Asking for a friend:

You're super nice for not posting my name, but I'm just learning so I fully expect--and can take the heat of--someone telling me why my question is dumb.

Just do it in small words, k? I actually am trying to understand.

I think you shouldn't leave big issues totally unexplored. Aliens could be caught in the SETI. Think it's a classic 'you don't know before trying'

I'm not great at math, but I wouldn't assume patterns of a random system would look like white noise just because you're looking in decimal for instance, and I assume SETI has already thought of this.

Base 10 is just notation. The integers are the integers, regardless of which base they are denoted in.

This is a really fair point.

Would they be able to recognize systems and patterns that didn't exist to us yet? Like, if binary only represents part of the picture, is or is not, and doesn't account for unobserved Flux state, then the pattern becomes infinitely more complex and hard to recognize.

Is that a super-flawed understanding on my part?

Also, I'm sure they have thought of all this, too. I was really hoping for an answer.

I think I understand your point. (Trying to anyway )
You think that it's completely insane to think there could be a system like the one I spit-balled where aliens intuit extrapolation and had different values, like us:them in earth integers would be 1:1, 2:2, 3:4, 4:8, 5:16...or something else entirely? Or that it wouldn't matter if there were?
That's just a random example of a possibility, obviously.

I thought that SETI was just trying to tune in to another civilization's Top 40 radio show.

To be fair, our check IS in the voyager mail




You're welcome, Universe.

"Two different values for its frequency have been given: 1420.356 MHz (J. D. Kraus) and 1420.4556 MHz (J. R. Ehman). The frequency of the Wow! signal matches very closely with the hydrogen line, which is at 1420.40575177 MHz. The hydrogen line frequency is significant for SETI searchers because, it is reasoned, hydrogen is the most common element in the universe, and hydrogen resonates at about 1420.40575177 MHz, so extraterrestrials might use that frequency to transmit a strong signal."
https://en.wikipedia.org/wiki/Wow!_signal

10000 twitter messages were beamed back a while back.

As for expecting aliens to demodulate and decode a digital signal and even use RF for communication, or to have a two way convo using radio with a civilization say only a couple hundred light years away seems unlikely...

I was more thinking about neutrino communication than RF for current binary applications.

I think there are some applications where a notation based on exponents is convenient to use. But that is still a case of another notation being used to apply the same number system, i.e. the integers.

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

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

Higher intelligence can detect the hell out of us no matter how funny or arbitrarily formalized our technology is.

All they need is to notice we are not random natural noise and it's game on. Tons of hints we are not normal background nature exist. After that they can science the hell out of any data from us and learn everything.

First of all they can see us from anywhere in this galaxy. Seriously! We will be able to do the same in 1-2 centuries.

What is the best telescope you can create? Well make it as big as your own solar system now, how hard is it if you have machines running all the labor on a fusion based economy or better? Create small size mirrors each one like a typical telescope and place them next to each other over many thousand km in space. Then control each one remotely very accurately to have its position behavior known. Use a supercomputer or 100 of them who cares to have all this work together and you can resolve now exoplanets better than anything we will have this century in the absence of a true physics breakthrough.


Look at some future telescopes already planned to see how we are only at the beginning of this;

https://en.wikipedia.org/wiki/James_...pace_Telescope

https://en.wikipedia.org/wiki/High-D...pace_Telescope

https://en.wikipedia.org/wiki/Europe...arge_Telescope


I mean i ask you why cant we make 1000 times larger telescopes if we wanted as a priority? Cost in a non scientific society model world is the problem! Right now its very expensive ($10 bil and more for these things say above as order of magnitude improvement next level) but how expensive will it be if it is financed by the labor of machines running on fusion exploiting the resources of an asteroid exclusively for this purpose?

Effectively we will be able to see a car size object at a distance of 100 light years eventually. I am not kidding.

http://www.centauri-dreams.org/?p=31358

http://www.huffingtonpost.com/seth-s...b_5691353.html

You can use interferometry over thousands of km and even millions of km creating thousands of individual mirrors and collecting data from all of them and then processing that data to reconstruct images resolving planets to say detail of 1/10 continent scale over thousands of light years effectively mapping the milky way at any region of interest that has been spotted from previous less accurate methods (the future equivalent of Kepler mission 1 mil times better).


