## Does infinity exist?

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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.
Serendipper
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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.
Serendipper
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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).
Serendipper
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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.
Serendipper
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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?
Serendipper
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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.
Serendipper
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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.
It is true that liberty is precious; so precious that it must be carefully rationed.
~ Владимир Ильич Ульянов Ленин

THE HORNED ONE

barbarianhorde
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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
It is true that liberty is precious; so precious that it must be carefully rationed.
~ Владимир Ильич Ульянов Ленин

THE HORNED ONE

barbarianhorde
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