Does infinity exist?

Extremely good insight. An arbitrary real number is essentially a number whose decimal digits are random, in the sense of incompressibility. There is no formula or computer program anyone could ever come up with that can crank out its digits.

By the way note that pi is NOT a random real number in that sense. Its digits are the output of many formulas and algorithms. Pi is a computable real number. Most real numbers are not computable.

I have said no such thing and I believe no such thing. You have made up a claim out of thin air an attributed it to me.

I’m perfectly well aware of modern set theory and the idea of cardinalities. I’m glad you looked it up and perhaps learned something; and if you have any questions about it, I’d be happy to answer them. I’m fully conversant with the theory.

Happy to discuss transfinite set theory. One of my favorite topics. In fact the cardinals are what everyone hears about, but the ordinals are even more interesting. Cantor discovered them too.

By the way, a historical note. What was Cantor doing when he discovered transfinite numbers? Did he wake up one day and say, “I think I’m going to revolutionize the foundations of math, piss off my mentor Kronecker, and have a nervous breakdown?”

No in fact that’s not what happened. Cantor was engaged in studying trigonometric series, the very series that arose from Fourier’s research into heat. In other words Cantor was led to discover transfinite cardinals and ordinals based on problems that arose directly from physical phenomena. Something to think about when contemplating the philosophy of the infinite.

Finally, again reiterating what I said earlier, the statement you attributed to me is NOTHING I said. Nothing at all. You just made it up then pretended to debunk it. I believe in the philosophy biz that’s called a strawman argument.

How would you describe an infinite set of oranges using that definition of infinite?

Good question. I came up with two separate answers.

a) It’s a theorem of set theory that every infinite set contains a countably infinite subset so it’s no loss of generality to simply assume your set of oranges is countably infinite. If the set is uncountable we can adapt the same idea. So we label the oranges 0, 1, 2, 3, 4, … We can do that since they’re countable, which means there’s a bijection between the naturals and the oranges. So we can number each orange by the natural number that maps to it in the bijection.

Now the entire set of oranges is in bijection with the set of even-numbered oranges, by the usual mapping n => 2n. Since the set of oranges can be bijected with a proper subset of itself, it’s an infinite set of oranges.

b) Set theory as currently understood is purely about mathematical sets, the sets of ZFC or some similar axiom system. In ZFC, everything is a set. We start with the empty set, and the set containing the empty set, and the set containing those two, and so forth, and the the powersets and unions of all those sets, and so forth.

So in math, there is no set of oranges. If you have two oranges, I do NOT CLAIM that there is a set containing the two oranges. I do not personally believe in set theory outside of the pure sets of mathematics! That’s essentially a formalist position. A formalist is a philosopher who maintains that math is simply about the formal manipulation of meaningless symbols according to arbitrary rules. It means NOTHING.

So there is no set of oranges. There are no sets of anything, other than the empty set and all the other sets that can be built from it via the axioms.

Either of those float your boat?

What I’m really driving at is the thing being perceived as infinite is really part of the one doing the observing and the perception of infinity is proof of that. If we are made of spacetime fabric stuff and we start inspecting the fundamentals of existence, then essentially what we are doing is looking at our own inner workings and self-inspection results in infinite regression, and self-inspection includes deductive means to peer inside which results in notions like infinite causality which should really mean lack of causality because the whole thing is one continuous thing giving rise to time itself as an emergent property rather than being subject to some objective time concept that would invariably have to be infinite.

A mirror can only reflect images larger than its wavelength, so it’s an illusion of infinity.

Yes we had an eclipse last year I think (the one Trump looked at). I can’t imagine what I would have thought about it if I didn’t already know what it is.

I’m not sure how definition b. describes an infinite set of oranges and how is definition a. any different than simply saying the number of oranges is unlimited/unbounded?

How would you describe infinite space? Would you say the number of sq inches can correspond to the number of sq feet? Again, how is that different from simply saying space is without end? Bijection couldn’t be possible if sets had ends.

Wtf sorry for reading you wrong in that case!
I didn’t look it up by the way but talked to my math teacher, an old friend. I had dinner at his house and we discussed infinity.
Then I went to corroborate some stuff online.

So since it seems we are somewhat in agreement again (I shift from side to side) let me ask you this, what is a subset of the irrationals that can be bijected with it?

Serendipper — yes I definitely agree with that. I meant this means that whatever infinity is it is “inside” of a circuit, such as an observer hooked into an observation, the two together forming a world in a way, as a kind of closed system. At least closed for escaping. Stuff can still come in.

Such a feedbacking system is by definition infinite from the inside.

So a thing which is observed is part of the observing system and becomes hooked up to infinity. Lol. I need coffee.

In (b) I’ve taken the position that even finite “sets” of oranges don’t exist, if by set we mean a mathematical set. And surely since there are only 10^78 hydrogen atoms in the observable universe, there can’t be infinitely many oranges, whether contained in a set or not. So your question seems ill-founded.

What on earth do you mean by conceptualizing infinitely many oranges? What are they made of?

If space is truly infinite, then the number of square (or cubic, or hypercubic) inches is the same as the number of square feet. Surely this is obvious. They’re both infinite, and of the same cardinality.

The entire real line is in bijection with the unit interval. The bijections in each direction are the tangent and arctangent.

Another way to see something similar is the function f(x) = 1/x. That maps the open unit interval (0,1) to the entire positive real line.

God is infinite

Okay thats great, Destiny. Thanks for your wonderful contribution.

Do you mean the line of real numbers?
But I guess you don’t because the real numbers aren’t a subset of the irrational numbers.

Whats the real line?

The real line is the real number line. The set of real numbers. Of course if you mean to exclude the rationals, the 1/x example still works since 1/x is rational if and only if x is.