Lets face it. If photons can come from there you can collect them and recover an image over time that may require AI manipulation to reconstruct and is no longer the conventional fast photograph we imagine but it will yield in the end after substantial computations the same result effectively as seeing it right away in high zoom lens.

Earth reflects back (albedo) say something like 30% of solar radiation received. Lets say this is like typically;

1400*0.3*Pi*6378500^2/(6.63*10^-34*3*10^8/(550*10^-9)))=1.5*10^38 photons per second in the visible spectrum.


That say goes to 50% of the sky in the directions opposite to the sun illuminating this.

Lets consider now a region that is 50000 light years away that is a typical distance in this galaxy given our location in it (i mean most other places in the galaxy are within that distance).


Imagine now a region there devoted to observing this solar system and in particular earth out of the 10^11 total they have studied by then lol (ie we are at the beginning of this of our own with the first 5000 exoplanets or so- we will keep learning more for each of these catalogued worlds now).

Lets say they have devoted to this task that is an immense computation effort the recording from an area of R=1000 km in space orbit around their planet that they collect all radiation and process it to see where it came from and remove uninteresting data. How many photons from earth alone do they get there out of this 1.5*10^38 per seconds we reflect back?

About

1.5*10^38*(Pi*1000000^2/(1/2*4*Pi*(50000*86400*365*3*10^8)^2)=335 *10^6 photons per second.

(ok that assumes there is no significant loss of these photons in the distance they have to travel due to interstellar medium etc but this is not a very bad approximation even if 1/10 of the photons only survived, its still a big number and in fact a lot more will survive)

Well you better believe it that they can see us then if they can receive this light and process it in time even if only at the pace of 1/1000 of that number by not necessarily having to fill the entire 1000 km space telescope and having instead vacuum in 99.9% of it and only occasionally placed mirrors or whatever receivers they design for this task within that 1000km radius.

They can watch us for an hour and then manipulate the image to take care of rotation and relative speeds that they can use other methods to understand and correct.

See what i mean? Its heavily processed optics really where gradually they obtain information from our system that it is used to further correct the data accumulation and processing until the data can be now properly processed the right way (to correct all details for all effects involved - something like optical forensics) to yield an artificially created image that is very close to the real thing though as if you had such a huge telescope that could see us live.

The end result is a true picture of our system from 50 thousand light years away!!! You better believe it!

Therefore with enough technology that is within reach in 100-1000 years, even without any further physics observation breakthrough, they can "see" us like we see eg Mars from Hubble below;




If you can do that then you can learn all kinds of things about us because you better believe it they will then special order processing of light from our system removing all other irrelevant light from other places that they have already recorded and studied.

If you can receive thousands of photons from a source every second and accumulate light for a minute you can do that every day for weeks and learn so much (from these individual 1 min exposures) .


So if they exist in this galaxy they already know about us because the chance they are of the same technological age as we are (just beginning) is insanely ridiculous. They are either thousands to many hundred millions of years ahead or behind in age. Behind is irrelevant but ahead is everything.

You see if we sensibly expand our civilization we (our AI and us) will be all over this galaxy in 100000 years (the same as the age of modern humans).

So the above technology is not even needed. They do not need to observe us from 50000 light years away. They can observe us from 100 light years away because they have millions of outposts with top technology resources all over the galaxy. One advanced station in every 1 mil systems is enough.

So this is why they know all they like about us already. Surely they see us as we were hundreds or thousands of years ago but they can read from this a lot.

And if they had seen earth thousands of years ago and spotted the activity of life and civilization on its surface from 100-1000 light years distances they would have sent probes here long ago to learn more.

This is why its entirely obvious that we are either alone within billions of galaxies with the next technologically advanced one in many billions of light years typical distance away or they are all strangely going silent and careful after they become very advanced (unlikely) or they go extinct( also a bit unlikely since they cant all fail before expanding and learning from local isolated demises).

