What is the standard definition of number ? — Wittgenstein

I can't give you a precise verbal description of the meaning of the word "number". What I can do is I can show you that people define the word "number" in such a way that it encompasses quantities that are not strictly finite, quantities such as Pi.

In a way, yes. ‘Ought’ is a sign that one rejects/resents the experience of being hungry in favour of a world without hunger. — Possibility

That's true. I refuse to die and prefer to live. But is that resentment? Most importantly, is that something negative? Consider the alternative, which is accepting reality as it is. What happens? I die.

Too much attention payed to the nature of words, not enough to the nature of the world. — fdrake

The world is full of obscurantists.

Even if you consider the "conceptual existence " we cannot construct infinity even with all the symbols and operations in a system. — Wittgenstein

The concept of infinity exists. That's what I meant by conceptual existence.

If you regard infinity as number, that implies that it is finite, since all numbers are finite. — Wittgenstein

That's according to your own definition of the word "number". You defined the word "number" to mean "a finite quantity". That's not the standard definition.

I am hungry, I ought to eat. According to you, that's a sign of resentment, a rejection of part of the universe in favor of other parts, or more bizarrely, in favor of a world that doesn't and couldn't exist.

However, the statement, " an infinite number of elements " is a contradiction in terms as infinity is clearly not a number but sets have definite number of elements and on other hand infinity is not definite.How can we justify the existence of infinite sets. — Wittgenstein

Depends on how you define the word "number". If you define it narrowly, to mean "a natural number", then you're right, infinity is not a number. But that's not how most people define it. Likewise, if you define the word "set" narrowly, to mean "a finite collection of elements", then you're right, there are no infinite sets. But again, that's not how most people define it.

Infinity is a number and infinite sets are sets and that's all there is to it.

How can we justify the existence of infinite sets. — Wittgenstein

Make sure you don't confuse conceptual existence with empirical existence. The existence of the concept of infinite set is one thing and the existence of infinite collections of physical objects out there in the world is another. The former kind of existence is clearly real, the latter can be disputed.

At the top of every hierarchy of goals there is a goal that is chosen freely in the sense that it is not chosen in order to attain some other goal. The choice of such a goal is certainly regulated by external factors (by the so-called nature) but it is not regulated by internal factors (such as your other goals.)

So if you don't want to die in a month, you better eat something. And if you want to eat something, you better think of ways to find food. Say you decide you want to buy some rice (I don't like rice but that's what came up first.) So if you want to buy some rice, you better find a shop that sells it. And so on and so forth. That's an example of a hierarchy of goals. At the top of that hierarchy is a goal -- to be alive in a month. You chose that goal independently from any other goal. You don't want to be alive in a month in order to attain some other goal . . . you just want to be alive in a month. It's an arbitrary choice mediated only by external factors.

No other goal is telling you it's best to be alive in a month. You might as well just choose to not be alive in a month. Most people don't because they can't -- the need to remain alive is too strong.

And what if they had the ability to choose to die in a month? What would happen? Well, they would all die, and with them, the drive for death. Such people can continue to exist only if they are created or birthed by people who do not think like them or if they give birth to other people before they die (but then again, one must ask, what is the probability that such a strategy will be successful given the environment we live in?)

I'm just rejecting the argument that an abstracted infinity implies there are real-world infinities. — Relativist

Well, just because we can imagine something, it does not mean it exists. Just because we can come up with a symbol such as "unicorn" does not mean there is a portion of reality it can represent. But I thought that your argument is that we need to count an infinite number of things in order for there to be an infinite number of things, or at the very least, in order for us to prove or justify that an infinite number of things exists. I don't think any of these two beliefs is true. We don't need to observe every human being dying in order to prove or justify our belief that all human beings are mortal.

If a point has no length it does not exist so the definition is contradictory. — Devans99

So if a symbol has no property called length assigned to it, it follows that there is no portion of reality that can be represented by it?

The word "point" is just that -- a word. It is a symbol we use to represent reality. And that symbol has no property called length and that's simply because we didn't define one. If you want, you can do so. But that won't change the fact that it does not follow that just because some symbol has no property called length assigned to it that there is no portion of reality it can represent.

A "sphere" (or "ideal sphere") is an abstraction, not an actually existing thing. — Relativist

We use abstractions, i.e. symbols, in order to represent reality. For example, the term "human being" is a symbol -- a written or a spoken word -- that can be used to represent certain portions of reality. We don't say human beings don't exist merely because the term "human being" is an abstraction. We only say that human beings don't exist if there is no portion of reality that can be represented by the term "human being".

