Does infinity exist?

I’m not rejecting math, except to the extent that it does not apply to physical reality, such as a completed infinity (ie the bounded unbounded). Your refusal define what you’re talking about is your problem if you intend to challenge my definition and the definition of every dictionary on earth.

You can’t say you reject my definition and then not offer a replacement and then continue to use the word. You also cannot offer a definition that relies on the very fact that you’re disputing (a set bijected with a subset of itself requires that the set be unbounded).

If infinity is not the unbounded, then simply tell me what it is… in plain english with good faith effort to convey meaning.

I reject ALL infinities as nonexistent because they are just as absurd as your assertion that anything could be both bounded and unbounded, in terms of cardinality, at the same time.

How can the cardinality of the set 0 and 1 be “uncountable” and also be bounded? And not only that, but be bounded by a bigger unbounded thing? This is absurdity!

What is unbounded is without borders and cannot be said to exist. How can we have a box with no walls or a container with no confinement?

And all this strife and struggle because you so desperately want this concept of infinity to exist.

This debate is exactly the same as the atheist/theist debates I hear on youtube.

I’ve heard of it, but have no interest in it because it has no practical use, at least not to me. Arithmetic, algebra, geometry, trigonometry, calculus all have uses that apply to the world I live in and I’ve never needed set theory. I’ve read that there are uses for it in computer programming to counter infinite loops, but I’m not a programmer and perhaps there could be another way which doesn’t involve set theory.

I follow math from the engineering perspective as it relates to reality. The purpose of this thread is to establish if infinity exists and not if it exists in manmade constructs. I could write a play containing pink elephants and then claim pink elephants exist and that is essentially what you’re doing by appealing to math.

For anyone interested, there is some controversy about this:

“I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.” Johann Carl Friedrich Gauss - a German mathematician and physicist who made significant contributions to many fields in mathematics and sciences. Sometimes referred to as the Princeps mathematicorum (Latin for “the foremost of mathematicians”) and “the greatest mathematician since antiquity”, Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history’s most influential mathematicians. en.wikipedia.org/wiki/Carl_Friedrich_Gauss

Who is more authoritative than Gauss? Only Euler and he existed too early to chime in.

“I don’t know what predominates in Cantor’s theory - philosophy or theology, but I am sure that there is no mathematics there.” Leopold Kronecker - a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor’s work on set theory, and was quoted by Weber (1893) as having said, “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk” (“God made the integers, all else is the work of man.”) en.wikipedia.org/wiki/Leopold_Kronecker

Can’t you see that makes no sense? If no infinite cardinality is the largest, then there is no bound; therefore it can’t be bounded; especially by something bigger that is also unbounded. This is like saying our infinite universe is bounded by a larger infinite universe when neither have any bound.

The reasons for that are pretty obvious once one takes into account the religious aspects.

I don’t see how 2+2 depends on infinite set theory. I don’t see how A=PI(r^2) depends on infinite set theory. I don’t see how ax^2 + bx +c = y depends on infinite set theory. I don’t even see how integral calculus depends on set theory.

Mathematician Solomon Feferman has referred to Cantor’s theories as “simply not relevant to everyday mathematics.” en.wikipedia.org/wiki/Controver … e_argument

NJ Wildberger published a rant proclaiming it nonsense and a religion researchgate.net/publicatio … ou_believe

No it isn’t.

“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” — Albert Einstein

I find the assertion that the unbounded exists in physical reality is a tough position to take, especially when it cannot be tested scientifically nor even conceived cognitively, much like god, and must be held on faith.

The notion of infinity should, in physics, always and only be understood as a place-holder for an unspecified finite boundary.

I already defined infinity several times. A set is infinite if it can be bijected with a proper subset of itself. That definition is due to Dedekind in the 1880s.

But nothing could be further from the truth. There are lots of bounded infinite sets. I gave the unit interval as one example. Another is the ordinal (\omega + 1). That’s the usual set of natural numbers 0, 1, 2, 3, … with an extra element at the end, like this: 0, 1, 2, 3, 4, …, (\omega). That’s an infinite set that contains a largest element. It’s bounded.