So either they know a ton about us and are remaining silent (very funny) or i am indeed correct (from past posts on this) and life is very rare and intelligent life that becomes technologically advanced is as rare as 1 in 10^20 to 10^25 planets. That explains the silence and we are one of the first locally and typically the first in our galaxy to observe it.

For it to be a colossal waste of resources, there'd have to be a colossal amount of resources being used on it. There isn't, so it isn't.

Looks like a Red herring.

In the first instance we are only looking for signals that do not appear be natural. The reasoning behind the signals is secondary.

Well in the sense that all pure science can be considered a waste of resources.

What would the practical applications be if we ever did manage to detect an alien race using SETI? Would they ignore us, help us or send an invasion fleet? Personally I expect the first option.

Here is part of the excelent book Godel, Escher, Bach: an Eternal Golden Braid, by Douglas Hofstadter, which might be interesting to consider.

An Unlikely UFO

We can get some perspective on this issue by considering a strange hypothetical event. A record of David Oistrakh and Lev Oborin playing Bach's sonata in F Minor for violin and clavier is sent up in a satellite. From the satellite it is then launched on a course which will carry it outside of the solar system, perhaps out of the entire galaxy - just a thin plastic platter with a hole in the middle, swirling its way through intergalactic space. It has certainly lost its context. How much meaning does it carry?

If an alien civilization were to encounter it, they would almost certainly be struck by its shape, and would probably be very interested in it. Thus immediately its shape, acting as a trigger, has given them some information: that it is an artifact, perhaps an information-bearing artifact. This idea - communicated, or triggered, by the record itself - now creates a new context in which the record will henceforth be perceived. The next steps in the decoding might take considerably longer - but that is very hard for us to assess. We can imagine that if such a record had arrived on earth in Bach's time, no one would have known what to make of it, and very likely it would not have gotten deciphered. But that does not diminish our conviction that the information was in principle there; we just know that human knowledge in those times was not very sophisticated with respect to the possibilities of storage, transformation, and revelation of information.

Levels of Understanding of a Message

Nowadays, the idea of decoding is extremely widespread; it is a significant part of the activity of astronomers, linguists, archaeologists, military specialists, and so on. It is often suggested that we may be floating in a sea of radio messages from other civilizations, messages which we do not yet know how to decipher. And much serious thought has been given to the techniques of deciphering such a message. One of the main problems - perhaps the deepest problem - is the question, "How will we recognize the fact that there is a message at all? How to identify a frame?" The sending of a record seems to be a simple solution - its gross physical structure is very attention-drawing, and it is at least plausible to us that it would trigger, in any sufficiently great intelligence, the idea of looking for information hidden in it. However, for technological reasons, sending of solid objects to other star systems seems to be out of the question. Still, that does not prevent our thinking about the idea.

Now suppose that an alien civilization hit upon the idea that the appropriate mechanism for translation of the record is a machine which converts the groove-patterns into sounds. This would still be a far cry from a true deciphering. What, indeed, would constitute a successful deciphering of such a record? Evidently, the civilization would have to be able to make sense out of the sounds. Mere production of sounds is in itself hardly worthwhile, unless they have the desired triggering effect in the brains (if that is the word) of the alien creatures. And what is that desired effect? It would be to activate structures in their brains which create emotional effects in them which are analogous to the emotional effects which we experience in hearing the piece. In fact, the production of sounds could even be bypassed, provided that they used the record in some other way to get at the appropriate structures in their brains. (If we humans had a way of triggering the appropriate structures in our brains in sequential order, as music does, we might be quite content to bypass the sounds - but it seems extraordinarily unlikely that there is any way to do that, other than via our ears. Deaf composers - Beethoven, Dvorak, Faure - or musicians who can "hear" music by looking at a score, do not give the lie to this assertion, for such abilities are founded upon preceding decades of direct auditory experiences.)

Here is where things become very unclear. Will beings of an alien civilization have emotions? Will their emotions - supposing they have some - be mappable, in any sense, onto ours? If they do have emotions somewhat like ours, do the emotions cluster together in somewhat the same way as ours do? Will they understand such amalgams as tragic beauty or courageous suffering? If it turns out that beings throughout the universe do share cognitive structures with us to the extent that even emotions overlap, then in some sense, the record can never be out of its natural context; that context is part of the scheme of things, in nature. And if such is the case, then it is likely that a meandering record, if not destroyed en route, would eventually get picked up by a being or group of beings, and get deciphered in a way which we would consider successful.

