Does infinity exist?

The conclusion all along was very old, that if you’re committed to abstracting empirical data all the way, unlike the Greeks were, the ideas of infinity and 0 become available.

Sets are just ultra lazy abstractions without any class or style.
Russell at least had class and style, which is why he pricked through set theory in one gesture, and liked Wittgenstein, who is a fledgeling philosopher in how he overcame his Tractatus.

Serendipper for the win because he exposed the grammatical naiveté which causes all the various perspectives unawareness of being various.

:open_mouth:

:laughing:

Whats cool about being all too realistic before infinity is that you’re probably very aware of the finitude of some pretty important shit. Its possible to arrive at the value of finite things, not of infinite things.
and since there is definitely value that can be attributed to the power to identify a definite value, I think you can’t do things like quantum computing on set theory.

Anyway, this is very interesting. Set theory fails. Type theory must take its place. Thats a lot of grinding fucking weight lifting.

Blimey if I cant see now why mathematicians always lie on couches so proudly and don’t walk around to think. Set theory.

Ah, lets say we have a set, blah, and a set blah, and oh what would they do together! Oh what a delightful rainbow of things!

No, not things. Dreams.

X apples available would be theoretical and not actual. You either have in possession/existence the number of apples or you don’t.

Why not? Do you mean to say there is a possibility that extra space could exist such that there would be insufficient apples to fill it? In that case there would not be infinite apples. Conversely, if there are infinite apples, then there cannot be extra space left over.

To say one infinity is bigger than another is to place limits on the smaller infinity which would then make it finite.

If time is infinite, then having a beginning is impossible. This is the same as my argument against having an infinite road that has a beginning.

The universe must be finite or else it wouldn’t be definite and therefore wouldn’t exist. Plus, there is no evidence to indicate the universe is infinite and lots of evidence to preclude it, such as: the conservation of energy which wouldn’t make any sense in infinite energy, the fact that a photon cannot be emitted until its partner is found across spacetime which wouldn’t be possible if parts of spacetime were infinitely far away, the fact that the speed of light is a definite (finite) number where time and space end.

So, in summary:

  • Lots of evidence suggesting a finite universe
  • No evidence suggesting an infinite universe
  • Yet people still believe the universe is infinite

Yep

Does infinity exist = Does infinity exist in relation to something that it’s not, as a function of something engendering it, inside of something, as a part of something?

Is existence infinite = is a relation to something infinite?

“Does infinity exist in relation to something” is not the same question as “is a relation to something infinite”.

You’re assuming time means causality, but it is not since causality cannot describe the speed of causality. Time is something that can speed up and slow down, but causality simply describes one thing leading to another with no speed component.

There can’t be beginnings or ends in infinite anything.

Time without a beginning is a problematic proposition.

How long is a moment? 1s/infinity? lol

Yes, subjectivity is the universe in reference to another part of the universe.

Chaos is deterministic, but sensitive to initial conditions. Randomness is the uncaused event.

What’s the difference?

I don’t understand that.

:confused: :-k

Why would the infinitely tall wall curve around the cosmos and touch its bottom?

@Serendipper

The same as you, but there can be an absolute infinity, and specific infinities.

Or unlimited in quantity here/now, but limited there/then, or unlimited in x qualities, but not in y.

Conceptually you can draw a line anywhere, but you can also conceive of a road ending one way, but not the other.

You can also conceive of an impenetrable wall that keeps everything this side of it from crossing over, not that such a wall is necessary for a road to end one way, but not the other.

Beyond the wall, there might be nothing, not merely empty space, but no MEST at all, or there might be stuff.

It’s endless backwardly, and endful forwardly.

You can make it longer forwardly, but not backwardly.

It has a boundary backwardly, but not forwardly.

Apples could unendingly sparsely populate the unending universe, and still be unending in number, which means some infinites could be bigger than others.

I’m not so sure, for example, if two things are both infinitely divisible, but, finitely multipliable, if you will, than one of them could still be bigger, stronger and so on than the other.

But even if things are necessarily finitely small, the smallest unit of matter, motion and space might still be centillions of times smaller than quarks.