You keep insisting that infinite sets must be unbounded, but there are so many counterexamples that you’re simply wrong, absent a compelling argument. I have no problem with you holding different opinions than standard math. I do have a problem that you can’t intelligently defend your position, or frankly even articulate it.

A set is infinite if it may be placed into bijection with at least one of its proper subsets. Galileo noted this fact about the natural numbers and there are references in Indian mathematics hundreds of years earlier. en.wikipedia.org/wiki/Galileo%27s_paradox. This is not a new idea.

I see. So earlier when you wrote …

… you were just kidding? You have contradicted yourself. You reject infinity and accept infinity.

Easy. We know its cardinality is uncountable, (2^{\aleph_0}) in fact. And its powerset has cardinality (2^{2^{\aleph_0}}), which is strictly greater.

You are not the first person to have felt that way. If you wish to reject all of modern mathematics from 1874 onward, that’s perfectly ok. Back in those days many people rejected these ideas. Today they’re almost universally accepted. But I did say “almost.” There are finitists who reject the axiom of infinity. The only problem with that doctrine is that it’s difficult and presently impossible to found physical science! You can’t do relativity and quantum physics without infinitary math. Now that is in fact a genuine philosophical problem. I don’t think you appreciate it though. You wave your hands and say, “absurd,” but you haven’t proposed how to salvage physics; and you are decidedly out of step with modern math. Nothing wrong with that, but you have the burden of justifying your position. Everyone else accepts infinitary math because it is (1) interesting; and (2) useful. Whether it’s “true” in any meaningful sense, I cannot say.

So you see I am not disagreeing with you. I am only pointing out some of the subtle issues with your position that you may not have considered.

I make no claim that any mathematical objects “exist.” Existence is a technical term. If we can prove something exists by our axioms, it has mathematical existence. I make no further ontological claims. You’re arguing with a strawman.

Do I claim the unit interval of real numbers exists in the physical universe? Of course not. Do I claim it exists in some sort of abstract Platonic sense? No not even that!

I only claim it has mathematical existence. I make the same claim for the number 3. That has mathematical existence. I have no idea if it has Platonic existence. You’re fighting a strawman argument that you’ve made up yourself.

No I make no claims of existence at all. Only of mathematical existence within infinitary set theory. That’s a very limited claim and it is all I claim. Just as when I play chess, I don’t claim the knight “really” moves that way. It only moves that way as long as I accept the rules of the game.

Mathematical infinity only works if we accept the rules of the game of set theory. But this game is (1) interesting; and (2) useful. So we play the game.

I believe this ontological point is the heart of our disagreement. I make no claims whatever for the “truth” or “existence” of mathematics. I only claim interestingness, usefulness, and the agreement of almost all modern mathematicians on these matters.

I make no claim about truth or Platonic or physical existence.

No it’s not. I have no “belief.” I only enjoy playing the game. If I accept the rules of chess, I may move my knight in such-and-so a manner when it’s my turn to move. If I accept the rules of set theory, I may prove various facts about infinite sets.

That’s all I’m claiming. You are the one fighting against a position I don’t hold.

Without infinitary set theory you lose quantum physics, which is based on a mathematical discipline called functional analysis. You simply lose QM. Relativity too. A century of physics, gone.

Ah. Well calculus can not be placed on a logically sound basis without infinitary set theory. That was the great advance of the 1880’s, what’s called in math history the arithmetization of analysis.

You don’t need set theory to do freshman calculus. You do need it to do higher mathematical analysis. That was the lesson of the 1800’s and 1900’s.

You are confusing the terminal point of your own education with the terminal point of human knowledge.

I have no problem with that. But why are you on a philosophy discussion forum talking about things you don’t understand?

From an engineering perspective? You’re several disciplines removed. From engineering to applied physics to theoretical physics to mathematics to philosophy.

You can not hope to elucidate the infinite from an engineering perspective.

But no. You could write a play about pink elephants and if your play was entertaining, people would enjoy it. And it would be useful in the sense that it brings joy to children. Again these two criteria: Interestingness and usefulness.

I appeal to the interestingness and usefulness of math. I make no claims about truth or existence.

Posting that hoary old quote from Gauss is the last refuge of someone with no argument to make. A lot has happened since the 1820’s when Gauss did his work.