Imaginary Spacescape

In asking about the meaning of a molecule of DNA above, I used the phrase "compelling inner logic"; and I think this is a key notion. To illustrate this, let us slightly modify our hypothetical record-into-space event by substituting John Cage's "Imaginary Landscape no. 4" for the Bach. This piece is a classic of aleatbric, or chance, music - music whose structure is chosen by various random processes, rather than by an attempt to convey a personal emotion. In this case, twenty-four performers attach themselves to the twenty-four knobs on twelve radios. For the duration of the piece they twiddle their knobs in aleatoric ways so that each radio randomly gets louder and softer, switching stations all the while. The total sound produced is the piece of music. Cage's attitude is expressed in his own words: "to let sounds be themselves, rather than vehicles for man-made theories or expressions of human sentiments."

Now imagine that this is the piece on the record sent out into space. It would be extraordinarily unlikely - if not downright impossible - for an alien civilization to understand the nature of the artifact. They would probably be very puzzled by the contradiction between the frame message ("I am a message; decode me"), and the chaos of the inner structure. There are few "chunks" to seize onto in this Cage piece, few patterns which could guide a decipherer. On the other hand, there seems to be, in a Bach piece, much to seize onto - patterns, patterns of patterns, and so on. We have no way of knowing whether such patterns are universally appealing. We do not know enough about the nature of intelligence, emotions, or music to say whether the inner logic of a piece by Bach is so universally compelling that its meaning could span galaxies.

However, whether Bach in particular has enough inner logic is not the issue here; the issue is whether any message has, per se, enough compelling inner logic that its context will be restored automatically whenever intelligence of a high enough level comes in contact with it. If some message did have that context-restoring properly, then it would seem reasonable to consider the meaning of the message as an inherent property of the message.

(By the way, on a different note: I know pretty much no physics, so I can't really comment too much about it, but the property of being prime on the integers is invariant under changing basis; for instance 3 is a prime integer because

(1) it is not 0,
(2) it is not invertible (ie, there is no "1/3" in the integers which solve the equation 3x=1; in fact the only invertibles in the integers are 1 and -1, which are their own inverses) and
(3) its only divisors are the invertibles (1,-1) and itself multiplied by these invertibles (3,-3).

This does not change if we decide to denote 3 by "11", in a binary notation (or any other base): (1),(2),(3) will still hold, except that now we write "11" for 3 and "-11" for -3 everywhere.

Similarly, 9 is not prime: 3 divides 9 since 3x3=9, and this fact is still true wheter or not you decide to rewrite it in binary: "1001 is not prime: 11 divides 1001, since 11x11=1001".

Facts about the integers like this are true in general: the inner structure of the integers does not change just because one decides to use different notation for them).

Let me expand a little bit on what do I mean by inner structure, by giving a different example then the integers (in particular, a finite one); this might be more enlighening:

Consider the subset of the integers {1,-1} with the operation of standard multiplication; we know that

1x1=1, -1x-1=1, 1x-1=-1 and -1x1=-1.

Consider aditionally also the set {e,o} (which you should think as "evens and odds") with some binary operation + which I'm going to completely define this way:

e+e=e, o+o=e, e+o=o and o+e=o

(this matches the idea, in the integers, that "even plus even is even", "even plus odd is odd", and so on).

You should note that the role of 1 in the first set with respect to its operation x matches the role of e in the second set with respect to its operation + (they are both identities) and similarly the role of -1 matches the role of o.

In other words, while we have different NOTATION for these things (1,-1,x versus e,o,+) they have the same INNER STRUCTURE (in mathematics one would express this fact as saying the both these sets with their respective operations are (group-)isomorphic; in other words, in a very strong sense, they are the same thing).

