## Does infinity exist?

For discussing anything related to physics, biology, chemistry, mathematics, and their practical applications.

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### Re: Does infinity exist?

wtf wrote:
Serendipper wrote:So you concede then that the infinite is boundless? How else could inch cubes and feet cubes correspond?

I responded to your own hypothetical about infinite space. Your response is disingenuous. You asked, IF space is infinite, is the number of cubic inches equal to the number of cubic feet. You posed a hypothetical. Jeez man.

Jeez man, answer the question: How else could inch cubes and feet cubes correspond?

Serendipper wrote:What I want to know is if there is any way sets can biject without being boundless.

Yes, the unit interval [0,1] bijects to the interval [0,2]. Both intervals are bounded.

Then what is the bound?

Show me the bound where the bijection ends.

If you cannot show me the bound that terminates the bijection, then you simply must concede that the set is unbounded.

There is no escaping this one: either show me the bound or admit it doesn't exist.
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### Re: Does infinity exist?

wtf wrote:
Serendipper wrote:Infinity comes first, then one performs the bijection. Obviously we can perform bijection before the set exists.

Galileo noted in 1638 that the set of counting numbers can be bijected to one of its proper subsets, namely the perfect squares.

This was 240 years before the advent of set theory. Bijection is a perfectly sensible notion even without a theory of infinite sets. The fingers on your hand are in bijection with the fingers on your glove.

This is irrelevant history and demonstration of finite bijection which still proves my point that the fingers must exist before the bijection.

The infinite set must exist as an infinite set BEFORE bijection with a subset of itself can happen, so bijection CANNOT be a definition for infinity.

The a posteriori cannot be the cause of the a priori.
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### Re: Does infinity exist?

Serendipper wrote:Jeez man, answer the question: How else could inch cubes and feet cubes correspond?

Consider the real number line. If we measure it in feet, it's infinitely many feet long. If we measure it in inches, the same. The cardinality's the same. How many copies of [0,1] are there in the real line? Countably many. How many copies of [0,2]? Countably many.

Serendipper wrote:
Then what is the bound?

The unit interval [0,1] is bounded by 0 below and 1 above. If this is unclear to you, please tell me what's unclear about it so I can attempt to explain.

Serendipper wrote:Show me the bound where the bijection ends.

In this particular case, the positive reals are unbounded above. (Though bounded below, by 0). Of course SOME infinite sets are unbounded. You're having quantifier problems.

Serendipper wrote:If you cannot show me the bound that terminates the bijection, then you simply must concede that the set is unbounded.

I've shown you this same elementary example, obvious to a high school student, half a dozen times already.

Serendipper wrote:There is no escaping this one: either show me the bound or admit it doesn't exist.

I've shown it over and over and over. You just keep coming back with the same question, as if you either didn't study any math after eighth grade, or just want to argue for the sake of arguing. I can't go back and forth with you anymore. You're denying high school math.
Last edited by wtf on Fri Nov 09, 2018 7:06 pm, edited 1 time in total.
wtf

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### Re: Does infinity exist?

wtf wrote:
Serendipper wrote:
wtf wrote:you can biject (0,1) to the set of positive reals via f(x) = 1/x.

I don't need it to show the reals are infinite, the (0,1) examples already do that.

How do you show infinite f(x) without having to first prove there are infinite x?

I don't understand the question. Do you deny that the unit interval (0,1) is in bijection with the positive reals via the map f(x) = 1/x? This is a fact familiar to high school students.

The chain of logic is as follows.

* I note that (0,1) is a proper subset of the positive reals.

* I note that (0,1) is in bijection with the positive reals via the map 1/x. If you doubt that you need to review your high school math.

* Since the set of positive reals are in bijection with one of its proper subsets. the positive reals are infinite by definition.

* I can do the same thing for the entire set of reals using the tangent/arctangent, but then you needed to have taken high school trig. The 1/x example has a more modest mathematical prerequisite so it's preferred for this conversation.

No, it's a very easy question: how do you show there are infinite f(x) without already having infinite x?

You must have, in your possession, infinite x in order to plug all infinite x into f(x) in order to show there are infinite f(x). So, first you must prove there are in existence infinite x and THEN you can move to the next step, which is proving there are infinite f(x). So your example (0,1) is moving the goal posts, which I pointed out on page 1 and have been trying to get you to respond to ever since.

You must prove there are infinite x before you can perform any function of x and that includes bijection with one of its subsets. First prove the set is infinite, then show the bijection, then show the f(x) is infinite.

- The bijection with a subset cannot exist unless the set is infinite, so prove the set is infinite BEFORE showing the bijection.
- The infinite set of x must exist BEFORE infinite f(x) can be shown, so prove there are infinite x before proving there are infinite f(x).
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### Re: Does infinity exist?

wtf wrote:
Serendipper wrote:Jeez man, answer the question: How else could inch cubes and feet cubes correspond?

Consider the real number line. If we measure it in feet, it's infinitely many feet long. If we measure it in inches, the same. The cardinality's the same. How many copies of [0,1] are there in the real line? Countably many. How many copies of [0,2]? Countably many.

