Serendipper wrote:You should submit a better definition in that case.

You did not appreciate my point. You said that "we all" agree that infinity is boundless. But as someone who's studied university math, I've seen so many examples of bounded, infinite sets that the idea that "infinity is bounded" is obviously false.

Therefor it is false that "we all agree." That was my point. So even if you disagree with me regarding infinity, that very disagreement makes my point! Because my point is that we do NOT "all agree."

That said, reading through your post it seems that you are arguing from a finitist or ultra-finitist perspective. Given that, it's perfectly sensible for you to deny infinite sets. In so doing you must also abandon a rigorous construction of the real numbers, which means you're going to lose the foundation of modern physical science along with most of modern math.

Now it's perfectly consistent to do this, and I have no objections to your making that choice. But it seems to me somewhat nihilistic, since it forces you to reject the whole of modern science along with physics, which turns out to be founded on infinitary and nonconstructive math.

So as I say I have no problem with the logical consistency of your point of view, though it does seem to limit the conversation. If you say, "Let's talk about infinity" when in fact you reject infinity, further dialog seems pointless.

All that said, you did mention a number of interesting and widely-believed fallacies and misunderstandings, which I will endeavor to correct for anyone who may be interested.

Serendipper wrote:Below, you're proceeding to talk about what you haven't defined. What do you mean when you say there are "infinite" numbers between 1 and 0 if infinity is not boundless?

As I say, if you deny that there are infinitely many real numbers between 0 and 1, that is your choice. I assume you must deny there are infinitely many rational numbers between 0 and 1 as well. This is your right, to adopt a finitist or ultra-finitist stance. It just makes conversation pointless.

When you were in high school and they showed you the real number line in Algebra I when you were 14 years old or so, did you complain that there can only be finitely many points between 0 and 1? I confess I don't understand this point of view.

And of course it is manifestly obvious that the unit interval is bounded. No member is ever less than 0 nor greater than 1. The unit interval is an infinite, bounded set.

But really, if you want to use your time on God's earth to post to an Internet forum that you think there are only finitely many points on the real number line between 0 and 1, more power to you. I won't stand in your way.

Serendipper wrote: The first step in an argument is to define terms clearly before using the terms to construct an argument, otherwise you're saying a+b+c=z but not saying what the variables are. Defining terms commits you to what you're saying.

Mathematically, a set is infinite if it may be bijected to a proper subset of itself. That was one of your dictionary definitions if I recall. Galileo noted this in the 1600's and various non-Western mathematicians noted it in the 1200's or earlier.

Serendipper wrote:Probably why 1+2+3+4+5+.... forever = -1/12

Oh my. You are a victim of some very unfortunate misinformation floating about the Internet.

The infinite series 1 + 2 + 3 + 4 + ... of course diverges. Or we can say that it converges to +infinity in the extended real numbers. Or we can say that it converges to Aleph-null as a cardinal, or omega as an ordinal. These are all different ways of saying that this infinite sum "converges to a point at infinity."

So, what is this -1/12 business about? Briefly, everyone knows that the sum of the positive integers is infinity, or undefined, whichever formalism you're using at the moment. But there's a thing called "Zeta function regularization" that says that something called zeta(-1) is -1/12. And you can choose to INFORMALLY interpret zeta(-1) as the series 1 + 2 + 3 + ... But it is NOT the same thing as that series. I'll give you some links that explain all this.

https://en.wikipedia.org/wiki/1_%2B_2_% ... larization

https://blogs.scientificamerican.com/ro ... equal-112/

Why some ignorant Youtubers decided to start an online campaign to confuse people I have no idea. But that's modern life. The Internet, which was supposed to make us all smart, has in fact made us all much more stupid. And this -1/12 business is a classic example.

Serendipper wrote:infinity doesn't exist. If it did, the sum of all natural numbers would certainly equal it.