You bring up another abstraction: the number of possible paths being infinite. This is hypothetical; in the real world, you cannot actually trace an infinite number of paths. So in the real world you cannot actually COLLECT an infinity. All you can do is to conceptualize. — Relativist

You don't need to be able to count an infinite number of things in order for that infinite quantity of things to exist. Things exist whether or not we are conscious/aware of them. Similarly, our beliefs are true or false regardless of whether we can justify them. Just because we cannot make an infinite number of observations does not mean we cannot come up with a theory that we can use to make an infinite number of predictions each one of which is true.

Calculus has problems too. For example the infinite series 1/2^n

1 + 1/2 + 1/4 + 1/8 ... = 2

Logically it’s incorrect to write =2 should be ~2. It’s only a small error but the sum of that series is always less than 2. — Devans99

You can say that's a somewhat careless use of the equality sign. The clean way to do it would be something like lim(1 + 1/2 + 1/4 + 1/8 + ...) = 2.

On the other hand, if you think that 1 + 1/2 + 1/4 + 1/8 + ... ~ 2 that means you are fine with infinite numbers.

Forget about human faces. Let's take a simpler concept -- the concept of natural number. Is the concept of natural number undefined? If not, does that mean the set of all natural numbers is a finite one and that you can show it to me? If so, will you please show it to me? Or can you just answer this simple question: what is the largest natural number?

- The concept of a human face can be defined
- The ‘set of all human faces’ is a finite list so in principle is also definable
- the description ,set of all human faces’ is not a complete definition of the set — Devans99

The set of all human faces is a finite one? Can you show it to me?

We are comparing two undefined things and we get nonsense. — Devans99

Is the concept of human face undefined? If not, how do you define it?

Apparently, the concept of human face is undefined, merely because noone ever listed every instance of the class represented by the word "human face".

- The concept of potential infinity is useful as an approximation of the very large and small. Potential Infinity exists in the material world. — Devans99

Are you saying the universe has no temporal end?

Wasn't it recognised several pages earlier that those insisting that there is a clear distinction between the terms 'Actual Infinity' and 'Potential Infinity' are Aristotelians, while the rest are not? Is there any hope of ever coming to a common understanding between Aristotelians and non-Aristotelians, given the fundamentals of their worldviews are so completely different? — andrewk

A quantity is said to be actually infinite if it is temporally bounded from both sides i.e. if it occurs between two points in time. Quantities that only have an upper temporal bound, such as the concept of infinite past, can also be included in the definition. Potential infinities, on the other hand, have no upper temporal bound; in this sense, they are never-ending, lasting forever.

There's hardly anything contradictory or otherwise non-sensical about these concepts. They may not refer to anything in real life but there is no way in hell you can say they are contradictory.

What we have here (which also applies to Zeno's paradoxes) is the classic case of not being able to understand that what we're aware of is only a small portion, a small subset, of what is "out there", and that just because we can never be directly aware of an infinite quantity using our finite consciousness does not mean that the concept of infinity is meaningless.

My initial points were that infinity isn't inherently off the table when talking about reality, as the OP and another user were arguing that infinity is a contradictory concept (which is just flatly untrue); so if anything in reality is infinite or not is an empirical matter, there's no strictly logical argument against it being instantiated. — MindForged

Yes, @Devans99 is not merely arguing that infinity does not exist in reality, he's also arguing that the concept of infinity is meaningless, non-sensical, undefined, contradictory, etc.

Give me one example of the Actually Infinite from the material world. — Devans99

You need to understand what actual infinity is before I can do that.

I understand that maths has tried to build a consistent logical structure around the logical fallacy of the Actually Infinite and has failed. The numerous paradoxes attest to that. — Devans99

I understand that you do not understand what actual infinity is.

I don't see how an instantiated infinity could ever be established empirically since we can't count to infinity. — Relativist

We need to agree that:

1. Things exist whether or not we are aware of them
2. We can make an infinite number of predictions and retrodictions based on a finite number of observations

Just because our awareness is finite, which means that we can never directly observe an infinite quantity, does not mean that infinite quantities do not exist. And just because we can never directly observe an infinite quantity does not mean we can't observe it indirectly via some finite number of observations.

- So I have infinity X and a copy X’.
- I add one to X
- then X > X’ by common sense — Devans99

I think you should spend less time trying to prove other people wrong and more time trying to understand what they are saying.

- Actual Infinity is larger than any other number — Devans99

Only in the sense that infinity is larger than any finite number. Otherwise, it is not true.

- Actual infinity plus one is larger than actual infinity

Only in the sense that some specific infinite number plus one is larger than that specific infinite number.

