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.)

Time is a construct created how? — frank

That’s a tricky question. If I had to construct time, I’d do it virtually using a computer simulation (think ‘The Sims’ - the game contains space and time but not our space and time and it’s created space and time). That is the basis of the simulation hypothesis (https://en.wikipedia.org/wiki/Simulation_hypothesis).

I think you'd enjoy a book about Aristotle and the philosophy of set theory.

What annoys me is that bringing Zeno up so much makes it seem like there's no standard resolution to most of Zeno's paradoxes, which is just ignoring the calculus.

bringing Zeno up so much makes it seem like there's no standard resolution to most of Zeno's paradoxes, which is just ignoring the calculus. — MindForged

- Calculus resolves Zeno’s paradoxes in a complex way.
- Denying Absolute Infinity (and thus implying discrete time) solves them in a simple way
- it also solves the other paradoxes of infinity (https://en.m.wikipedia.org/wiki/Paradoxes_of_infinity)
- Occams Razor simple solutions are better than complex ones.

There is no natural number with the property that you can keep subtracting one from and never reach zero. Hence actual infinity does not exist.

Calculus doesn't solve Zeno's Paradox.

Calculus resolves Zeno’s paradoxes in a complex way.
- Denying Absolute Infinity (and thus implying discrete time) solves them in a simple way
- it also solves the other paradoxes of infinity (https://en.m.wikipedia.org/wiki/Paradoxes_of_infinity)
- Occams Razor simple solutions are better than complex ones. — Devans99

They are not on par. Occam's razor is to be used when all else is equal. Denying infinite time is not simple, there's really no independent reason to posit time as finitely indivisible, so it's just unnecessary since we can eliminate most so-called paradoxes involving infinity.

There is no natural number with the property that yoacan keep subtracting one from and never reach zero. Hence actual infinity does not exist. — Devans99

That's just a misunderstanding. Infinity is not a member of the set of natural numbers, so of course there's no natural number of which you can indefinitely subtract from without reaching zero. But the cardinality of the set of natural numbers will.never reach zero just by subtracting members from the set. So your conclusion does not follow.

But my point is actual infinity is not a number so Actual Infinity is undefined in mathematics (except in set theory which merely declares that it exists as an axiom).

But the cardinality of the set of natural numbers will.never reach zero just by subtracting members — MindForged

You can’t take one from from Undefined

But my point is actual infinity is not a number so Actual Infinity is undefined in mathematics (except in set theory which merely declares that it exists as an axiom). — Devans99

Again, false. Infinity is not a natural number, but there are many kinds of infinite numbers. Namely, those which are the cardinalities of the innumerable infinite sets. The natural numbers have a set size of aleph-null. Take one member out of that set and it's size is still the infinite number aleph-null. Transfinite cardinals and ordinals are infinite numbers, so you're just wrong.

You can’t take one from from Undefined — Devans99

It's not undefined, it is literally defined.

Look someone made an error in set theory with the axiom of infinity and a lot of complex math has built up due to the complexities introduced by the logic error.

You have given no evidence of that and you have fundamentally misunderstood many aspects of these issues. Transfinite cardinals and ordinals are infinite numbers so an actual infinity is perfectly coherent. Not a single known contradiction can be derived in standardly studied mathematical formalisms containing the Axiom of infinity. And infinite sets are defined, not undefined.

This has been a waste of time IMO. You haven't dealt with the actual definitions and elucidations of these things as done in modern mathematics.

The set of natural numbers is not constructable through any known operation hence it does not exist as a completed set.

A set is not a basket that we fill with members. It's criteria.

The definition of a set from Wikipedia starts with:

‘In mathematics, a set is a collection of distinct objects, considered as an object in its own right.’

A set is an abstract object. It's criteria. Don't think of it in spacial terms.

A set is an abstract object. It's criteria. Don't think of it in spacial terms.

