Huh? Infinite doesn’t mean maximized or maximal.
Pretend you’re talking to a 5-year old kid who doesn’t know what infinity means. If you say “we have an axiom of infinity”, the kid will look at you stupid. If you say, “we can align the set with a subset of itself”, the kid will look at you stupid. If you say, “the infinite is the unbounded, unlimited, unending” then the kid will say “Ooooh!” You cannot do any of these acrobatics (axioms, bijections) until you make it clear what infinity means.
There is no way I could perform a bijection of a set with a subset without already understanding the set is unending. So I would need to know what infinity means before I could understand the definition.
The next step back from the axiom of infinity is the Peano axioms. Here we have each of the natural numbers 0, 1, 2, 3, …, but we do not have a completed “set” of them; where a set is defined as something that obeys the particular axioms you’re assuming.
It was in this context that Galileo made his famous observation that the counting numbers may be placed into bijection with a proper subset of themselves, in particular the squares. That is not a “appeal to authority” as you put it, but rather a statement of historical record. 1638, in his final work, Two New Sciences. You can find a copy online these days.
More generally if I assert that the planets revolve around the sun, I am not making an “appeal to authority” when I accept this universally agreed upon conclusion of the greatest scientific minds in history. “Oh yeah, Newton and Einstein, you’re just appealing to authority.” Is that the argument YOU are making here? That Galileo was some jerk and YOU know better?
I accept that planets revolve around the sun not because of authority, but because it’s the most sensible scenario.
If you take a step back and view your position with objectivity you will agree that the burden of proof is on you.
The infinite is something that cannot be observed (much like god) and there is no evidence for (much like god), yet I am required to prove that it doesn’t exist? There is also no evidence of a teapot orbiting the earth, so do I have to prove that doesn’t exist too?
Before analyzing the case of Peano, let me first cover the third alternative: You deny that all of 0, 1, 2, 3, … exist. You claim that at some point, there aren’t any more. You deny not only infinite sets, but mathematical induction too.
What’s the biggest number you can think of? Now make it bigger: square it, factorial, define new symbols to reduce the size and continue on and on until you run out of room on the forum to write that number down. Regardless what you devise, you will find a biggest number, but you won’t be able to do anything with it other than bask in its glory.
I completely agree with you that from an ultrafinitist position that it’s meaningless to talk about a bijection. There’s no map that inputs n and outputs 2n. At some point you put in a big n and it says, “Sorry Dave, I can’t do that.”
The issue isn’t whether the bijection is possible, but whether it defines infinity.
Ultrafinitism is a really interesting idea. There have been a couple of serious ultrafinitists, though many more adherents are cranks. That said, ultrafinitism is useless. Even if it’s true it’s useless. You can’t do math with it and if you can’t do math you can’t do science and then we’re back to living in caves and throwing rocks. If you reject mathematical induction you lose all of finite math, combinatorics, everything. First you threw out calculus, and now basic probability theory?
I can integrate an area over a height to yield a volume without using infinity. I can add my grocery bill without infinity. What do you need infinity for?
What’s the limit of 1/x where x → 23+10^10^10^10^10^10^10^10^10^10? Zero right? If it’s not zero, then how far am I off? No machine that could ever exist will be capable of discerning the difference, so from the perspective of practicality, we don’t need infinity.
- Ok. Back to Peano. We have 0, 1, 2, 3, … and each one of them “eventually” exists. We have the law of induction; which says that if 0 exists; and if whenever n exists, n+1 exists; then all natural numbers exist. Of course “exist” just means mathematical existence.
Induction isn’t the same as deduction and if n exists, you cannot say with certainty that n+1 exists. However, if n+1 exists, you can say with certainty that n exists (deduction).
Do you see my point? In order to define the map that sends n to 2n, I do not need any “a priori” unbounded or infinite sets or collections. All I need is a FINITE string of symbols that represents the operation of a Turing machine (or a Python or Java or Javascript program, same thing) that inputs the number n, and outputs the number 2n. And such a thing exists.
Yes I understand your point, but we still must have a priori understanding that the turing machine never ends. Once again, infinity must be understood before it can be defined with the turing machine.
Huh? Infinite doesn’t mean maximized or maximal.
Yes it does because there are no boundaries. If there are boundaries, then it’s not infinite.
If you claim there are infinite apples and I find a place with no apples, then you don’t have sufficient apples to go around in order to fill every place and therefore the quantity of apples is limited or else there would be apples in every place that apples could exist.
If there could be an apple here, but there is no apple here, then the only reason to explain that is insufficient apples.
If I would claim there are infinite apples available, then yes that would apply. Also with one apple. But to have infinite apples in an infinite universe would not necessarily guarantee any apples within reach.
Like if time is infinite, which it is since beginnings are part of the concept time, then even if you are born now, some of your actions will still have effects for an infinite duration, unless the universe collapses into a singularity which I think is fantasy.
So your deeds would (do) resound in the infinite future, but not in the also infinite past.
Of course it would be easy to negate infinity if you start out with the assumption of a finite universe.
You can just say “the universe is finite, thus it isn’t infinite, thus nothing in it is infinite”.
