Does infinity exist?

Yes. It had some adherents in the 1930’s then lost out on mindshare to set theory. It’s making a comeback via approaches like homotopy type theory, and on the philosophical side, neo-intuitionism.

Yes I’m aware of type theory. Maybe I don’t get where you’re coming from. Of course there are alternatives to set theory. The fact that foundations are in a constant state of turmoil, and that there are interesting alternatives, doesn’t invalidate set theory, nor disturb its place as the dominant paradigm up to the first couple of years of grad school. In areas like abstract algebra, differential geometry, and algebraic geometry, set theory has already been supplanted by category theory.

Alternative approaches are perfectly fine. I get the feeling that you think we are having a difference of opinion over whether type theory exists or whether set theory may or may not be supplanted by a better foundation. I agree to all of it.

Can you just explain to me where you’re coming from? The point is that if you, or anyone, out of a sense of what, contrariness or whatever, wants to argue that set theory is some kind of evil plot designed to suppress the Truth, well I just can’t hold up my end. Set theory is a tool. Math is a big toolbox with many tools.

No I disagree with you. There is intuitionist philosophy that corresponds to type-theoretic math. And there’s also philosophy that corresponds to full infinitary math. I do not believe you are correct that all the philosophers are intuitionists or finitists. That’s simply not the case.

That expression is undefined. When I speak of mathematical infinity I speak of no such thing. A set is infinite if it may be bijected with a proper subset of itself. That’s the working definition.

Nonsense. The reification of reciding? What the hell does that mean?

I don’t care if circles don’t exist. Mathematical circles do, and they’re interesting and useful. And by Putnam’s indispensability argument (linked earlier), circles are entitled to abstract existence by virtue of the fact that they are indispensable in understanding the world.

Take it up with Putnam and Quine, not me.

ps Here is the SEP link. These are high fallutin’ philosophers making the case that because abstract mathematics turns out to be indispensible – as I’ve been putting it, interesting and useful, but indispensability seems to be a higher standard – that abstract mathematical objects have a claim on existence.

plato.stanford.edu/entries/mathphil-indis/

Yes, thats what I meant but damn thats nicely phrased. So an object is infinitely grounded in itself. A set that describes a function such as a rational number sequence ids infinite in its potential reflection of itself on itself, where each reflection produces another integer, but there is not any infinitude of integers given unless that set is taken as the vessel. So the infinity opt the set is always infinity+1, the infinity of the number brings along the notion of the set. Which already shows its is not really infinite in capacity.

Damn, I dont now everyone can follow my thought here. Capacity as different from potential… well, like the capacity of a hose tied to an opened hydrant, and the potential of the closed hydrant and the rolled up hose.

Yes, a number only acquires a capacity to mean anything under certain circumstances, such as existence.
lol.

Pi is one of these numbers.
An irrational number. That speaks volumes. Its infinity is not a neat row or axial system, but more like snow on a tv. It is a better infinity if you want to come close to existence.

I think this is attained in the mirror loop metaphor for the set, where the set is a thing which is infinite inside but has no infinite capacity to change things, which would be infinite existence, which would mean infinite divisibility of meaning.

Im compelled by this image of the trees that cause the wind I must say. Thats pretty damn cool. Yes, as kids we clearly have a lot more touch with the contradictions that are thrown at us, the way things are set against each other.

Out of the blue, it reminds me of my first solar eclipse when I was just a 4 year old kid walking home with my friend from getting some candy, I don’t know why I was allowed, it was the 80s, and it suddenly got dark. It was a partial eclipse and no one was paying attention (it was the 80s) but for a moment I had the distinct sensation of “well that was it folks!”. Later if you know what an eclipse is, it loses most of its capacity. Unless you’re not in a horde of morons (humans) but in a field where suddenly every being is holding its breath, and you realize what you thought was silence was actually deafening noise.

wtf - I can now begin to undermine your claim that all infinity is just basically the same thing. All I need to do for this is follow the threads. A philosopher is a detective.
Let us start in the context of bijection on good ol wikipaedia…

“Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers.”

But I can add to this to make it stronger. Different infinities are also distinguished in terms of how they can be overlaid. If they can be overlaid, like the set of all even numbers with the set of all natural numbers, then it is the same type of infinity. The same goes for rational numbers, they too can be overlaid with the natural numbers. But irrational numbers can not. They form a different class in which even the distance between 1 and 0 contains more numbers than all of the natural or (thus) rational numbers.

This is orthodox math, so where do you want to go from here?

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.