It seems weird…asymmetrical to me the universe could be infinitely big, but not also infinitely small, and if a thing could be infinitely big, and not infinitely small, than why couldn’t a thing be finitely big, and also infinitely small?

The possibility of anything we can imagine existing is endless and infinite :wink:

George W was not the first president of America, and Ahab was a fictional character, so the first premise is technically not true… the second is fictionally true, of things that we read in books…

I do? :confusion-shrug:

You and I have discussed the axiom of infinity in this thread. Perhaps I’m misremembering. If we have discussed the axiom of infinity, then your remark is disingenuous. If I’m mistaken and we haven’t discussed the axiom of infinity, I’ll try to remember that I’m talking to a bunch of 5 year olds. That actually explains a lot.

That is a very interesting remark. Of course if you took freshman calculus, you can do that using a rote procedure, say by taking an antiderivative of the kind of elementary functions you see in calculus class. Integrand is (x^2) so antiderivative is (\frac{1}{3} x^3) kind of thing.

But if you studied the subject more deeply, you would realize that in order to form a logically rigorous definition of an integral, you require modern infinitary set theory. In calculus they don’t show you that. Perhaps you remember that when they defined the Riemann integral, they defined lower and upper sums relative to a partition, and then you took the LIMIT over all possible partitions. To formalize that requires the full apparatus of ZF set theory, including the axiom of infinity.

So to me, the fact that you DO believe in Riemann integration (aka freshman calculus integration) tells me that you’ve seen the rote procedures, but not the underlying theory nor all the weird counterexamples and corner cases that made 19th century mathematicians realize they needed a rigorous theory. Infinitary math is essential to define an integral and do freshman calculus. They just don’t tell you about this until you take a more advanced course in real analysis.

No infinitary math, no logical foundation for freshman calculus. No axiom of infinity, no Riemann integral.

ps – Let me give a concrete example. You mentioned integrating an area over a height. How about if you have a rectangular metal plate with a temperature at each point and you want to integrate the temperature over the area of the rectangle to determine the average temperature. You could integrate the vertical slices then the horizontal ones or vice versa. This is multiple integration as in second year calculus. But how do you know when the order of integration matters and when it doesn’t? How do you know whether it makes a difference if you integrate the x’s and then the y’s, or first the y’s and then the x’s? This can be a very tricky business, especially with a weird or pathological integrand or temperature function. This is when you have to drill down to the rigorous, set-theoretic definition of the integral to prove theorems on reversing the order of integration. In other words the moment you go beyond the simplest examples you need some theory; and the theory of integration requires infinitary set theory, or my name’s not Guido Fubini!

en.wikipedia.org/wiki/Fubini%27s_theorem

The end of the Wiki article gives specific examples where reversing the order of integration gives a different answer.

Because that would be the only place it hasn’t yet touched. To say otherwise would be to put a limit on the wall and make it finite.

What’s the difference? Either there are infinite amounts of something or not.

Sure I’ll go along with having unlimited x qualities, but limited y, such as oranges are unlimited in quantity, but not size, shelf-life, color, flavor, price, etc.

If a road were truly infinite, there would not exist a place void of road. To say there is a place without road is to place a limit on the length of the road.

What’s north of the north pole? Nothing. Not because the wall is impenetrable, but because there is no place to exist north of the north pole; there is no there there.

Forwardly, the road would continue through every planck cube in the universe before finally terminating at its beginning for lack of having anywhere else to go. An infinite road would occupy every planck volume in the universe, in the forward direction, even if the universe is infinite in size.

Unendingly sparsely? What does that mean?

I don’t know what that means. Do you have an example?

This is the debate I had with James a year or two ago. He asserted an infinite amount of smaller particles and I asked why we occupy this tranche instead of some other. If there are infinitely smaller particles, then no size of particle has any special significance. Why is an atom the size that it is instead of some other size? If every particle were composed of infinitely smaller particles, then no particle would have less than an infinite amount of particles, regardless how big it is, and no particle would be any different from any other, and size would have no meaning.