How is that not a hypothetical? Or can you prove there are infinitely many real numbers? If you can't prove it, then it's no different than my hypothetical oranges or space.

Serendipper wrote:
Then what is the bound?

The unit interval [0,1] is bounded by 0 below and 1 above. If this is unclear to you, please tell me what's unclear about it so I can attempt to explain.

If it's bounded at 0 and 1, then it's not infinite. Show me how it's infinite if it's bounded.

You're denying high school math.

And you're appealing to the ridiculous.
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### Re: Does infinity exist?

Serendipper wrote:Or can you prove there are infinitely many real numbers?

I already did, three different ways: The tan/arctan bijection, the 1/x bijection, and Cantor's beautiful bijection between the reals and the irrationals. How many more proofs do you need?

Serendipper wrote: If you can't prove it, then it's no different than my hypothetical oranges or space.

True. However I DID prove it (three different ways), so the antecedent of your implication is false.

Serendipper wrote:f it's bounded at 0 and 1, then it's not infinite. Show me how it's infinite if it's bounded.

It's infinite and it's bounded. It's an infinite set, and none of its elements are less than 0 or more than 1. Did you take high school math? Are you sitting here claiming the real numbers are finite? What's the largest one?

Serendipper wrote:And you're appealing to the ridiculous.

Someone claiming there are only finitely many real numbers (or that the unit interval is unbounded) surely has the burden of proof.

Peace, brother. All the best.
wtf

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### Re: Does infinity exist?

wtf wrote:
Serendipper wrote:Or can you prove there are infinitely many real numbers?

I already did, three different ways: The tan/arctan bijection, the 1/x bijection, and Cantor's beautiful bijection between the reals and the irrationals. How many more proofs do you need?

I need one proof. You haven't proved "The tan/arctan bijection, the 1/x bijection, and Cantor's beautiful bijection between the reals and the irrationals" until you prove there are infinite x.

First prove that, then you can prove the others.

Serendipper wrote:f it's bounded at 0 and 1, then it's not infinite. Show me how it's infinite if it's bounded.

It's infinite and it's bounded. It's an infinite set, and none of its elements are less than 0 or more than 1. Did you take high school math? Are you sitting here claiming the real numbers are finite? What's the largest one?

You didn't show anything, but simply called me stupid.

x = 1,2,3,4,5

2x = 2,4

The subset 2x cannot be bijected with x BECAUSE the set x is finite, but what does finite mean? (limited/bounded)

x = 1,2,3,4,5,...

2x = 2,4,6,8,10,...

The subset 2x can be bijected with itself BECAUSE the set x is infinite, but what does infinite mean? (unlimited/unbounded)

What do the three dots (...) mean? <-- you must define that BEFORE conveying what you mean by "bijection with a subset of itself".

The three dots mean "unbounded".

It is only after I understand that the three dots mean the set is unbounded that I can then proceed to understand what you mean by being bijected with a subset of itself, so the bijection cannot be the definition of the three dots.

It doesn't mater what set we choose:

Here's your (0,1) which is f(x) = 1/x where x>=1, so x = 1,2,3,4,5,....

f(x) = 1/1,1/2,1/3,1/4,1/5,....

You still have to show what the three dots mean before showing anything else.

Therefore, whatever you show as a result of understanding what the three dots mean then cannot be used to define what the three dots mean.

Your definition of the infinite is invalid due to circular reference.

IOW, if someone never encountered the concept of the infinite, they could not possibly understand the concept via that definition because already having an understanding of the infinite is required in order to understand the definition.

Serendipper wrote:And you're appealing to the ridiculous.

Someone claiming there are only finitely many real numbers (or that the unit interval is unbounded) surely has the burden of proof.

You said before there can't be infinite oranges and you specifically said "What on earth do you mean by conceptualizing infinitely many oranges? What are they made of?"

But yet you can conceptualize infinite numbers lol

Well, to quote you again, "what are they made of?" Nothing? So then with "nothing" as your basis, you claim infinity exists in some way? Well they aren't nothing; they are concepts and your mind is finite.

Infinity cannot be conceptualized; we can work with the implications/ramifications of it, but we cannot conceptualize infinite real numbers.

Further, any universe in which anything can exist (and that includes all universes in imagination) there cannot also exist infinities because something cannot be conceived/beheld/perceived as a universe in imagination or reality without being bounded/finite/definite. The infinite makes existence impossible.

Peace, brother. All the best.

You're brother to someone who doesn't understand high school math?
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### Re: Does infinity exist?

Serendipper wrote:You didn't show anything, but simply called me stupid.

I did not call you stupid. But if you insist, consider it done.
wtf

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### Re: 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.
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### Re: Does infinity exist?

wtf wrote:
barbarianhorde wrote:I was pushing this topic a bit because I had been waiting for a good reason to get somewhat deeper into math. That worked very well, I now also have a better perspective on Russell, because I looked down on him pretty dramatically

I don't think any of us are in a position to do that! He was a great man in multiple endeavors. I don't know much about him, but why do you dislike him? Is it the pipe?