Likewise Kronecker. Kronecker was wrong, and Hilbert was right when Hilbert said, “No one shall expel us from Cantor’s paradise.” And quoting historical figures out of context really doesn’t make you look clever. It makes you look like someone who can surf Wiki but hasn’t got any background or knowledge of the historical developments.

You don’t even know what the word bounded means. Any infinite cardinal is bounded by that of its powerset.

LOLOLOLOL. I stopped reading here. Wildberger’s ideas on infinity are universally regarded as cranky.

All the best. I get that you like to surf Wiki. You would be better off learning some math or philosophy.

Therefore you agree with me that the infinite is the unbounded.

Nothing can be closer to the truth. The ONLY way to biject a set with a subset of itself is to have an unbounded set. You cannot do it with a bounded set. Therefore the definition of the infinite is unbounded and your definition is just another way of saying that.

If it’s bounded, it’s not infinite; if it’s infinite, it’s not bounded. That remains true until you come up with a new definition for infinity that does not require unbounded sets.

Because they are called infinite. If infinite doesn’t mean unbounded, then what does it mean?

I have no argument against dogma.

First you say I suck at math and now my linguistics are bad too?

I’m sure Galileo would have been smart enough to see that in order for bijection to apply, the sets would need to be unbounded.

Even though I assert infinity doesn’t exist, I can still talk about it in the context of an example. For instance I don’t believe Yahweh exists, but I can recite many things he said in the bible.

In your example, the category is the set of numbers between 1 and 0 of which there are infinitely many.

You didn’t answer the question. If the uncountable is bounded, then what is the bound? At what point must I stop counting because I hit your bound?

I don’t have to reject all of modern mathematics just because I don’t believe in absurdities.

We don’t need infinity for relativity and I’m not a big fan of quantum physics.

Just because we use a concept to arrive at a conclusion that matches reality does not mean that the concept exists in reality.

I’m not the one asserting the existence of something that cannot be tested.

I don’t really believe any of that, but I’ll remember you said it for future reference.

If you had simply claimed that infinity exists only within a mathematical construct like the knight exists only in chess, then I would have agreed. The point of the thread was that infinity can’t be said to exist in what we call reality.

I had 6 courses of calculus in college and don’t remember needing set theory. I looked in my book and found no reference to set theory in the table of contents nor the index. This is my book amazon.com/Calculus-Analyti … 0201163209 (I can’t believe it’s still worth $50 lol)

It’s a good textbook and has about 1200 pages. Inside on page 70 it says “While there is no real number ‘infinity’, the word ‘infinity’ provides a useful language for describing how some functions behave when their domains or ranges exceed all bounds.”

Also this book amazon.com/Elementary-Diffe … 0471509973

And this one amazon.com/Numerical-Method … 812030845X

Number theory sucked. I just wasn’t interested for lack of having practical purpose or maybe I was burned out on math by that point… or both.

Why would I talk about it if I understood it?

“Those who know, don’t say; those who say, don’t know.” - Lao Tzu, Confucius or someone like that.
“Writing is the act of discovering what you believe” - David Hare

So you go from that ^

To this:

Looks like you’re making claims to me.

I don’t need to rely on Gauss to substantiate my argument, but as an aside I was showing there are those who disagree with the concept of infinity, including the princeps mathematicorum.

Then define it.

How does a big thing limit a smaller thing?

Arrogance and ignorance go hand in hand.

Which means nothing. More appeals to popularity on your part. Are we teenage girls seeking to be fashionable?

Wildberger was on the OP. If you were going to stop upon seeing Wildberger, then why did you begin?

You would be better off updating your insults. At least make them funny.

I didn’t read any further in your post. I’ll just leave this here for people to read and let them draw their own conclusions.

You know it occurs to me that perhaps you don’t know what the word unbounded means. An ordered set (S) is bounded (above, say) if there is some element (x), which may or may not be in (S), such that if (s \in S) then (s \leq x).

