So what about an "infinite set"? Well, to begin with you should say precisely what the term means.
Okay if you don't, at least someone should. Putting an adjective in front of a noun does not in itself make a mathematical concept.
Cantor declared that an "infinite set" is a set which is not finite. Surely that is unsatisfactory, as Cantor no doubt suspected himself. It's like declaring that an "all-seeing Leprechaun" is a Leprechaun which can see everything. Or an unstoppable mouse is a mouse which cannot be stopped. These grammatical constructions do not create concepts, except perhaps in literary or poetic sense. It is not clear that there are any sets that are not finite, just as it's not clear that there are any Leprechauns which can see everything, or that there are mice which cannot be stopped.
Certainly in science there is no reason to suppose that "infinite sets" exist. Are there and infinite number of quarks or electrons in the universe? If physicists had to hazard a guess, I am confident that the majority would say: No. But even if there were an infinite number of electrons it's unreasonable to suppose that you can get an infinite number of them all together as a single data object
So what about an "infinite set"? Well, to begin with you should say precisely what the term means
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>a thread died for this
>neither I, nor my friends understand the concept, therefore the concept doesn't exist
>pooinloojuan
>enlightened
Have you guys read this essay by Zeilberger? Entitled “Real” Analysis is a Degenerate Case of Discrete Analysis
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Assume no infinite set exists
Integers are therefore finite
Integers are well ordered
Therefore you should be able to tell us what the maximum integer is, no?
how does the final question follow?
>Cantor declared that an "infinite set" is a set which is not finite. Surely that is unsatisfactory
Why? That's often how words that are grammatically the opposite of another word are defined. An irreducible polynomial is one that's not reducible, a connected set is one that's not disconnected, etc.
>It is not clear that there are any sets that are not finite, just as it's not clear that there are any Leprechauns which can see everything, or that there are mice which cannot be stopped.
But I can show you an infinite set whose existence doesn't contradict any of the commonly accepted axioms of mathematics. Watch this: [math]\mathbb{N}[/math]
>Certainly in science there is no reason to suppose that "infinite sets" exist.
When are infinite sets used in science?
Q.E.D fuckers.
But in all seriousness, how is it that the concept of infinity is so hard to grasp for some people? Math deals with abstract constructs, it's illogical to try to bring it to the real world context to, somehow, prove that it doesn't make sense. Numbers are the best way to understand the concept of infinity, don't try to look at it in "real world glasses".
ebin
The maximum integer exists: it is the largest number ever conceptualized or ever to be conceptualized, either directly or indirectly. That does not mean that I can tell you what it is because I don't know and, in fact, it is impossible to know.
zeilberger has gay opinions
>it's not clear that there are any Leprechauns which can see everything
what did he mean by this?
Define conceptualize?
Does that mean that whenever I think of a number larger than anyone has thought of before, the natural numbers grow larger?
Whatever number you think, no matter how big, you can always sum "1" and get a bigger number. I suggest you take a look at the principle of induction, maybe it helps you grasp it a little bit better.
(the largest number ever conceptualized or ever to be conceptualized, either directly or indirectly)+1 = ?????????
Y'all niggers need von Neumann.
Define 'define.'
The subset of natural numbers which have presently been conceptualized grows. But time is finite, too, so if you prefer to think of the natural numbers in terms of all time you can do that too.
You "can" but you won't. There will always be a largest number and there will be a largest number of all time.
>That does not mean that I can tell you what it is because I don't know and, in fact, it is impossible to know
So your finitist approach only works if you wave your hands and say "It's just, like, really big I guess."
>Define 'define'
I asked a very reasonable question. Are you going to answer it or are you just trolling?
Godel already proved that there are such things as unprovables. There's no hand-waving involved. DO you know the largest number ever imagined? Do you think that it doesn't exist? I mean the numbers are well-ordered aren't they? So either you're suggesting human thought is infinite or you content that there is a largest number ever imagined. Do you believe that you can know what this number is? How the hell could you know?