Now, going back to the integers and inteligent aliens:

Whatever we call "the integers" is a reference to a structure composed by some set with a certain amount of (infinite) elements, (say {...-4,-3,-2,-1,0,1,2,3,4...}, or, if you wish to use binary notation, {...,-100,-11,-10,-1,0,1,10,11,100,...}) and a pair of operations, which we generally call addition and multiplication, which we know (or at least have a very reasonable intuition) of what they are; the important part is not the NOTATION given, it's the inner structure. In fact, if some other combination of a set with some operations does not have the same structure the integers (say the two two-element sets above certainly do not have the same structure as the integers; they don't even have the same amount of elements) then well, this is structure won't be what we call "the integers" (incidentally, these two original sets above with those operations is what we call in mathematics Z_2, "the integers mod 2").

Finally: if one expects that a sufficiently inteligent life form actually exists (whatever "inteligence" may mean), these guys should probably somehow be familiar with THE inner structure of THE integers (or so I would expect, but what do I know), and it probably won't be too hard for them to figure out our notation for describing the integers, if they happen to get ahold of one of our math books (or so would I expect).

See also this https://en.wikipedia.org/wiki/Kolmogorov_complexity


Our messages will appear very intriguing in that they would have eventually some complexity in them that makes them very different from random noise or whatever natural background.

Look at a well used keyboard and see after such significant usage that certain keys are no longer as visible as others (their ink, what key it is each). Obviously that reflects their relative higher usage in text (and decay rate of their surface - imagine how much dust our activity creates all day by the way and what cancer implications this may carry as in how much of this fine dust we eat and breathe eventually but that's another topic).

So they would be able to see our text or information messages have such patterns or streaks that make no statistical sense if they were originating from random "natural" (or lower complexity) processes.

We would have tons of repeating patterns that make no sense to be appearing so often.

Example the word "pattern". What is the chance to see "pattern" formed from random letters?

Its like 1/40^7 if you use all 26 letters (ignored capitals) and basic symbols (say " ", ",", ".","!", 0-9 ) observed so far or ~1 in 1.6*10^11. Use full ascii system (eventually becoming some binary code equivalent of course) and its even worse.

Obviously "pattern" appears in human texts a lot more often than 1 in 1.6*10^11 kind of frequency.
Also things like " a " or space a space appear very often , a lot more often than 1/40^3= 1 in 64000.

Just see how many times it did already in this post lol.

Over time they can learn a great deal about us and decode a lot if they have access to any observations in detail even if they do not yet have intimate connection to the inner details and properties of our brains because they have different biology and even chemistry. But keep in mind that we have math and physics as common language and their AI is already the first alien they have encountered hahahaha (teaching them to see things from a very universal perspective and even anticipate our chemistry and biology as one solution of the general abiogenesis problem) and probably we will see their AI interact with us first rather than them.

Of course see also my argument that intelligent technologically advanced life could be as rare 1 in 10^25 systems or worse chance even, so we are very rare (locally the first) and separated by more advanced aliens and their AI by billions of light years if not alone (which is also unlikely to be alone - see past posts on this i have in other threads about the claim intelligent technologically advanced life is likely >99% certain to appear somewhere in the universe even without us if we disappeared) and this rarity explains both the "silence", the way the universe looks, the lack of being colonized already and our own existence of course given the size of the universe known so far. It also likely makes suspect/questionable the exotic sci fi physics ideas about wormholes and other structures. Such methods if they existed would make universe look very active in terms of aliens and their AI and the consequences of their expansion everywhere.

Back to universal patterns even our music across different cultures and languages obeys universal laws.