By that definition the closed unit interval ([0,1]) is bounded above by 1, which is in that set; or by 2, which isn’t. Either way it’s bounded. Likewise each cardinal up to that of the real numbers, (2^{\aleph_0}), is less than the cardinal of the powerset of the reals, (2^{2^{\aleph_0}}). And the third example I’ve given is the ordinal (\omega), an upper bound for the set of natural numbers. It’s an ordinal number greater than each of 0, 1, 2, 3, 4, … The first example should be obvious to anyone who made it out of high school analytic geometry. The latter two examples are less familiar but not very difficult.

I’m starting to think that you mean something entirely different by the word bounded. Because a set may be bijected with a proper subset of itself yet still be bounded by my definition. I just gave three examples. The definition I gave is the standard one in math.

In math, the attributes of bounded and infinite are not equivalent. In fact if a set is finite, it’s bounded. So you do have an implication in one direction. Finite implies bounded. But the other direction fails. A bounded set MAY be finite; or it may be infinite.

To sum up what I said earlier, you don’t need to accept modern math. But you do have to give it its due. If you have some different definition for the word bounded, by all means provide it.

That says it all!

You know everything; therefore what I say is inconsequential.

Unbounded means having no bounds or limits; unlimited; infinite.

X is an arbitrary starting point in an infinite series that has neither intrinsic beginning nor end.

Then it’s not infinite. The infinite cannot have a beginning nor an end.

Insisting there is a cardinality greater than that of all natural numbers is placing a limit on natural numbers that is transcended by the greater cardinality and if there is a limit to the natural numbers that is transcended by another cardinality, it would be a cinch for you to display it right here: what exactly is the biggest natural number that is transcended?

You’re struggling in transparent desperation to show that the absurd is true, that the unlimited has a limit that can be transcended by a bigger unlimited thing.

On the contrary. I know nothing. You don’t realize that about yourself; and thereby render your own opinions inconsequential.

I have said about all I can say here. @Serendipper, Thank you for an interesting and stimulating chat.

So infinity exists as a concept. Like we can all think the thought of infinite steps between 1 and 0, Serendipper showed he can do it too. But not as something you can hold in your hands.

I think Serendipper conceded infinity exists when he couldn’t give the exact maximum number between 1 and 0 except as infinite.

That doesn’t mean we can hold infinitely many apples in our hand.

But it does mean, infinity can be held in the mind.
Only to a mindless person that is nonexistent.

Well, like Gauss said, “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.”

Infinity can never be anything that can exist like forever can never arrive for when the boundless finally has bounds (infinity is completed, forever has arrived), then it’s no longer boundless.

That is illustrated by the infinite series 1-1+1-1+1-1+1-1+1-1+… which is either 1 or 0 if we stop to check, but continued forever and it’s assumed to be 1/2 (equidistant between states).

The maximum number of conceptual steps between 1 and 0 is limited by the size of the universe or medium used for writing the number down (or remembering it). To determine a hard ceiling, find the volume of the observable universe in terms of the number of planck cubes. (Wildberger’s point is that (((((((((10^10)^10)^10)^10)^10)^10)^10)^10)^10)+23 cannot be factored because the universe isn’t big enough.)

The maximum number of divisions between 1 and 0 on a ruler would be the number of quarks contained, or if I’m wrong about that, then there will be some other point where locality loses meaning and there will be no way to distinguish one point from another because there will be no “thing” at that resolution to act as a reference.

Either way, conceptually or realistically, there is a limit which makes the divisions finite.

Phaer enough. So lets put this in terms: there are infinite possible numbers between 1 and 0, but not actually an existing infinity of them.

I agree that if a number exists it should be possible to arrive at it.
But we can argue that to arrive at infinity you just have not divide by zero.
I know it is brutish but it kinda works.
Still I won’t say this overthrows the point you’re making, because 0 doesn’t “exist” like 1 does of course.

Of physical matter surely you are 100% right. There are no physical infinitesimals, there is a minimal thinkable and thinkably also a minimal existable quantity of quality. But there may be infinite amounts of qualities. This is how you really arrive at infinity in an additive sense, if you look at the ways in which things are intertwined with each other, how many of them are there, because as soon as you understand something about them you intertwine with the web of intertwined ways of relating and then you see true infinity. But then you also see that infinity is what the world must become when it is held up by one perspective. So there can’t be an infinite number, but there can be a number of infinities.

