Or is it just one and two?

The continuum? There are many continua.

https://en.wikipedia.org/wiki/Continuum_(topology)

Have you been skimming a book about Cantor recently?

Maybe crack open a logic 101 book?

Iirc someone proved that the continuum hypothesis isn’t provable under zfc or something like that which is where the size of the reals would equal the power set of aleph naught (or aleph-1), something Cantor was trying to prove before he died. That should hold if zfc is consistent.

I think it was one of hilberts turn of the century problems or the millennium prize problems. Some of the math giants here will surely know and I think there’s been threads on it before

I think I was incorrect in my reference to aleph 1 but the reals just can’t be between aleph 0 and 1, it has to be 1 or greater so yes it could be 3 if I understand.


I think this does a decent job of looking at this material

I think it's easy to see how the power set of the naturals can be put into 1-1 correspondence with the reals. For any subset of the naturals map it to the binary decimal with 1's in those positions k where k is in the subset; 0's otherwise.

I never learned the alpha, omega stuff. Just that the power set gets you to something bigger.


PairTheBoard

Perhaps the most meaningful contribution at this point would be if DS could actually state what he thinks he means when he says "the continuum" and have it come out as something sensible.

I thought that the continuum hypothesis was involving whether the number of real numbers was the next higher infinity after aleph null and that it was proven that it is OK to assume it is or isn't. So I wanted to know if it is also OK to assume that there are two rather than only one uncountable infinity smaller than it.

How do mathematicians determine what's ok to assume and what isn't?

You should practice asking questions like normal people more often.**

https://en.wikipedia.org/wiki/Cardin...uum_hypothesis

Note: c is the cardinality of the real line (ie, the continuum).

Edit: ** This is especially true since

It's impossible to tell whether you're asking a meaningful question or if you're merely uttering gibberish using technical mathematical language.

Try thinking also about cardinality of all real functions or Hilbert spaces to go above reals. 2^R

Until finally someone recognizes that this universe doesnt support any of this! Because with finite methods to represent things you cant have stupid infinities anywhere because you cannot realize a big object that requires more to define that the system has to offer in resources.

I predict QM fails because of that reason re Hilbert spaces only correct in the limit.

Well they don't like contradictions much.

A few years ago I wrote that even though I have no right to have a strong opinion about it, it seemed kind of nonsensical to portray infinities other than the number of integers and the number of real numbers on a number line as existing in any kind of meaningful way. Bruce Z and others did not take kindly to that thought. Now I am wondering if the things you are saying means you kind of agree with what I said.

So to use words I understand: "yes"

SMP: Infinity War

I actually agree. And you can have similar disregard for the math Phds who aren't smart enough to come up with groundbreaking new work but only smart enough to understand the (usually unuseful esoteric work) of others, only to wind up teaching calculus to unappreciative undergraduates.

But the same is not true for science, engineering and advanced medical and computer work. That blows almost all the stuff they talk about on the politics forum out of the water.

Infinity war:

https://en.wikipedia.org/wiki/Actual_infinity

You mean this post?

https://forumserver.twoplustwo.com/s...&postcount=138

But see also


https://en.wikipedia.org/wiki/Finitism


"The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets. While all natural numbers are accepted as existing, the set of all natural numbers is not considered to exist as a mathematical object. Therefore quantification over infinite domains is not considered meaningful. The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic."

I do not have a problem necessarily with the discussion that follows the assumption of existence of infinite sets and the higher levels of infinity produced by such discussions like Aleph numbers regarding the cardinality of infinite sets. Create all kinds of axioms and then good luck to you i say. Math eventually becomes linguistics above a certain level and i say good bye to the party unless you show me i need to pay attention because physics has to learn something. I assume the theoretical work there is consistent and not garbage or worthless plus i am not equipped to even have such position in terms of pure mathematics understanding at a personal level. I do enjoy math i can appreciate in terms of applications to the real world a lot more or topics that relate to tangible ideas immediately available in the "simple real" world.