For example, the set of even natural numbers S = {2, 4, 6, ...} plus 1 equals {2, 4, 6, ..., 1}. Clearly, these two sets aren't equal. The resulting set has a greater number of elements than the set S.

- Hence there is no number larger than all other numbers

Which does not follow. An infinite number is a number that is larger than any finite number. An infinite number is not larger than any infinite number.

I am convinced this thread has never been about empirical matters. Instead, it's entirely about conceptual matters (which is why OP talks about Zeno so much.)

That would mean a physical system passing through an infinite number of states in a finite period of time. — Devans99

What's wrong with that?

Humans can't perform an infinite number of operations because human consciousness is finite. But that does not mean that it is impossible for an infinite number of events to take place within a bounded period of time.

You cannot arrive at infinity through a finite number of operations. If you want to arrive at infinity, you need to perform an infinite number of operations.

Yes, you will get a different number, that's true, but that's not the point. Noone is saying you'll get the same number. More than Two things + 1 = More than Two Things does not mean you will get the same number. Infinity + 1 = Infinity does not mean you will get the same number. It means you will get a number that belongs to the same class of numbers. When you have an infinite set of elements and you add another element to it, you'll get a different set of elements, one that is larger, but one that is nonetheless infinite.

It's nonsensical to you, because you don't understand it. When I say More than Two things + 1 = More than Two Things what I am saying is that if you take a number greater than two and add one to it you will get a number that is greater than two.

Yes, when you have two apples and you add another apple, you get three apples. You don't get three oranges or three bananas. But you nonetheless get something that belongs to the same class that three oranges and three bananas belong to. And that class is the class represented by the word "three". In the same way, when you add 1 to a number that belongs to the class of numbers greater than two, you get a number that belongs to the class of numbers greater than two. You can hardly deny this. So yes, More than Two Things + 1 = More than Two Things.

So if you have a number greater than 2 and you add 1 to it you don't get a number greater than 2?

Interesting.

2nd proof that Actual Infinity does not exist:

- Actual infinity plus one equals Actual infinity
- but X+1 <> X for all X
- So Actual Infinity is absurd — Devans99

Fabulous argument.

Here's a variant of it:

- More than Two Things plus one equals More than Two Things
- but X + 1 <> X for all X
- So More than Two Things is absurd

My point is that you do not understand what the word means. If you know what actual infinity means, there is no way in hell you can take Zeno's "paradoxes" seriously.

The Actually Infinite has no place in the material world. — Devans99

You have to understand that actual infinity is just a concept, nothing more than that, and that you have to understand it.

I don't think that you understand it.

Not quite. You can have an infinite number of one and the same thing. Or you can have an infinite number of completely different things. Or you can have a mix of the two. The concept of infinite time is not defined in such a way that it means that everything that can happen will happen an infinite number of times. That's simply NOT how the concept of infinite time is defined.

For example infinite time implies anything [that] can happen will happen an infinite number of times which is absurd. — Devans99

The concept of infinite time does not imply that things will repeat an infinite number of times. You can have an infinite number of completely different moments.

You are making things complicated. Zeno’s paradoxes disappear if we assume time is discrete for example (IE then Archiles only has to cover a finite number of steps to catch the tortoise).

Don’t you get it, logical contradictions like Zeno’s, Hilberts Hotel etc... exist because we have an absurdity (Actual Infinity) at the core of our reasoning — Devans99

Not quite. Zeno's "paradoxes" exist because people do not understand the concept of actual infinity. That's the problem. For even if you accept that time and space are infinitely divisible and that in order to move from point A to point B you must cross every point in between the two points, Zeno's conclusions still do not follow.

You are assuming time is continuous.

- Assume we have a system
- Watch it evolve over a finite time period
- Will we observe it pass though an actually infinite number of states?

My gut feeling is no so time is probably discrete.

You say that you believe time is continuous but you don’t give an argument why.

I say time is discrete because otherwise we get logical contradictions. — Devans99

If time is infinitely divisible, it follows that it is impossible to experience what happens at every single point in time. So yes, you cannot observe a system passing through an actually infinite number of states. But just because you cannot be directly aware of something does not mean that that something does not exist. We are only ever aware of a small subset of what is "out there".

I want to understand why you think that the belief that time is continuous (= infinitely divisible) leads to logical contradictions.

If it is, it’s a potential infinity rather than an actual Infinity (you do understand the distinction?).

The division of space takes time, first we must cut one inch, then 1/2 an inch, then 1/4... No matter how many cuts we make we never get to actual infinity, just some small number. — Devans99

It's not possible for a human being to count an infinite number of things in a finite number of steps.

However, that DOES NOT mean that it is impossible for an infinite number of things to happen between two points in time.