But who will think this makes sense? Only people who already agreed with you on faith.
You can’t prove an end to the universe, or to a straight line, so to insist it does have one (i.e isn’t infinite) is like being really ambitious without any means.
So basically, an infinitely large basket isn’t going to be filled even with an infinite amount of apples. Because there still is space for an infinite amount of pears.
The whole thing is boringly easily solved when you see that infinite is a predicate and not a proposition.
So “infinity” of itself just means either everything or nothing. But any meaningful statement with infinity in it still gives infinity as a limited part of the proposition, where quality x or y or z is required to make the proposition.
“Does infinity exist” must mean “is existence infinite?”
And if it is, then all things must somehow reach into infinity too. And they do, by their consequences in time.
AH
fuck I understand.
the whole idea of a variable implies infinity.
So the Arabs who came up with algebra (with the contributions of a guy called Al Jabr) and also with 0 destroyed the Euclidean and Parmenidean idea of numbers as elements of a world, and made them magical appearances defying apparent reality yet working very well with the brain.
But this defiance can be because whats defied is what wasn’t proven yet. Maybe they were like hold on, its taken pretty long now that we sought for the end-all, and we didn’t find it so lets just suppose it goes on forever. Then at least we can be free of this supposed end-all and appreciate what we really see.
In my understanding, both 0 and infinite are predicates, and for X=existence they give respectively false and true because otherwise the concepts are contradicted.
If the axiom is “something exists” then for 0 to be true there needs to be a thing that is withdrawn from another.
For infinity to be true there needs only be one thing, it just needs to be infinite. So 0 is a higher order operator than infinity.
Still, infinity is a higher order principle to 1.
2 and uncountability are the same by implication because to get from 1 to 2 you need an assumption, which is that things are separate and not unified, and if you do that there is certainly no way of demonstrating any limits to the number of things that can be listed. And infinity is a higher order function of uncountability.
Only after all this is fixed we get to 0.
Or while its not being fixed but then 0 is seen as the basic depth of the thing where it is actually the summit.
Infinity is the root of all hypothetical numbers, including 0. 1 is the only non hypothetical number because it is the only number which can contain all others.
If the axiom is “something exists” then for 0 to be true there needs to be a thing that is withdrawn from another.
???. Doesn’t parse at my end.
For infinity to be true
Infinity can’t be true or false. It’s not a thing that can be true or false. A set being infinite might be true or false. Precision is critical.
there needs only be one thing, it just needs to be infinite. So 0 is a higher order operator than infinity.
Doesn’t make sense to me.
Still, infinity is a higher order principle to 1.
Order of what? Haven’t seen order defined.
2 and uncountability are the same by implication
Nonsense.
because to get from 1 to 2 you need an assumption, which is that things are separate and not unified, and if you do that there is certainly no way of demonstrating any limits to the number of things that can be listed.
More nonsense. I have one apple, I have two apples. I have no idea what you are talking about. Have you a reference so I can have some clue as to what domain of discourse you’re working in?
And infinity is a higher order function of uncountability.
Doesn’t parse. Says nothing. Word salad.
Only after all this is fixed we get to 0.
After all what is fixed? What’s broken?
Or while its not being fixed but then 0 is seen as the basic depth of the thing where it is actually the summit.
You often make sense. This post of yours does not make any sense.
Infinity is the root of all hypothetical numbers,
What is a hypothetical number? What other kinds of numbers do you have in mind that aren’t hypothetical? The root? Like the root of a polynomial, or a square or cube root? You’re just throwing out random words. This is unlike your usual posts, which are generally connected with reality and sense.
including 0. 1 is the only non hypothetical number because it is the only number which can contain all others.
Bullpucky.
So “infinity” of itself just means either everything or nothing.
Isn’t the collection of even numbers infinite? They’re clearly only a part of something larger. They’re not everything. They don’t include the odd numbers, for example.
I see you wrote several posts, not just one. But you seem to have decided to wake up this morning and post strings of word salad, devoid of meaning or sense. I don’t mean for that to be an attack. Only an observation. I’ve come to expect sensible posts from you. If you only posted nonsense I wouldn’t bother to mention it.
So basically, an infinitely large basket isn’t going to be filled even with an infinite amount of apples. Because there still is space for an infinite amount of pears.
You might enjoy Hilbert’s hotel. Or then again, maybe this will only confuse the issue.
the whole idea of a variable implies infinity.
No. Consider a variable x that ranges over the set {1, 2, 3}. Consider basic finite probability theory. The roll of a single six-sided die. You use variables to stand for things like “I roll a 3,” or “I roll and even number.” There is no implication of infinity.
In my understanding, both 0 and infinite are predicates,
No. What can you possibly mean by that?
and for X=existence they give respectively false and true because otherwise the concepts are contradicted.
Word salad. Makes no sense. I’m disturbed by the fact that I formerly thought you were making some level of sense in your posts, and now I wonder if I missed this strain of illogic. Can you put your morning flood of posts into context? It all seems … well, not good.
lol, yeah this is why you’re not a philosopher.