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

http://www.pnas.org/content/109/10/3716.long

"Much of our enjoyment of music comes from its balance of predictability and surprise. Musical pitch fluctuations follow a 1/f power law that precisely achieves this balance. Musical rhythms, especially those of Western classical music, are considered highly regular and predictable, and this predictability has been hypothesized to underlie rhythm's contribution to our enjoyment of music. Are musical rhythms indeed entirely predictable and how do they vary with genre and composer? To answer this question, we analyzed the rhythm spectra of 1,788 movements from 558 compositions of Western classical music. We found that an overwhelming majority of rhythms obeyed a 1/f^β power law across 16 subgenres and 40 composers, with β ranging from ∼0.5–1. Notably, classical composers, whose compositions are known to exhibit nearly identical 1/f pitch spectra, demonstrated distinctive 1/f rhythm spectra: Beethoven's rhythms were among the most predictable, and Mozart's among the least. Our finding of the ubiquity of 1/f rhythm spectra in compositions spanning nearly four centuries demonstrates that, as with musical pitch, musical rhythms also exhibit a balance of predictability and surprise that could contribute in a fundamental way to our aesthetic experience of music. Although music compositions are intended to be performed, the fact that the notated rhythms follow a 1/f spectrum indicates that such structure is no mere artifact of performance or perception, but rather, exists within the written composition before the music is performed. Furthermore, composers systematically manipulate (consciously or otherwise) the predictability in 1/f rhythms to give their compositions unique identities. "

A mathematical structure they'd be likely to recognize would be the group of permutations. For example, you could send all the ways of reording (a,b,c), all the ways for (a,b,c,d), all the ways for (a,b,c,d,e), etc. Seems like there ought to be something you could do with that.

PairTheBoard

To talk with them we need just Math language and Physics/Chemistry.

They would instantly recognize for example prime numbers and symmetry patterns/group theory etc.

We could teach them what our symbols mean with examples like 3 = 3 , 5 = 5 *=* then say a triangle symbol ∇=∇ then maybe 3 ≠ 5 , ∇≠O

That shows what = and ≠ are.

Then what is the symbol for true or false T/F so 3=3 T 3=5 F etc.

We can basically show them the symbols for logic and numbers, set theory, etc.


We could then use these symbols to talk to them about hydrogen atom and so much more that are universal and would serve to verify their understanding of language.


A message that is definitely less than 100000 characters can teach them a ton of things. Then logically by knowing what others know you can infer a lot more about what else they know in order to make sense with their examples offered.


So if you can do this you can eventually communicate everything even emotions.

Here is an emotion. Describe to them a solar system that shows a civilization there in all its glory (with symbols) and then suddenly present nearby in 4 light years distance at the same timeframe a supernova explosion. That right there is so meaningful to them that instantly conveys the message of fear or stress such explosion implies for life. They can relate. Of course they already have the solution for this problem too but they know once they didnt.

"We" does not include Gauss, for example.

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

You might find this interesting:

http://math.stackexchange.com/questi...about-infinity .

Here are a few of comments in this link which I think are very important:

"To be fair to Gauss you should consider also what his contempories thought about completed (vs. potential) infinity. To properly evaluate Gauss' remark requires extensive knowledge of the mathematics of that era (and an ability to effectively "forget" what you know of today's math when need be). Neither of these are commonplace."

"Probably would have been horrified for 15 minutes. Might have taken him a day or two to really make good use of the new tools."

"It makes no sense to write something like [Gauss] "would have condemned Cantor's ideas". Almost 80 years passed between Gauss' Disq. Arith. and Cantor's work on set theory. If Gauss had been a contemporary of Cantor and, so, had knowledge of mathematics of that later era, then he may well have praised Cantor's work. But no one can know. It is disrespectful to Gauss to write such highly speculative remarks. We don't need more romanticized math history in the style of E.T. Bell. Nowadays much is on line, and you can read the true history."

Lets try this. Tell me what has come out of actually accepting the reality of actual infinity in terms of spectacular theorems of results that apply to finite mathematics or any mathematics that relates to tangible science even other abstract mathematics that their development through these concepts has lead to other mathematics and applications that can be seen as success of the completed infinity concept.

Basically i want to see a result (and maybe i am missing something trivial but i would imagine i am ok with limits everywhere to avoid any problems of actually completing the concept of infinity) i am failing to instantly recall at this point.

Is there any subject where we cannot essentially reproduce results with limits and multiple limits etc?

Technically in terms of real physics there is not such thing as 2^(1/2) distance between 2 points etc. We have a finite universe with finite number of states etc. In any case we have finite resources to perform representations etc. So we cannot actually see anywhere in nature the concept of infinity realized. (other than limits etc ie basically ultimately calculus is not proper for physics and re-framing is inevitable, infinitely divisible space-time distances makes no sense for example in terms of measurements).