Perhaps the fact that it can be imagined is all that infinity needs to exist, or does a concept need to manifest into reality in order to. :-k

Even the finite can be infinite…

Question: Do purple flying elephants exist? I can imagin e them.

Bonus question: Is Ahab captain of the Pequod?

Second bonus question: If George Washington was the first president of the US, and Ahab was the captain of the Pequod, are those statements equally true? True in the same way? Both Washington and Ahab have an equal claim to existence? Along with the purple flying elephants?

I hope you can see that you need to greatly qualify your remark about imaginability being sufficient for existence.

Hard to know how you can allow one possibility without the other. First, of course we are not talking about the physical world. Numbers are abstract objects. So how can you allow a “potential” infinity of counting numbers 1, 2, 3, 4, … but then deny the actual infinity of the set {1, 2, 3, 4, …}?

In terms of set theory, we make the leap from 1, 2, 3, … to {1, 2, 3, …} via the axiom of infinity, which says that an infinite set exists. It’s much more useful to allow infinite sets so we generally accept the axiom of infinity.

But if you deny the mathematical existence of the set of natural numbers, yet allow for infinitely many natural numbers, that seems like a philosophical quibble that you would need to justify.

And again, note that NONE of this has anything to do with the physical world. Numbers are abstract. By what philosophical principle can you say that infinitely many numbers 1, 2, 3, … exist, but that I’m not allowed to put set brackets around the list?

It is not a mathematical quibble but a philosophical one which means it could end up telling mathematics what to do, after all philosophy still trumps mathematics by being the arbiter of what constitutes true correspondence. So what is actually in play here is what a “set” means, not what infinity means.

I can now see how Serendipper considers this as strictly speaking an unwarranted shortcut. But I don’t contest that it is useful. Shortcuts are very often useful, look at the Panama Canal.

I personally don’t deny the mathematical set. I just deny the philosophical set. I mean I deny that this set of infinitely many numbers has any meaning outside of how the set is being made useful.

I know that. So the question is how far we want to allow mathematics to operate in defiance of physics.

Because philosophy is about reliability and not about speed. The power gained from seeing sets as potentially having infinite size may come with a drawback of making it doubtful if sets can be trusted, if they can still logically correspond to another set.

I guess what I mean is all questions like, how does the set of integers correspond to the set of real numbers? Does the fact that the second is infinity squared make the former into the root of infinity? If this can’t be addressed there is a logical problem with the infinite set, even if it can still do mathematical work, creating a subprime mathematics bubble.

You’re allowed, but all you can do is use that for a specific mathematical operation. That you put brackets around it to mean it goes on infinitely does not mean it actually goes on infinitely. What do I mean by actually, if not physically?

I mean in the sense that it identifies infinity. Thats where philosophy begins to not be bored.

I don’t necessarily agree, but I will stipulate for sake of discussion that philosophy trumps math. BUT philosophy has perfectly well accepted infinitary math. Wasn’t Russell a philosopher? The philosopher of math Penelope Maddy has written Believing the Axioms (parts I and II. I linked to part I). She walks through each axiom of ZFC (Zermelo-Fraenkel set theory with Choice), places it in historical and philosophical context, and describes the principles by which we accept it today.

So it is NOT true that philosophy says one thing and math another. Some philosophers argue for finitary math, I suppose, just as some mathematicians study finitary math. It’s interesting to study! But mainstream math accepts infinity and the mainstream philosophers of math do too.

So I don’t agree with your conclusion even if I accept your premise. There’s no dispute between philosophy and math when it comes to the mathematical infinite.

Yes that is very true and also insightful. In elementary education we tell people that a set is a “collection of objects.” But of course that was Frege’s idea, demolished by Russell. A set is, in fact, a highly technical gadget with no definition at all. A set, in mathematics, is any object that obeys the axioms of set theory. And what axioms are those? Any axioms you like, subject only to consistency and interestingness. And we don’t even know for sure if our axiom systems ARE consistent.

If anyone wants to argue that mathematics is based on a pile of sand, you will get no argument from me. It’s the job of philosophy to explain why all this obvious nonsense is so damn useful.