So 1/x bijectively maps the set of irrationals in the unit interval (0,1) to the set of positive irrationals. And the irrationals in (0,1) are a proper subset of the set of positive irrationals.

You mean to include ignite between 1 and 0 - yes but thats the same as why naturals and rationals are the same class.
what I try to figure out is how you can show one set (irrationals) is infinite by mapping a greater set (reals) onto it.

No, completely different proof and idea. Well related, but not really the same.

I did. The set of all positive irrationals can be bijected onto one if its proper subsets, namely the irrationals strictly between 0 and 1.

Are you confused about the reals versus the irrationals? The reals include both the irrationals and the rationals. You can use the same proof idea for the reals or the irrationals. If you only care about the irrationals, you need to exclude the rationals.

Please tell me which part of this isn’t clear. It’s clear in my mind so perhaps I’m not understanding your question.

We show the reals are infinite by mapping them onto a proper subset.

We show the irrationals are infinite by mapping them onto a proper subset.

You could map the reals onto the irrationals bijectively, but it’s a bit tricky and not worth the trouble.

Remember to show a set is infinite, I only have to biject it to SOME proper subset of itself. I don’t have to biject it to any particular proper subset.

I do in fact know how to biject the reals to the irrationals, but it’s a tricky construction and not worth going into detail about unless you want me to.

“I do in fact know how to biject the reals to the irrationals, but it’s a tricky construction and not worth going into detail about unless you want me to.”

This bijecting the reals to the irrationals is indeed what I was inquiring about, since what you said earlier about the real line hinges on it. The rest was clear to me before.

No that is not true, and it’s a point you seem unclear on. Please take a moment to engage with this point, it’s important.

A set is infinite if it can be bijected to at least one of its proper subsets. So the naturals are infinite because they can be bijected to the even naturals, or the odd naturals, or the primes, or (as Galileo noted in 1638) the perfect squares.

en.wikipedia.org/wiki/Galileo%27s_paradox

Likewise the reals are infinite because (0,1) is a proper subset and the tan/arctan functions biject the reals to (0,1). Or if you haven’t taken trigonometry, you can biject (0,1) to the set of positive reals via f(x) = 1/x.

Please I request that you spend some time to understand this point.

Now, bijecting the reals to the irrationals is a curiosity. I don’t need it to show the reals are infinite, the (0,1) examples already do that. But bijecting the reals to the irrationals is an interesting exercise, and shows how in general to get rid of a countable set within an uncountable one without altering the cardinality of the uncountable set.

So, here’s a function that maps the reals to the irrationals.

First, the rationals are countable so they may be placed into an order like this: (q_1, q_2, q_3, \dots)

Now we need to choose any countable sequence of irrationals. It doesn’t matter which one we choose, but for definiteness let’s pick the sequence (\pi, 2 \pi, 3 \pi, 4 pi, \dots)

We define our function (f(x)) as follows. If (x) is rational, it’s one of the (q_n)'s.
We map each rational (q_n) to (2 n \pi). That is, we map (q_1) to (2 \pi), (q_2) to (4 \pi), and so forth.

If (x) is irrational and one of the (n \pi)'s, we map it to ((2 n - 1) \pi). For example (\pi) goes to (\pi), (2 \pi) goes to (3 \pi), etc.

Finally, if (x) is anything else – that is, if it’s irrational and not one of the (n \pi)'s – we map it to itself.

If you think this through (and I don’t claim that’s easy, this takes some work), you will see that we have a bijection between the reals and the irrationals.

This proof is due to Cantor. He used the irrational sequence (\frac{\sqrt 2}{2^n}) in order to show a bijection between the unit interval of reals and the irrationals in the unit interval.

See this thread for several variants. The example I showed is based on the answer by MJD (fifth and final answer on the page). math.stackexchange.com/question … -irrationa

Again, please note that showing the reals are an infinite set does not depend on this somewhat complicated example. We know the reals are infinite because we may biject them to (0,1), a proper subset of the reals. But if you ever need to show that there’s a bijection between an uncountable set and that same set minus some countable set, this is the construction to use.

Ah, this is the sort of reply I was hoping for. Yes, I will take some time. Thanks wtf.

You’re very welcome. It’s such a great construction and I enjoyed reviewing it myself.

I would feel better if I understood why you think it’s important, because of the reasons I already mentioned … that it’s the (0,1) example that shows the reals (or the irrationals) are infinite, and the real → irrational bijection is just a curiosity, although a nice one. But if you’re happy I’m happy, and I’ll stand by for questions. I’ll be off the air the rest of the day but I’ll be back tomorrow.

I was moved to make this little sketch. I never understood this construction so clearly before. Thanks for asking about it.

I really like it when abstract things are worked out to the concrete detail. It always, always always turns out to be valuable. Because it takes real effort from real minds.

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 but I never knew of his type theory. I think that may be a bit underestimated, so far. I think it may become important in the future.

And yes I understand the proof now!
I wish I was good enough at math to compare this to type theory.
Still, I can begin to work at it now.

Yes I agree. And it’s cool that Cantor himself came up with this proof. It also serves as the standard technique for getting rid of countably many pesky problems. For example in Cantor’s diagonal argument, we are listing the decimal representations of real numbers, but some real numbers have two distinct representations. For example .5 = .49999… So when we form the antidiagonal, how do we know that even if it’s not on the list, its OTHER representation is?

The way this is usually handled is that we vigorously wave our hands (that part is essential) and say, “There are only countably many such pesky dual-representations so it doesn’t matter.” A thoughtful person might respond, “How do you know it doesn’t matter?” Our example shows how to prove that the dual representations don’t matter.

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?

How does that change your estimation of Russell?

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.

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).

h

Yay! Me too! I had a vague idea about it before and now it’s beautifully simple.

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.