It is found everywhere of course in calculations but you know these things are never as accurate as you want if you wish to be fair and technically the same actual "accuracy" is achieved with finite number of steps or perfectly safe limiting approximations to the real thing. I mean that eg calculus works where its used because the fine details do not typically become relevant to produce a different answer but the pathology remains there in our theories in certain situations and re-framing is essential to avoiding infinities in our calculations (ie no singularities in black holes is a given, spacetime is not continuous etc but its not a grid either ;-) ).

I'm not completely sure how to respond here, but I'll try my best:

It seems you are coming off from a perspective where you want to know "why is a certain subject important?" by asking "why is a certain important in certain selected areas I choose to care about". Well, my guess is that logic/set theory has no any direct applications ("direct" in a sense you would find reasonable) to anything physics (or, whatever it is that you mean by "real" physics) would care about , but then again, I know no physics (and I don't know that much set theory anyway).

What I can tell you to try to motivate you on this subject is this: elementary set theory starts with the amazing insight of realizing we can extend our intuitive notion of counting finite things, to a better one which both

(1) works in general and
(2) agrees with our intuition when restricted to the finite case.

It is the notion of pairing off objects, ie, establishing bijections between sets. It is clear this ideia is the correct one, from an intuition point of view, for the finite case: when you count the number of objects in say {a,e,i,o,u}, you pair each one of them of with the objects in {1,2,3,4,5} (you form the bijection, or pair them off, by saying it out loud, say "one" while pointing at "a", "two" while pointing at "e", "three" while pointing at "i", "four" while pointing at "o", and "five" while pointing at "u") and when you are done you say, "well, first set has five elements". At some point someone realized that the idea of forming bijections is one we can work with for infinite sets as well, and then someone realized things which were intuition defying such as that there are bijections between certain infinite sets which are properly contained one in the other (say there are bijections between integers and rationals) and there are no bijections between certain infinite sets (and we might express this by saying "there are different sizes of infinites"). Then someone realized that we can construct an extension of the natural numbers, the cardinal numbers (of which the naturals are a particular subset), which we can then use count infinite sets (and things work perfectly).

Anyway, you might already know this, you might not, but the point is that to me (and, apparently, to the majority of mathematicians, among many othe people), this is interesting on itself. It tells us something about how to handle the notion of "infinity", a concept which interested mankind for a long time.

Again, I don't think any of this is of much importance to anything in physics,. BUT, it certainly has A LOT of aplications within mathematics, of that I'm sure: particularly in topology and analysis, but certainly also in algebra and number theory. MANY constructions in mathematics depend on the fact that (modern) set theory became a thing; say I'm going to give a few rather random and out of context examples, but its what I have in mind right now:

In algebra, to prove that every field is contained in some (algebraically) closed field (for instance the rationals form a field and they are contained in the field of complexs which is algebraically closed, ie, every polynomial of degree n with complex coeficients has n roots in complex numbers, counting multiplicity; the fact that the field of complexs is closed is what we call the Fundamental Theorem of Algebra) one invokes Zorn's Lemma (which is equivalent to the Axiom of Choice) for obtaining a set which is a maximal ideal of some ring out of a certain ideal of that ring.

In topology, we have a central theorem (called Tychonoff's Theorem) which states that every product of compact topological spaces is a compact topological space; to prove that one invokes, again, Zorn's Lemma to obtain an certain set which is an ultrafilter out of a certain filter.

There are uncountable things like this in mathematics, there is nothing particularly important about these particular two examples I gave (except that they are fresher on my mind for whatever reason).

Anyway, as usual, I better leave a link of someone who understands math far better then me talking about this:

http://math.stackexchange.com/a/1075461/104587

Doesn't Quantum theory require Complex numbers, and don't the probabilities for quantum events require Real numbers?

There may actually be the kinds of examples you're asking for. I don't know. But I think the simple answer is that doing math the modern way makes doing the math much easier. We get results easier and we've gotten results we would not have gotten (at least not this quickly) doing finite math. The results may just approximate reality (what doesn't?) but at least we have the results.

PairTheBoard