Great example! And I don’t disagree with Serendipper on this point either. There is no logical or moral reason why we should prefer one assumption over another, when it comes to allowing the actual infinite into math. All we have is a centuries of experience that when it comes to understanding the physical world, mathematics is indispensable. SEP has an article on this indispensability argument.

Ok. But will you stipulate at least that most philosophers accept modern math? The axiom of infinity dates back to Frege and Russell and Zermelo and all those other ancients.

Ah. Ok. Meaning outside of utility. Well, tell me this. A Martian physicist comes to earth and sees a traffic light. She can tell red light from green light by the wavelength. But she cannot tell you which is stop and which is go. That’s a socially constructed fact that has meaning only because it’s useful. We could make green mean stop and red mean go, and that would be just as valid a choice. There’s no inherent meaning in the colors.

So I would say that it’s true there’s no “meaning” to set theory outside of how we use it. But so what? Most of reality is that way. Civilization is one abstraction piled on another. None of it has any meaning outside of how we as humans use it. Your criticism of set theory is a criticism of the foundation of civilization: namely, the human power of abstraction. Our ability to make the abstract real.

But it’s not. A lot of math comes directly from physics. Physics finds modern infinitary math indispensable. Even though the universe might be discrete, the math used by the physicists is infinitary. That may be a puzzle; but it is also a FACT.

Your beef is with the physicists, not the mathematicians! The mathematicians invented this crazy non-Euclidean geometry, but it was the physicists who decided it was the best way to understand the world. I hope you see my point!

Well maybe set theory can’t be trusted. Make your case. What does that mean? What if it can’t?

The integers are a proper subset of the reals. Additionally, the reals can be set-theoretically constructed from the integers. That is, if all we had was the integers, we would first create the rationals as certain equivalence classes of integers; then we’d create the reals as certain subsets of the rationals.

I would not say the reals are infinity squared and in fact that’s wrong. What is true is that the cardinality of the reals is the same as the cardinality of the set of subsets of the integers. Is that what you meant?

I hope I addressed it. There is no sense in which “the reals are infinity squared” is meaningful. The reals are in fact essentially the same set as the collection of subsets of the integers. You can encode each as the other.

All the philosophers of math I know accept mathematical infinity. I must be reading the wrong philosophers. What does “believe in” mean? Just that we accept it for being useful; and we rely on experience that the history of math is the history of weird stuff that someone realized was actually useful. Negative numbers, complex numbers, irrational numbers, non-Euclidean geometry. So … what is the meaning of the mathematics of infinity? Perhaps we’ll know in a hundred years.

Yes but his paradox (of the set which includes every set which isn’t included in itself) caused him to find set theory debunked and invent type theory, didn’t it?
So philosophy through Russell doesn’t redeem the infinite set. Im not aware of philosophers that do redeem it…and to be honest im not so sure physics rely on it except string theory?

I do understand of course that much if not most computer programming requires the tool of the infinite set.
But this doesn’t mean lets say that we can hypothesize an infinitely large bus which could compute an actual numerical infinity. Type theory is a bit more of an engineers thing than set theory.

I need to verify some stuff to get back to you on some of the other points.

I’m sure Russell believed in infinite sets but I am not a Russell scholar. Do you know which philosophers deny mathematical infinity? Is this point important?

Penelope Maddy would be one such. But I’m no expert on what philosophers think.

Quantum physics uses Hilbert space, a part of functional analysis, which is full of infinitary mathematics. Likewise relativity relying on differential geometry. You can’t do any modern physics without infinitary reasoning. For that matter, calculus (eventually formalized via infinitary math) was invented by Newton to describe gravity.

Computers are finite and do not require infinitary mathematics.

Not sure where you’re going with this.

Did you look at type theory at all? Unlike set theory it integrates with Peano. So it doesn’t ignore the machinations of the logic of integers. That’s a first step to draw this anywhere near philosophy.

The infinity you mean is just 1/0.
That’s not actually an infinite number of values but a functional limit, a horizon.
This is not a nominal infinity, it is just the reification of receding.

A straight line is infinite in length and in infinity it forms a circle. But to say that this circle exists is not philosophically valid.