And I can google a very reasonable answer. You're just trying to divert. It's not insightful.
>conceptualize - form a concept or idea of (something).
What do you mean "I won't"? I can't really figure out what you're trying to say.
I think user is asking for a formal definition, in the mathematical sense of the word. You wouldn't want to give a formal definition of the natural numbers in terms of some very vague informal notion, after all.
>The maximum integer exists: it is the largest number ever conceptualized or ever to be conceptualized, either directly or indirectly. That does not mean that I can tell you what it is because I don't know and, in fact, it is impossible to know.
So did you just give an incomputable definition of the natural numbers, to avoid problems with incomputable operations on real numbers?
So every time someone thinks of a bigger number the naturals grow larger? But we can't know what other people are 'conceptualizing' so the natural numbers could just be growing and growing as we speak? What problem do you think this outlandish approach actually solves?
That's because you can't understand a finite world. You don't understand the concept of discreteness
In you're theoretical infinite world of circular logic you could say "if I knew the largest number I can imagine a bigger number by taking that number + 1", but that's like saying "if 1 = 2, then 5 = 7." It's a vacuous truth that has no bearing on reality.
The idea that there is a largest number is pointless and not well defined.
Jewish infinity
vs stronk aryan potential infinity
largest number is 1488. Praise jeebus. Justice to liberty.
I already answered this question. What problems do assuming the world is actually infinite solve? It's just different interpretations. Both discrete and continuous mathematics have their uses, but that doesn't mean the world is either discrete or continuous any more than the fact that hammers were invented means the universe is made of nails.
It's extremely well-defined: it's the number that exists that is larger than all other numbers that exist. This is not only well-defined in a finite world, but it is easily possible to be well-defined in a continuous world. I agree that it's pointless. I'm not the one who first brought it up.
Except 1 = 2 would directly contradict any standard construction of the natural numbers. You don't need any extra axioms to show there isn't a largest natural number. Finitists insist on claiming there is a largest natural number, yet can't tell us what it is or how to find it, and even seem to struggle with what exactly the implications of its existence are.
I just don't see the point of adding an extra axiom, that's not even clearly defined, if we can't justify it without appeals to nature or finitist hand-waving.
If we take the example of the natural numbers, we don't assume they are finite. The fact that they aren't follows from the axioms we use to construct them. You would have to add an axiom if you wanted to claim they are finite.
1) It takes far more axioms to form the concept of infinity than the concept of finite sets. If it's a question of Occam's razor I think I know which theory is in danger of getting excised.
2) What does any of this have to do with axioms? Do you need a mathematical axiom to tell you that you can't know what number someone from 1921 thought of?
I really don't care what number someone from 1921 thought of. I don't think it's relevant at all because I don't insist on imposing finitist assumptions on mathematics.
It's actually a product of mathematical induction, which already assumes infinity. There are more axioms you're using than the algebraic ones.
But are you insisting on imposing infinite ones? I mean if you're not taking a stance on anything then why even contribute?
I definitely do insist that you can always find a larger number (eg by adding 1 to whatever you thought was the largest number). I don't believe we should only define addition up to an arbitrary number.
That is now the largest integer.
That's not what he proved you dimwitted fuck. Stop trying to half ass quote shit you don't understand and have never read as common knowledge.
OK, then I challenge you to find a number larger than any number that has ever existed. You insist that it's possible, so do it.
It's just the contrapositive of the direct statement of his theorem. "No consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers" => "All consistent systems have truths that are unable to be proved about the arithmetic of the natural numbers."
>He doesn't understand what "ever conceptualized or ever to be conceptualized" means
Whatever the current largest number is plus 1.
>A set that can only be constructed by using the axiom of infinity is finite because I reject the axiom of infinity
>It is more useful to define this set as "all the numbers below the biggest number someone has thought of (for now)" because numbers that people haven't thought of don't exist.
>This is somehow reasonable because (???...???) Godel's incompleteness theorem
Well, at least Tychonoff's theorem is trivial and doesn't imply AC in this universe.
What is his endgame?