My main question here to all is this;

Has anything that ever required higher levels of infinity other than the concept used in Calculus produced any tangible progress in math that resulted in insights and proofs in other areas and applications in science?

There is to my knowledge absolutely nothing in Physics that infinity makes possible that would be not true otherwise as an approximation of the truth that infinity never improves actually.

Can physics predict the point near the end of the universe when the laws of physics stop working? If so, does it need any of these exotic understandings of infinity to do so? Or, is it purely mathematical in the sense that the current models of cosmology predict this demise but there is no proof and physics has no opinion on its own life cycle?

Funny, I was going to come in here to just post “time” as a reply but your post says it better

What do you mean by end of the universe? It has no ending. Unless you mean the singularity at the beginning. This is not really singular just not well understood in the actual starting point yet. Standard cosmological models predict various things for the first second and then after that all is reasonable.

It's like the surface of a balloon without the room inside which you imagine it. If you imagine the universe as the surface of the balloon then there is no end.

I have no example in physics where a result is correct only if you require infinity to exist. All infinities that cancel with calculus calculations (from infinite line charges to quantum field theory) are simply big numbers that cancel each other at the limit but the theory or conditions "break down" at the limit anyway so it doesnt require the actual existence of infinity to get there. You do not need to have the perfect result of the integration literally. You can stop 1 km out and you will get almost the same answer within experimental error and other phenomena you ignored.

In fact insisting to literally go to infinity gets you in trouble in many things and i imagine even in the Hawking radiation calculations or all the attempts to answer to it also on the other side of the argument that take black hole horizons so literally rigid to make sense to take limits. It doesnt, you need to take care of them better there!

The universe has a size, beyond which there is no universe. Black holes have multiple horizons, defined by the laws of physics.

The thread is about infinities which are exotic to me, and my question in the thread was posed as a response to the floor being open as to whether or not any of this is practical or required within physics to arrive at a correct answer to a physics problem.

If the answer to the puzzle of dark energy is impractical in the sense that no human being will ever be alive to observe the eventual consequences of a hyper-expansive universe, then so be it.

One possible solution is that a vacuum contains an infinite energy reserve, and that our universe was born from such a reserve.

Well the point is that i do not recall anywhere a tangible application of infinity or irrational numbers to begin with without which you would not get the correct physics in some way as a limiting process that eventually goes beyond the accuracy of the theory and therefore is not literally accepted anyway as approximation.

The Calculus used in Physics is an approximation of the truth. You do not have infinitely divisible spacetime or any realization of numbers like pi or even e .

Sure equations are full of them but they are all limits that fail eventually when you get to levels beyond desired accuracy of the theory.

Talking speculatively about physics (like negative infinite energy ? what is that?) doesnt answer my quest for a counterexample!

Basically i take the wild position irrational numbers have nothing to do with the real world. Hence infinity is unrealized and irrelevant to the real world. I hold suspicious in Physics value any discussion of higher infinities as well. But i am open to changing my mind.


Keep in mind calculus is all over physics and the main reason for the world you see out there today and impossible without irrationals but it is only an approximation of the real world not the literal laws of nature.

This is where the next breakthrough will come from, the emergence of calculus and geometry as a limit of the actual process not the background where all is played.

The conflict starts with luck itself (and wavefunctions). If probability is to work as QM experiments have shown so far then do we need the continuum for the probability or we do not understand it well yet? If probability is discrete ultimately then what determines which path is followed each time? There is no reason though to think its a continuous probability distribution at play, only in the limit behaving as continuous roughly.


Basically i have absolutely no conviction in irrational numbers in science. Take me down from that position. I dare you!

Surely i know what you mean by 2^(1/2) in the real line using Pythagorean theorem ( using basic geometry but that Geometry is not a realizable thing in nature). Now can you show me a natural system that uses in anything 2^(1/2) or Pi or e precisely for if it werent that, something massively important fails?