people who already agreed with you on faith.

You can’t prove an end to the universe, or to a straight line, so to insist it does have one (i.e isn’t infinite) is like being really ambitious without any means.