Well, no worries, the universe is finitely big and small :slight_smile:

Lots of good things have been said about being like a child. Jesus said it was conditional to get to heaven :wink:

Being childish is generally regarded as being petty, vindictive, and perhaps stupid, but children are open-minded and every thought is outside the box because they haven’t formed a box yet. Only a child can learn perfect pitch, which is to say that only a child can learn to accurately perceive aspects of our world. youtube.com/watch?v=816VLQNdPMM

Has it been proven that integration cannot be formalized without infinity?

How can we assume an axiom and then claim anything is proven because of that axiom? Since infinity is an axiom, then it can substantiate nothing.

The proof is in the pudding: it gives the right answers consistently and doesn’t require notions of infinity to implement, which is my point: we do not need infinity to “do math.”

Because in advanced math you’re studying applications only to math instead of the real world. Advanced math is for people who have exhausted the practical uses of math and have graduated to the study of math for the sake of math.

So I have no logical foundation, but you simply assume the axiom of infinity and use that assumption to claim your foundation is more logical than mine?

I don’t see why it would make a difference whether we integrate in the x or y first so long as the function accurately describes the temperature variation. If the right answer is only coming out in one direction, then some more-fundamental assumption is probably flawed. Perhaps you could explain the temperature problem in more detail and show why it matters in one direction vs the other, then maybe we can see why.

This reminds me of those hotly debated arithmetic ordering of operations puzzles that I hope you’re familiar with because I can’t find a good example at the moment, but some will fill the comments section with debates and ultimately have no concrete resolution.

“No axiom of infinity, no Riemann integral.”

No Jesus, no Sacrament.

I understand the openmindedness of children, but the subject we are discussing is better served by assuming the participants are intelligent adults who have perhaps been to school or maybe read and thought a little bit about things.

Now that is a very good question! What is true that not only calculus but all of physical science is currently founded on infinitary mathematics. But, it is this a necessary or a contingent fact? I’m pretty sure it’s contingent. Foundations go in and out of favor. Netwon got results using math that’s not regarded as rigorous today, and in fact required another 200 years to logically formalize.

But is all of modern mathematics, including nonconstructive math and uncountable sets, necessary to found physics?

There are researchers trying to find weaker logical structures in which to do math and physics. Finitism (No axiom of infinity, but still with mathematical induction); and ultrafinitism (not even induction); are far too radical and I no of nobody who claims to be able to found physics on finitary principles.

However, constructive foundations are a subject of great interest. In constructive math and physics, an object is said to exist only if it is the output of a Turing machine. I discussed this earlier. So there is no axiom of choice, no uncountable sets, no noncomputable real numbers.

en.wikipedia.org/wiki/Construct … athematics

In set theory we have the full powerset axiom, that says that all of the subsets of a given set exist. In constructive math, only the contructible sets exist. These are the sets whose elements can be cranked out by a computer program (as exemplified by a TM). So the even numbers exist as a subset of the naturals. But most sets in standard math no longer exist because their elements can’t be computed.

You may be unhappy with this, because we do have infinite subsets of the naturals and for that matter we have the full set of naturals, infinitely many of them. So constructivism still needs “a little infinity,” but far less infinity than full set theory.

That is the state of the art today. If you wish to hold out hope of a glorious future in which all of physical science can be founded on ultrafinite or finite principles, that is your right. But why? How are you going to express the differential equations of biology? Why is it so important to you?

I quite agree. Nobody thinks the axiom of infinity is “true” in any meaningful sense. Rather, infinite sets are USEFUL to mathematicians, and infinitary math is useful to physics. Whether it’s necessary, we don’t know.

Put it this way. We could play chess without the queen. The game would be very different and much more dull. So we keep the rules the way they’ve evolved.

The axiom of infinity is like that. It’s a more fun and usesful rule so we keep it in the game. Why does that bother you?

We don’t need the queen to play chess. So what? But you’re wrong on the facts. Without the axiom of infinity, at the very least the constructible sets, you can’t develop the theory of the real numbers sufficiently well to do modern physics. Sure someday someone MIGHT find a way, but in the meantime are you throwing out all of science back to before Newton?

But no, this is quite false on the facts. Differential geometry and non-Eucidean geometry were mathematical curiousities in the 1840’s, and became the mathematical foundation of relativity aftter Einstein.