The set of natural numbers is not constructable through any known operation hence it does not exist as a completed set. — Devans99

Sure, if we just ignore standard mathematics you can believe that.

A set is an abstract object. It's criteria. Don't think of it in spacial terms — frank

The selection criteria for the set (for example the set of yellow cars) is different from the actual set (ie a number of distinct yellow cars). The later contains more information for example.

The definition of a set from Wikipedia starts with:

‘In mathematics, a set is a collection of distinct objects, considered as an object in its own right.’ — Devans99

Why don't we look a bit further than the first sentence, yeah?

A set is a well-defined collection of distinct objects. The objects that make up a set

[...]

There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description:

A is the set whose members are the first four positive integers.
B is the set of colors of the French flag.

The second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets

Sets are not like baskets, I don't need to engage in a temporal process in order to "make" a set. If I talk about the set of red things, it's not like I had to go out and get all the red things and put them somewhere, my specification covers them all immediately. And so too with infinite sets.

Discribing the set is not the same as the set itself. The description is incomplete unless all members are iterated.

selection criteria for the set (for example the set of yellow cars) is different from the actual set (ie a number of distinct yellow cars). — Devans99

Membership in a set is like membership in a club. Don't think of it in spacial terms.

Discribing the set is not the same as the set itself. The description is incomplete unless all members are iterated. — Devans99

Describing a set is how you populate it with members, e.g. Set "A"={1,2,3}

Putting non-existent things in a set in no way commits one to their existence (goodbye existential import). The set of Harry Potter characters is only populated by non-existent things. — MindForged

That's false. To put something into a set is to assign it some sort of existence. If Harry Potter characters are non-existent then the set of Harry Potter characters is an empty set. If you assert that the set of Harry Potter characters is not empty then you assert the existence of Harry Potter characters.

It's question begging because no one is using your definition of infinity which is defined in a way so as to preclude being actual, nor does the definition of a set preclude it from being infinite. — MindForged

Yes, as I explained, the definition of "set" precludes the possibility of an infinite set. A set is a collection. It is impossible to collect an infinite number of things Therefore an infinite set is impossible. Some people, like you, just like to deny the obvious. That means that you are in denial.

There's no understanding "the" definition because there is no one definition. — MindForged

Well, so much for your "clear" definition of infinity in mathematics then. You seemed to be so certain of that point. I'm glad you now see that you were wrong about it.

Incorrect. If two things hare a property they share it whether or not I judge them to. Two red objects share the property of being red even if no one exists to recognize such. So to speak of sets having members based on a shared property in no way requires a judgement to make it so. — MindForged

As I said, that something has the property of being red, is a judgement. Whether an object is red or not requires a definition of "red", and a judgement as to whether the thing fulfills the criteria of being red. That definition, and that judgement, are necessary because "having the property of being red" is a relation between the universal "red", and the particular object which is said to be red. Otherwise "red" might be defined in any way, and any object might be red. Or do you think that "red" has determinate meaning without a definition?

You're doing it again. It's not a mechanistic process that occurs over time nor is it necessarily done by an agent. Sets don't exist in the mind. The "set of numbers greater than 500 trillion but smaller than 1 quadrillion" is simply too large to be conceptualized in the mind, but it's obviously a perfectly legitimate set. — MindForged

It seems like you're redefining "set' to suit your purpose. No longer does "set" refer to a collection, it refers to things which are collectible, potentially collected. That's the issue of the thread, things which can be potentially collected together do not make an actual collection. And in the case of infinity, an infinite number cannot even be potentially collected together, because the definition of infinity makes collecting an infinite number impossible. So all you are doing with your "infinite set" is asserting that the impossible is possible. That's nonsense.