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.

barbarianhorde wrote: but I never knew of his type theory.

How does that change your estimation of Russell?

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

barbarianhorde wrote: I think that may be a bit underestimated, so far. I think it may become important in the future.

There is no question that type theory is making a comeback as a potential foundation for math. The drive in this direction is from automated checking of proofs. There's a big movement called homotopy type theory (HOTT) that's the buzzphrase and Wiki page to know.

Thank you, this is very interesting indeed!

One thing I know from type-theoretic approaches to foundations is that they are associated with denial of the law of the excluded middle. I know very little about this entire area. It makes sense that this idea would gain currency in our age of computers; since there are many sets of natural numbers such that neither they nor their complement are computable. So neither is "true" in a world where truth is whatever can be determined by an algorithm.

The mathematical philosophy of all this goes under the name of intuitionism (if it's classic 1930's Brouwer type) or neo-intuitionism (if it's more recent).

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.

barbarianhorde wrote:I wish I was good enough at math to compare this to type theory.
Still, I can begin to work at it now.

What leads you to your interest in type theory? Just curious. Well if HOTT eventually gains majority mindshare, I will think of that as Brouwer's revenge. There is another alternative
foundation that's made huge inroads into much of modern research-level math, Category theory. Attitudes toward foundations are a matter of historical contingency.

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.
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### Re: Does infinity exist?

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!

serpens iam procul est
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### Re: Does infinity exist?

Serendipper wrote:
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.

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. https://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?

Serendipper wrote:You're welcome to submit a new one.

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

Serendipper wrote:5. Mathematics: not finite. (of a set) having elements that can be put into one-to-one correspondence with a subset that is not the given set.

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.
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### Re: Does infinity exist?

wtf wrote:
Serendipper wrote:
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.

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.

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.

Galileo made this same observation in 1638. https://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 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.

Serendipper wrote:You're welcome to submit a new one.

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

Serendipper wrote:5. Mathematics: not finite. (of a set) having elements that can be put into one-to-one correspondence with a subset that is not the given set.

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.
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### Re: Does infinity exist?

@Serendipper

Gloominary: For example, a road could be endless in one direction, but endful in another.

I attacked this in the OP.

Infinities cannot have beginnings or ends because those are boundaries and we said in the beginning that infinity has no boundaries.

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.

We cannot divide infinity in half and say infinity is bounded by this finite location and extends to infinity in that direction; it's nonsense and breaks our definition of infinity being boundless.

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

You: Here is a road that starts here and goes to infinity.
You: What do you mean?
Me: Where you're standing... we could add more road and since we could add more, then it's not infinite, but bounded by the location where you're standing. If we could add more road, then the amount of road we have is not infinite.

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.

I don't know what MEST is.

Matter, energy, space and time.

Yes it does because if there is a place that we could add another apple, then the amount of apples is not infinite. If there are infinite apples, then apples exist in every place there is to exist.

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

We can divide down to quarks, but to divide a quark pair will create another quark pair from the energy used to divide them. There is no such thing as smaller than a quark because smaller than a quark is smaller than the thing that defines size.

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".

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### Re: Does infinity exist?

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.

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### Re: Does infinity exist?

Serendipper wrote: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 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.

Serendipper wrote: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.

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.
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### Re: Does infinity exist?

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?
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### Re: Does infinity exist?

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.
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### Re: Does infinity exist?

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.
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barbarianhorde
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### Re: Does infinity exist?

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.
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### Re: Does infinity exist?

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.
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### Re: Does infinity exist?

Gloominary wrote:@Serendipper

Gloominary: For example, a road could be endless in one direction, but endful in another.

I attacked this in the OP.

Infinities cannot have beginnings or ends because those are boundaries and we said in the beginning that infinity has no boundaries.

I never agreed to that.

Then how do you define infinity?

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.

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.

We cannot divide infinity in half and say infinity is bounded by this finite location and extends to infinity in that direction; it's nonsense and breaks our definition of infinity being boundless.

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

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?

You: Here is a road that starts here and goes to infinity.
You: What do you mean?
Me: Where you're standing... we could add more road and since we could add more, then it's not infinite, but bounded by the location where you're standing. If we could add more road, then the amount of road we have is not infinite.

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.

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.

I don't know what MEST is.

Matter, energy, space and time.

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

Yes it does because if there is a place that we could add another apple, then the amount of apples is not infinite. If there are infinite apples, then apples exist in every place there is to exist.

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

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.

We can divide down to quarks, but to divide a quark pair will create another quark pair from the energy used to divide them. There is no such thing as smaller than a quark because smaller than a quark is smaller than the thing that defines size.

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

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.

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.

I think so.

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

I don't know... maybe.
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### Re: Does infinity exist?

Gloominary wrote: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.

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.
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### Re: Does infinity exist?

Huh? Infinite doesn't mean maximized or maximal.
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### Re: Does infinity exist?

wtf wrote:
Serendipper wrote: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 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.

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.

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?

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

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

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?

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.

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.

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."

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

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?

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.

* 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.

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).

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.

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.
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