My mistake, your point is well taken. It should be said (somewhat contradicting what I said before) that even in something as seemingly dry and precise and abstract as mathematics, the actual process of coming up with a mathematical theory may start with somewhat vague, intuitive idea of what it is that you want to see, and then you evaluate your formal construction against that idea. Thus, we have intuitive, pretheoretical, or just pragmatic ideas of what a set should be, what an arithmetic should be, and then we axiomatize those ideas, giving them definite, precise form (and there can be more than one way to do that, some better, some worse, some just offering different possibilities).

so you make a distinction between something you call "Absolute" infinity and any other sort of infinity. I don't know what that difference is, — SophistiCat

https://en.m.wikipedia.org/wiki/Actual_infinity

Please read the definitions of ‘Actual’ and ‘Potentially’ Infinite are very helpful.

- 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.

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.

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.

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 understand that you do not understand what actual infinity is. — Magnus Anderson

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

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 it:

https://en.m.wikipedia.org/wiki/Actual_infinity

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?

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.

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

- The concept of actually infinite is not useful and does not exist in the material world.

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.

- 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?

No I believe the universe has an end: The universe is a material object and material objects have starts and ends.

BTW the universe is a macroscopic object, the question of whether it has starts and ends is a macroscopic question. Quantum Mechanics is a microscopic theory so has no bearing on this particular question.

Well that's a definition of 'actual infinite' that is clear and understandable. One can actually work with a definition like that. I suspect however, that that is not the definition that would be accepted by Aristotelians, given my past experience of what they say about it, and of what it says at the link posted above by Devans. That link doesn't use the word 'bounded' at all and is full of woolly, undefined words: 'given, actual, completed'. I particularly like the middle one - an infinity is actual if it is actual. Ah, I see!

- Potentially infinite is the process of continued and potentially endless iteration (IE a limit).
- Actually Infinite is the result of an unbounded number of iterations; IE NOT DEFINED (IE an infinite set)

It can't be all that obvious, since so many mathematicians and scientists have failed to observe the contradiction, and some of them have been reputed to be quite bright. — andrewk

Honestly, I don't think mathematicians care about contradiction within they're work. What is important is that the prescribed methods work. Mathematicians, and scientists, follow like sheep, the methods taught to them, without questioning the underlying principles, that is there discipline. Without that discipline there would be no such thing as mathematics or science. It's not an issue of how bright they are. Philosophers are wont to question these things, but it takes a major shift in strategy for a philosopher to tell a mathematician what to do.

Do you believe that "infinite" refers to an indefiniteness, and that "set" refers to a definiteness. If so, you should see the contradiction. Do you not think that it is contradictory to say that the same energy moves in the form of a wave, and in the form of a particle, at the same time (wave/particle duality)? Do you not think that the way that classical mathematics treats zero and the negative integers is contradicted by the way that "imaginary numbers" treats these? These, amongst others, are contradictions which are employed by very bright people in their daily practise.

We must all be grateful that this thread has finally come to light, so that the said mathematicians and scientists can be freed from the delusion under which they have been labouring. — andrewk

The problem is that there have developed philosophies such as dialectical materialism, called dialetheism, which support the acceptance of contradiction. So in principle, the use of contradiction is justified. From the angle of philosophy, many philosophers who recognize the existence of such contradictions, instead of trying to root them out, and replace them with acceptable principles, instead produce epistemologies which justify, and allow for the acceptance of contradiction.

Really MU? There's no such thing as a sphere? — tim wood

That's right, "sphere" is conceptual only. Take any object which appears to you to be a sphere, and examine it under a high power microscope and you will see that it really is nota sphere.

Infinite sets very obviously contradictory? How about the set of numbers greater than two? The set of irrational numbers between zero and one? — tim wood

Correct again, such named "sets" cannot really be sets by way of contradiction. Do you agree that a set is a "well defined" collection of objects, and accordingly is an object itself? An object has definite boundaries and cannot be infinite. Objects such as "numbers greater than two", and "irrational numbers numbers between zero and one" are not well defined because the cardinality is unknown. You cannot have a "well-defined" set in which the cardinality is an unknown factor.

The 'actual' and the 'infinite' would appear to be infinitely incompatible.

M

Honestly, I don't think mathematicians care about contradiction within they're work — Metaphysician Undercover

I can say that as someone who has studied dialetheism and paraconsistent logic, and has mentioned such to things to friends of mine doing their grad degree in math (or just taking higher maths courses) that this is flatly untrue. If there are any actual, provable contradictions in standard mathematics, the law of explosion entails every sentence becomes a theorem. This is obviously not a good conclusion to draw in normal mathematics, just look at Russell's Paradox before we had ZF set theory.

Really, there's no evidence any of standard mathematics entails a contradiction, provided you actually use the definitions mathematicians actually use.

https://en.wikipedia.org/wiki/Measure_problem_(cosmology)

Another infinity paradox. In this case cosmologists are plugging in Actual Infinity for the size of the universe into probability and getting nonsense like ‘two headed cows are as likely as one headed cows’

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

Actually no. Cantor's set theory is totally rigorous and logical. It doesn't fall into the paradoxes. And ZF-logic, basically developed in response to the paradoxes, is also sound. It has as an axiom of infinity.

What is sound about the ‘set of all sets does not exist’? It exists as much ‘as the set of Naturals’ yet it does not exist in set theory.

But anyway, neither of the above are fully defined sets. You have to list all the members to fully define a set.

What is sound about the ‘set of all sets does not exist’? It exists as much ‘as the set of Naturals’ yet it does not exist in set theory.

But anyway, neither of the above are fully defined sets. You have to list all the members to fully define a set. — Devans99

What is a non-existent set? If you've defined it, it exists. So that's just the assertion of a contradiction.

You don't have to list out all the members of a set to define it. Seriously, sets are defined intensionally all the time.

Actually no. Cantor's set theory is totally rigorous and logical. It doesn't fall into the paradoxes — ssu

Cantor's set theory did fall into numerous paradoxes because of the naive comprehension scheme. It was, as you said, ZF that avoided them using the separation and foundation axioms.

[Mods please delete this, it double posted]

You don't have to list out all the members of a set to define it. Seriously, sets are defined intensionally all the time. — MindForged

But a set is a list of elements, if you don’t list the elements you are missing out the definition of the set.

When we say ‘the set of bananas’ we are not defining a set, just specifying the selection criteria for the set which is a different thing from the actual set.

For example the actual set of bananas has a cardinality so clearly the actual set definition contains more information than the selection criteria.

But a set is a list of elements, if you don’t list the elements you are missing out the definition of the set. — Devans99

A set is not [merely] a list. A list can contain members of a set. The set of real numbers is unlistable (uncountable), but it's still a set. Listing out the members is only one way to define a set.

When we say ‘the set of bananas’ we are not defining a set, just specifying the selection criteria for the set which is a different thing from the actual set.

For example the actual set of bananas has a cardinality so clearly the actual set definition contains more information than the selection criteria.

The selection criterion is used to define the set. That's what an intensionally defined set is. The whole point of such definitions is that extensionally listing things is often not possible to do when defining something, especially a set. I can't list all the even numbers, but I can intensionally define their set.

https://en.wikipedia.org/wiki/Set_(mathematics)

So you are allowed to define a set:

- intensionally. By specifying selection criteria
Or
- extensionally. By listing each member.

These are two different definitions of the same core concept ‘set’. Using one label ‘set’ for two distinct concepts is bound to lead to confusion.

Intensional definition also allows an incomplete definition of a set such as ‘the set of all bananas’ - that is only a partial discription so the set is UNDEFINED.