And quantum physics lives in the mathematical framework of Hilbert spaces, a highly abstract infinite-dimensional vector space studied in a field called functional analysis.

So you’re just flat out wrong on the facts here. Advanced abstract math is indispensible for modern science. Not all of advanced math, but much of it. Sure there is math that’s “out there” today, but who is to say it won’t be essential to the study of the real world a century from now?

Not more logical. More useful. If you’d discuss what I write and not the words YOU put in my mouth, this would be more productive. You are constantly arguing against positions I’ve never expressed.

I don’t say the axiom of foundation is more logical than its negation. On the contrary, they are both equally logical, each being consistent with the rest of the axioms. The axiom of infinity has proven itself more useful so most mathematicians adopt it. There are constructivists, finitists, and ultra-finitists among mathematicans. Especially in the past few years, there’s renewed interest in constructivism due to the influence of computers and automated proof checking.

I linked a Wiki article that contained counterexamples, and I explicitly called out that fact. The Wiki article on Fubini’s theorem contains examples of functions whose integral depends on the order of integration.

Those puzzles only demonstrate the poor teaching of the order of precedence of the arithmetic operators. And the poor understanding of this topic even among elementary school teachers. They don’t hire elementary school teachers for their math acumen. God knows I wouldn’t spend my days among a bunch of ten or twelve year olds.

I don’t see how you can make this comparison. If you think those silly puzzles are anything like the discussion of the axiom of infinity, I don’t think you’ve given the matter enough thought.

No bacon, no BLT.

Bacon isn’t a hypothesis that is only validated by the sandwich its in. It can be eaten (made sense of, valued, used) for its own properties.

Explicating; consider “no humans, no railroads”.
Would the railroad suffice as a justification of the human?
In a philosophic sense, I mean.

Oh I see your point. Infinitary set theory has been validated by over a century of mathematical practice. Surely you would at least grant me this historical fact, easily confirmed by a study of the mathematical literature.

I still don’t follow your religious analogy, perhaps you can explain it to me. Is the Sacrament a validation of Jesus? How so? I don’t know anything about Christian theology past the Lord’s prayer, which (at the time I went to school) we were required to say every day, along with the Pledge of a Lesion, and to the republic for Richard Stands.

Lets say Mohammed then. No Mohammed, no Islam. And Islam is the desirable thing, evidently - to a muslim. Like the Riemann integral is to a mathematician. That is why Mohammed is holy, why infinity is “true”. Its not like islam is holy because Mohammed is holy. He was made holy (rendered into a desirable idea) by his service to Allah.

Infinity is an idea required to have certain other ideas possible, as you point out. Thats why it exists.

The same goes for Jesus, it is an idea required to make some moral systems work, moral systems which are the criterium. There is no evidence of Jesus directly. Whats more, the idea is that he exists simply to redeem mankind. I see a strong resemblance with the idea of infinity. It exists to make set theory and some other desirable ideas possible. Both are ideas justified by their making other things possible.

A better argument for you to make here would be “no bacon, no pig”.
Bacon tastes perfectly good without the BLT. But does the pig serve without the bacon? Would we keep pigs if we didn’t like bacon?

I have to admit I didn’t follow all of that. I am simply making a utilitarian argument for mathematical infinity. I suppose the official name for it is indispensability. plato.stanford.edu/entries/mathphil-indis/ However I’m not sufficiently knowledgeable about that set of ideas to say if that’s exactly what I’m saying, or just influenced by it. My thoughts arise from the ideas of philosopher of math Penelope Maddy, who makes a similar argument in Believing the Axioms. She refers to a philosophical principle she calls MAXIMIZE, which says that given a choice of axioms which allow us to do less, or that allow us to do more, we choose the axioms that allow us to do more. [My paraphrase, not necessarily Maddy’s literal words].

The Jesus and Mohammad analogies are inexact, in the sense that J and M were the founders of their respective religions. But we had mathematical integration before we had a theory of infinity. Infinity is currently (but perhaps not necessarily) essential to the foundation of mathematical integration. Whereas Jesus and Mohammad are necessary to the founding of their respective religions, because they were, after all, the founders.