Oh my. As I said, it's perfectly ok for you to reject completed infinity That's finitism. You can even reject potential infinity. That's ultra-finitism. Those positions are logically consistent (though there's no known axiomatic basis for ultra-finitism). But they're nihilistic, in the sense that you have just kicked the rug out from not only math, but physics as well. I'd say that's an intellectual challenge for your position, to rebuild physics on finitistic or even constructive principles. There actually are people doing the latter.

But now it seems you are drawing your finitistic conclusion based on a misunderstanding of bad Youtube videos. Every mathematician knows that 1 + 2 + 3 + ... diverges to infinity. The interpretation of that sum as -1/12 is a distortion and abuse of a very sophisticated bit of mathematics in higher complex number theory. Somebody made a misleading video and a horde of people just ran with the wrong idea. It's awful frankly. But if that's your evidence for rejecting infinity, your argument is refuted. Please read the links I gave you above to put the -1/12 business into its proper context.

Serendipper wrote:The sum of all squares = 0

The sum of all cubes = 1/120

https://www.youtube.com/watch?v=w-I6XTVZXww

https://www.youtube.com/watch?v=0Oazb7IWzbA

More nonsense on the same lines I imagine. The explanation of -1/12 in terms of analytic continuation and zeta function regularization is all over the Internet. For you to choose the completely wrong interpretation and then expound on your thesis of infinity based on it is truly ... well, it's your right. But I question your motivation. After all, the correct information is just as easy to find as the falsehoods.

Serendipper wrote:You're making divisions where there are none, or did you forget that you cut it? You're creating differentiations on the fly and will need an infinite amount of time to finish, so you're assuming infinity in your proof of it. This was spelled out nicely in the article I posted http://theorangeduck.com/page/infinity-doesnt-exist

Here you're again arguing that the unit interval's not an infinite set. That's not a serious intellectual position.

Serendipper wrote:Keep Adding One

The obvious argument for infinity existing is that, given some number, it is always possible to add one to get a new number. Therefore there must be an infinite number of numbers.

The problem with this argument is that it presupposes infinity exists already. More specifically it assumes that a process can be repeated an infinite number of times. If you can't repeat a process infinitely, and there isn't infinite time, it isn't possible to continue adding one forever.

If you reject taking successors, that makes you an ultra-finitist. Fine with me. You not only reject the real numbers but the counting numbers as well. Whatever. So what happens to physics in your theory? You can't do modern physics without the real numbers and the modern theory of infinite series.

Serendipper wrote:Likewise, you're just adding new numbers between 0 and 1 and you're reliant upon infinite time to complete the process, but if you stop to inspect along the way, the answer will always be finite.

The real numbers are a mathematical abstraction. They don't require time or space or energy or any physical resource to construct.

I would agree with you that PHYSICAL infinity may well not exist. I don't believe it does. But we are talking about MATHEMATICAL infinity. Mathematical infinity does not require any physical resources to construct.

Serendipper wrote:That is why 1-1+1-1+1-1... forever = 1/2 because if you stop, you get either 0 or 1.

It's tragic that people can get so many bad ideas from the Internet, yet won't take the time to learn any actual math.

Serendipper wrote:No, you've defined it to be so. You're not observing, but conjuring.

Did you have a bad math teacher when you were young? Where is this coming from?

Serendipper wrote:You've defined the universe to be a fraction 1/x where x >1, so you've just moved the goal posts from simply counting forever to counting forever and plugging into a function... it's the same problem of using infinity to prove infinity. You need an infinite amount of x to prove there are infinite amounts between 1 and 0. Why bother? Just prove there are infinite x >1.

That, I'm afraid, is incoherent.

Serendipper wrote:You can only cut something so many times before the knife becomes larger than the thing you're cutting and that fact will become self-evident once you begin and persist in cutting.

The only way to "have" infinity is to stop the process, but if you stop the process, the result will always be finite.

How did you get these ideas? I'm curious how someone goes down this path of rejecting all of modern math and physical science.