If you’re saying what I think you’re saying (bolded part), then you’re hitting the nail on the head for the point I’m ultimately driving at which is there is no way to make an observation without affecting the thing being observed and, per Goethe, observation includes deduction. So, the fundamental of whatever we behold invariably will be perceived as infinite due to the infinite regression involved by affecting the thing being beheld, but that doesn’t mean there is an existent infinity, but it simply means the subject and object are the same thing. So I would consider infinity to be proof of unity (the camera observing its own monitor) since the only alternative is to concede infinity exists as a completed incompletion, which is too nonsensical to get my head around and if we open the door to nonsense, how will we know where to draw the line.

When I try to picture an infinite plane in my mind, it can only curve back on itself because I’m trying to grasp the full extents of it as one thing and when I do that, it turns concave until it eventually joins with where my mind is calling the center. An infinite plane that extends forever without end isn’t something I can imagine. I can fool myself into believing I can, but I’m lying if I claim such ability; the best I can do is make the edges fuzzy and call that infinite (that’s cheating). But if I REALLY make a plane that doesn’t end, then there is no other place to go than where it started. The only way to exist without also having beginning nor end is to be a loop (and why the wedding ring is a symbol of eternity and also a symbol of unity).

I think there is a way we can work with notions of infinity by working with the inevitable ramifications without actually conceptualizing infinity as a thing, but even that doesn’t pan-out in practice. For instance the Thompson Lamp where the switch is turned on after 1 min and off after 1/2 min and on after 1/4 min and so on. Eventually the speed of the switch will exceed the speed of light, so whatever state the lamp was in before crossing the velocity threshold (probably on) will be the final state of the lamp since the switch would be moving too fast for electrons to react. So we can say in our heads that the final state of the lamp should be half-on and half-off, but it can’t work in practice due to our finite universe. Even infinite velocity doesn’t make sense since speed can’t be faster than instant, which is c. Otherwise things could arrive before they left, and not only that, but arrive infinitely sooner than they left (whatever that means).

Conceptualizations of infinity are like that dream I had as a kid where a cat had its head in its own mouth (back in the days that I didn’t realize bicycle spokes held the bike up and trees didn’t make the wind blow); we probably can’t admit to ourselves infinity is absurd because at such an old age, we shouldn’t be that silly anymore, so it’s denial. Compounding that, people really want infinity to exist since it’s a good substitute for god and answers so many questions. The incentive to cheat (not be scientifically objective/unbiased) is high. And there is no proof or even good reason to believe an unbounded thing can be beheld by either our hands or our minds or even be said to exist, much like zero, which is also the bounded unbounded thing: limitless nothingness in a tidy package. Obviously we can think about “nothing” as the absence of something, but that’s not nothing. I don’t think anyone can truly think about nothing because there’s nothing there to focus on, conceptualize, and observe.

What do you consider to be a philosopher of math? Is it someone who first accepts infinity then is bestowed with the title of philosopher of math while anyone who rejects it is labeled a crank? I’d be willing to bet. Much like the ones accepting the hiv causing aids hypothesis get the funding while deniers are lost to obscurity; or the sat fat / cholesterol deniers get ostracized; or climate change deniers. Science is hardly more than fashion: you either support what they tell you to support or they’ll pull your funding and call you names.

All The Great Geniuses in Art, Music, Philosophy, Science and Literature Believed in God amazon.com/Greatest-Minds-B … B00K598JF4

Even modern-day geniuses believe in god. Chris Langan (IQ 200) is hard at work trying to prove it. Show me a famous genius who isn’t a theist. There might be one.

So does that prove god exists?

They do to escape loops. Actually, that’s the only practical use for set theory that I could find.

mathoverflow.net/questions/1033 … heory-have

When such programs involve loops and recursive calls (self-reference), we need methods for showing that the loops and recursive calls terminate, i.e., that the program won’t run forever. The usual induction principle for natural numbers suffices for showing that a single loop terminates, but we need double induction for double loops, triple induction for triple loops, etc. The whole business can get very complicated when the program is more than just a simple combination of loops. Set theory helps sort it all out with the principle of transfinite induction and the calculus of (infinite) ordinal numbers. Transfinite induction covers all possible ways in which one could show that a program terminates, while the ordinal numbers are used to express how complex the proof of termination is (the bigger the number, the more complicated it is to see that the program will actually terminate).