That's false. To put something into a set is to assign it some sort of existence. If Harry Potter characters are non-existent then the set of Harry Potter characters is an empty set. If you assert that the set of Harry Potter characters is not empty then you assert the existence of Harry Potter characters. — Metaphysician Undercover

That's just not true, no axiom in set theory entails this nor in classical first order logic. Objects quantified over are not assumed to exist. The set of Harry Potter characters has members, but the members do not exist as real things. To call it an empty is to say that there are Harry Potter characters, in which case the books are entirely gibberish.

Yes, as I explained, the definition of "set" precludes the possibility of an infinite set. A set is a collection. It is impossible to collect an infinite number of things Therefore an infinite set is impossible. Some people, like you, just like to deny the obvious. That means that you are in denial. — Metaphysician Undercover

The "collection" is not created through a temporal process. A collection does not entail finitude. Again, you are literally just proving my point: It's the definition you and only you are using, it is not the definition used in modern mathematics. A set is a well-defined collection, often characterized by sharing some property in common or holding to some specified rule. The set future moments is a perfectly comprehensible set, as is the set of natural numbers. Our condition for what makes certain things members of those sets is in the conditions themselves, (e.g. being after some specified time "t", or being a natural number (zero and greater)).

Well, so much for your "clear" definition of infinity in mathematics then. You seemed to be so certain of that point. I'm glad you now see that you were wrong about it. — Metaphysician Undercover

"My" definition (in actuality, the mathematical definition) of sets are clear and they allow for infinity. My point was there's no one definition to which you (as you did) can appeal to to claim that sets are "by definition" finite collections. You're just not wanting to acknowledge that such is a limitation of one definition of a set which you use, not the one mathematicians use.

As I said, that something has the property of being red, is a judgement. Whether an object is red or not requires a definition of "red", and a judgement as to whether the thing fulfills the criteria of being red. That definition, and that judgement, are necessary because "having the property of being red" is a relation between the universal "red", and the particular object which is said to be red. Otherwise "red" might be defined in any way, and any object might be red. Or do you think that "red" has determinate meaning without a definition? — Metaphysician Undercover

You are confusing determining if an object belongs to a set with whether or not the object does in fact belong to a set. Judgements are made by agents, sharing a property in common has no dependence on people's judgements. What the property "being red" corresponds to has nothing to do with people. Intensionaly defining sets is not about going about and determining what specific objects go in the set, it's simply a way of specifying what criteria is entails being a member of some set. Like if I say "the set of even numbers" the objects which satisfy this can be determined, but it's not like I actually have to ascend up the natural numbers to know which ones will be in the set. It's all in the definition, I already know what makes a number an even number.

It seems like you're redefining "set' to suit your purpose. No longer does "set" refer to a collection, it refers to things which are collectible, potentially collected. That's the issue of the thread, things which can be potentially collected together do not make an actual collection. And in the case of infinity, an infinite number cannot even be potentially collected together, because the definition of infinity makes collecting an infinite number impossible. So all you are doing with your "infinite set" is asserting that the impossible is possible. That's nonsense. — Metaphysician Undercover

You are making up definitions of sets, I'm literally using the standard mathematical definition which in fact captures many of our intuitions about collections and does so without any contradictions. It's not about being potentially collected (whatever that means, sounds like you're again assuming everyone is using your definition). What I am asserting is that sets are well-defined collections which can be defined intensionally or extensionally. The former allows one to easily define infinite sets without any contradictions. It's not asserting the impossible, it literally has (as I've already given) perfectly clear examples which are infinite and which to not result in any contradictions within the standard math formalism. Show the contradiction from the actual mathematical definition of a set or else you're just ignoring mathematics.

- Actual Infinity is larger than any other number
- Actual infinity plus one is larger than actual infinity
- Hence there is no number larger than all other numbers
- Actual Infinity does not exist

https://en.wikipedia.org/wiki/Galileo's_paradox

There are less squares than numbers because not all numbers are squares. Yet each number has a square so the number of numbers and squares must be the same.

He is trying to compare two actually infinite sets, IE comparing two undefined things. A set definition is not complete until all its members are interated.