I managed to keep it extremely simple for you, go along in your little baby steps, doing a bit of theatrics, that was when you thought I was making sense.
I should take offence at your radical laziness but I know mathematicians hold this for some sort of virtue.
Ive been ahead of you constantly, drawn your proud definitions and drawings out of you by pretending I didn’t understand so well, telling you the difficulties along the way. This is all because I don’t think inside of language but just use language where it is constructive.
Now Ill leave you to your graceful temper.
No. The idiotic thing in your approach is the assumption that these issues have been satisfactorily figured out, even though in your discourse your examples all point to the opposite.
The only thing that was ever to learn here is Russells type theory which validates him as a philosopher, and elevates him above the ballroom of mathematicians.
I think set theory stands refuted at this juncture.
If philosophy is to be concerned in any case.
Which I think should gradually become an issue for mathematics.
When a philosopher takes a couple of steps running the mathematicians all stop walking, instead of trying to catch up. It is unfortunate.
in baby steps:
Infinity is something that is a condition.
That means it applies only to things that have already been defined.
Do you get this?
This is where you will have to put on your thinking cap.
Infinity has a habit of eternally popping up in debates, so I figured I’d put together a thread that is easily referenced upon such occurrence that will dissuade folks from religiously promulgating the concept of infinity as an explanation for the unexplainable.
First, what is it?
infinite
[in-fuh-nit][b]adjective
- immeasurably great.
- indefinitely or exceedingly great.
- unlimited or unmeasurable in extent of space, duration of time, etc.
- unbounded or unlimited; boundless; endless.
- Mathematics: not finite. (of a set) having elements that can be put into one-to-one correspondence with a subset that is not the given set.
Origin of infinite
1350–1400; Middle English < Latin infīnītus boundless.[/b]According to definition #1, there is a sense in which the infinite can describe merely what is not measurable, so in that light, finite amounts can be so large that they are not measurable, yet are still finite. This is not the sense that I intend to deal with when talking about infinity. Finite numbers that are so large that they couldn’t be represented on the entirety of the observable universe, even if written on the planck scale, I’m defining those as “dark numbers” because they are finite, but unrepresentable (dark/unseen) within the universe.
Hopefully we can all agree that a good working definition of infinity is “boundless”, “without bounds or constraints: either physical or conceptual”.
Now, can the boundless exist? Well, what does it mean to exist? This dot exists —> . because there is something that is not-dot providing contrast and context (the white background), so existence is the relationship of the dot to the not-dot because if either are missing, then there is no dot and no dot could be said to exist. Existence is therefore dependent upon relationship and relationship precedes concepts of existence.
Now, what if the dot had no boundary? Well, immediately we can surmise that it would have no contrast because if the dot had no boundary, there would be nothing that is not-dot to provide the contrast in order to underpin existence. And if there were something to provide contrast, then obviously the dot would have a boundary. So right off the bat we can say infinity isn’t anything that can exist, but I’m just getting started.
Infinities are said to contain things, because they contain infinite things, but how can a container contain anything with no walls (boundaries)?
Infinities cannot have beginnings or ends because those are boundaries and we said in the beginning that infinity has no boundaries. We cannot divide infinity in half and say infinity is bounded by this finite location and extends to infinity in that direction; it’s nonsense and breaks our definition of infinity being boundless. Zero is not a boundary, but is just an arbitrary starting point on an infinite number line extending in both directions and we could just as easily started at -2,-1,0,1,2,3,etc or 5,6,7,8,etc. A line that is not infinite is a segment because all lines are defined to be infinite within the construct of mathematics; therefore a line with a beginning (such as a timeline) is not an example of infinity. Further, if time had a beginning, infinite time could not be said to exist until forever arrived, and forever means never because forever can never be realized, so infinite time could never exist if time had a beginning.
A better conceptualization for eternity is absence of time instead of infinite amounts of it, but really they both mean the same thing since in both cases time would have no relevance.
I disagree here. Eternity can be taken to be implicit in the concept of time, given that this concept includes all beginnings and ends.
“Time began” is a problematic idea.
Eternity is an infinity of moments, the idea of an unfolding dimension made subjective, tied to a reference frame.
Absent reference frames there is not eternity but chaos.
Since infinity has no starting point/reference point/unique edge, then infinite computer memory would equate to having no memory and anything written to memory could never be found again. Where would allocation start? Afterall, the memory stick would take every bit of space in the entire universe because to say it wouldn’t would be to limit the size of it. Where would an origin/center be placed and how could it be found again?
The infinite is the ubiquitous, omnipresence.
Ah! This is where I really disagree.
First of all, “The” infinite is not the same as Infinity.
Infinity is a predicate to a given, it is not a starting point. If you presume it and ask if it exists, thats the wrong order.
You must observe things and then ask if they are infinite to ground it in reason.
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.
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.
If I would claim there are infinite apples available, then yes that would apply.
X apples available would be theoretical and not actual. You either have in possession/existence the number of apples or you don’t.
But to have infinite apples in an infinite universe would not necessarily guarantee any apples within reach.
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.
Like if time is infinite, which it is since beginnings are part of the concept time,
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.