Actual Infinity in Reality?

[SteveII]

I bring in set theory because you have postulated an infinite collection of rooms. That collection of rooms *is a set*. That's all that sets are: collections of objects. In particular, the collection of rooms in the HH is an infinite set. Because of this, already adopted the axiom of infinity when you start asking these questions. So, it was not I that begged the question, but you.

And right there is the problem with ALL of your reasoning to date on this subject. An infinite number of rooms is not a set. The rooms are rooms. Distinct objects in their own right. There is no justification to collect the rooms into some mathematical abstract object and treat them as a block with a different set of rules. Sets only exist on paper--in mathematics. 

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}.  [opening sentence in the article: https://en.wikipedia.org/wiki/Set_(mathematics)] emphasis added

[polymath257]

I bring in set theory because you have postulated an infinite collection of rooms. That collection of rooms *is a set*. That's all that sets are: collections of objects. In particular, the collection of rooms in the HH is an infinite set. Because of this, already adopted the axiom of infinity when you start asking these questions. So, it was not I that begged the question, but you.

And right there is the problem with ALL of your reasoning to date on this subject. An infinite number of rooms is not as set. The rooms are rooms. Distinct objects in their own right. There is no justification to collect the rooms into some mathematical abstract object and treat them as a block with a different set of rules. Sets only exist on paper--in mathematics. 

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}.  [opening sentence in the article: https://en.wikipedia.org/wiki/Set_(mathematics)] emphasis added

So we cannot look at the collection of all the rooms? Why not? Whether or not it is 'mathematical', does it make sense to talk about the collection of rooms?

[SteveII]

And right there is the problem with ALL of your reasoning to date on this subject. An infinite number of rooms is not a set. The rooms are rooms. Distinct objects in their own right. There is no justification to collect the rooms into some mathematical abstract object and treat them as a block with a different set of rules. Sets only exist on paper--in mathematics. 

So we cannot look at the collection of all the rooms? Why not? Whether or not it is 'mathematical', does it make sense to talk about the collection of rooms?

Why would we have to add all the rooms to a set? When we are counting and dividing up bricks to build four sides of a house, do we have to put them in a set to discuss them? What principle are you applying that requires it? Also, set theory is a specialized mathematical concept governed by a whole set of axioms. Why would you have to apply a mathematical concept to a simple accounting of rooms in a hotel? We wouldn't for a 100, 1000 or even 10^10 rooms.

[RoadRunner79]

[polymath257]

[RoadRunner79]

[SteveII]

Why would we have to add all the rooms to a set? When we are counting and dividing up bricks to build four sides of a house, do we have to put them in a set to discuss them? What principle are you applying that requires it? Also, set theory is a specialized mathematical concept governed by a whole set of axioms. Why would you have to apply a mathematical concept to a simple accounting of rooms in a hotel? We wouldn't for a 100, 1000 or even 10^10 rooms.

Yes, of course you put them in sets: the collection of bricks for each side. You don't *have* to, but you can then discuss the bricks that will go to each side. That is a collection, i.e, a set. 

Arithmetic is a specialized mathematical subject governed by a whole set of axioms. To justify those axioms requires the use of collections, i.e, sets.

So, when you add, what are you doing? You are taking two collections (or whatever sizes) and putting them together into a single collection and counting the number of things in the new set. That is how addition is *defined*.

The same goes for multiplication: How do you define multiplication? You repeat a set of one size, one copy for each element in a different set. Then we look at how many total elements there are. So to define multiplication, you need some sort of set theory.

But, more, to show those operations are well defined (that they give the same result no matter how you rearrange things) is crucial and also depends on set theory.

So, yes, if you are even attempting to *define* 10^10, you will need the concepts of set theory.

Not only are you correct when you say "you don't *have* to" put the rooms in sets, but why would you? The desk clerk is not pulling out paper and using {G1, G2, G3...} to move the guests around. He is making changes with real rooms and real guests and does not have to resort to creating abstract objects by grouping them together. 

You do not need sets to preform any basic operations like addition, subtraction, multiplication or division. I'll give you a chance to back away from that assertion. 

The only reason you desperately want to hang on to putting those rooms into sets is so you can apply unwarranted external constraints (set theory) to them and dismiss the contradiction and absurdities that would otherwise surface.

[Edwardo Piet]

I answered yes. I see no reason why an infinite number of things can't logically exist. Whether there actually are an infinite number of things, I have no idea at all.

I think they're not actually physical possible so I voted no but there's no reason to say they're not logically possible, yes

[GrandizerII]

However this doesn't address the dichotomy paradox by Zeno. It's not about motion, or answering how we get from position 0 to position 1.

The whole point of Zeno's paradoxes is that they are arguments against motion, continuous or discrete.

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I love, by the way, how Steve has not responded to my last response or two and is just repeating the same old bullshit that has long been debunked. So much for intellectual honesty.

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Since (2) is of the class of axioms that are not self-evident, they are assumptions on which further mathematical equations can be developed (useful in calculus for example). To be clear, this axiom is not reasoned into--it is just assumed as a foundation for the subset of infinite set theory in mathematics. See this earlier post. 

This is CLEARLY a question-begging argument and therefore invalid.

That, or you still haven't grasped (or maybe you don't want to grasp) the exact argument we're making. We're not arguing that infinity is logically possible, therefore it is. We're arguing that you haven't provided any good reductio ad absurdum arguments to disprove the logic of actual infinity. You've failed with all the analogies and paradoxes you've posted about thus far.

And even if it was a question-begging argument, which it isn't, it would still be a valid argument anyway. Circular yes, invalid no.

And about the B-theory of time, William Lane Craig himself said the KCA is predicated on the A-theory of time and that if the B-theory of time is true, then the argument is useless. Well, we know which theory of time is backed by the science, and it's not the A-theory of time.

[GrandizerII]

Yes, of course you put them in sets: the collection of bricks for each side. You don't *have* to, but you can then discuss the bricks that will go to each side. That is a collection, i.e, a set. 

Arithmetic is a specialized mathematical subject governed by a whole set of axioms. To justify those axioms requires the use of collections, i.e, sets.

So, when you add, what are you doing? You are taking two collections (or whatever sizes) and putting them together into a single collection and counting the number of things in the new set. That is how addition is *defined*.

The same goes for multiplication: How do you define multiplication? You repeat a set of one size, one copy for each element in a different set. Then we look at how many total elements there are. So to define multiplication, you need some sort of set theory.

But, more, to show those operations are well defined (that they give the same result no matter how you rearrange things) is crucial and also depends on set theory.

So, yes, if you are even attempting to *define* 10^10, you will need the concepts of set theory.

Not only are you correct when you say "you don't *have* to" put the rooms in sets, but why would you? The desk clerk is not pulling out paper and using {G1, G2, G3...} to move the guests around. He is making changes with real rooms and real guests and does not have to resort to creating abstract objects by grouping them together.

Because no matter how you put it, you are grouping things together in your favorite analogy. And you represent that mathematically with a set, just as you represent mathematically you grabbing 5 sticks from here and then grabbing 5 more sticks from them and counting them with 5 + 5 = 10.