Actual Infinity in Reality?

[RoadRunner79]

You're not giving much specific here to work with... I would encourage anyone to look it up for themselves.

Yeah, because all these people participating must be so ignorant compared to you that they don't know what it means, and when to spot it.

I looked it up again.... I don't see a problem with my example or the way that it is being used.   I do see a problem in your assertion, in that the reasoning was not that the properties of the parts are to be applied to the whole as I explained.

[polymath257]

I also am amused that 'thought experiments' are done in the 'real world'. LMAO.

[The Grand Nudger]

Recall, the impetus behind all of this was to rescue a poor argument that's truth status is not necessary to the existence of a god or a requirement of belief. No new ground has been covered in or by this thread compared to that old thread, it serves only as a place to deny every little tidbit of the thread over again on a clean slate. Another thread just like it, inevitably..will be made. If af existed in perpetuity...there would be an infinite number of actual math denial threads on gods behalf.

Talk about pounding sand.

[Amarok]

[SteveII]

1. Again, question begging. By axiom, you assume something exists. That cannot be then used as proof of that thing existing. You did not get to the assumption by logic, therefore you cannot say that it is logical.  


2. What?!? Conflicting answers (Hilbert, Galileo), impossibilities (Ross-Littlewood, Thomson), and obviously false (Zeno) is not just "counter to intuition". Your bar is set really, really low for metaphysical impossibilities. Your reasoning is that we don't assume mathematical non-logical axioms--therefore we can't make sense of the paradoxes. That is clearly question-begging. 


3. I have no idea why you might think that Graham's number has a logical problem. It has none at all. Ironically, there are an infinite amount of numbers that could not be counted to in any age of any universe. The fact that you think this is a point is puzzling.

1. One standard way to show the impossibility of something is a proof by contradiction. If you assume the existence and derive a contradiction, you have established the non-existence. But, in spite of many attempts to show a contradiction in the notion of actual infinities, no such contradiction has ever been found.

It is utterly confusing to me why you can't see that your entire #1 is exactly what I have done with my list of 6. 

2. What conflicting answers? Be specific. There are two notions of size relevant to sets: containment and one-to-one correspondence. They are different ways to describe size and yes, they can give different answers. That isn't a contradiction any more than the fact that volume and mass can give different answers to the question of 'how much?'. All that is required to resolve this 'absurdity' is more precise language.

Your answer to Hillbert's hotel is that "with infinite sets...". I have shown conclusively that any argument that contains the words "infinite set..." is question begging. You have assumed what you are trying to prove. You need to look that up if you are fuzzy on that. 

The impossibilities of Thomson and Ross-Littlewood are not in the notion of infinity, but the fact that the activities required cannot be done because of relativistic effects.

That's nonsense. Relatively has nothing to do with Thomson's two minutes of light switching or Ross-Littlewood's 30 seconds of ball tossing. 

Zeno's paradoxes were *solved* by the introduction of infinities! The infinite divisibility of both space and time nicely solve ALL of the Zeno paradoxes.

Except that the Dichotomy paradox example of moving a distance can never start because you always need to traverse the first fraction of the distance--but that fraction is infinitely small. Yet we reach our goals with quite regularity in the real world. So, it would seem you can and cannot traverse an infinite number of points. A contradiction solved by deciding that infinities do not work the same in division as in multiplication, addition and subtraction. It illustrates another aspect of infinity does not translate well into the real world. 

3. Well, one of your objections to the notion of an actual infinity is that it cannot be counted to (which is, truthfully, irrelevant). Neither can Graham's number. So why do you accept one as a possibility and not the other?

What? Do you really think that Graham's number has the same properties of an actual infinity? Of course it is metaphysically possible to count by 1 to Graham's number.

---

Steve,

Sorry to be late in the conversation.  But I really am puzzled as to why you think ifinities pose a logical contradiction as opposed to being hard to show in reality. 

I think the hotel idea sounds intuitively wrong to you merely because it's mathematical.   Infinite odd and infinite even numbers may feel wrong but isn't.  Think of it without numbers.  Suppose there is an infinite number of smoking rooms and an infinite number of non smoking rooms.  Two infinities of different kinds of things.  Together, both are infinite, yet you have twice as many room choices should you  be ambivalent to smoke.  

If that's still too close, consider infinite cats and infinite dogs .  Both infinities together are infinite pets.  Still infinity but with more choices.  

No logical contradiction.  Just more kinds in the omega.

Start with these. 

1. You cannot get to infinity by successive addition. That means the actual infinite cannot be built--it must already exist. 
2. You get absurdities when you propose an infinite number of actual objects (Hilbert's Hotel).
3. You get contradictions about how many squares and square roots there must be (Galileo's paradox)
4. Is the vase full or empty in the Ross–Littlewood paradox?
5. Is the lamp on or off in the Thomson's lamp paradox?
6. It seems we cannot traverse even a finite distance in Zeno's paradoxes

and then give me one good reason why I should ignore all these paradoxes and absurdities and believe that an actual infinite is possible. 


BTW, you are missing the whole point of Hilbert's Hotel:

Imagine a hotel with a finite number of rooms. All the rooms are full and a new guest walks in and wants a room. The desk clerk says no rooms are available.

Now imagine a hotel that has an infinite number of rooms. All the rooms are filled up so an infinite number of guests. A new guest walks up and wants a room. All the clerk has to to do is to move the guest in room #1 to room #2 and the guest from #2 to #3 and so on so your new guest can have a room #1. You can do this infinite number of times to a hotel that was already full.

Now imagine instead the clerk moves the guest from #1 to #2 and from #2 to #4 and from #3 to #6 (each being moved to a room number twice the original). All the odd number rooms become vacant. You can add an infinite number of new guests to a hotel that was full and end up with it half empty. 

How many people would be in the hotel if the guest in #1 checked out?

If everyone in odd number rooms checks out, how many checked out? How many are left?

Now what if all the guest above room number 3 check out. How many checked out? How many are left?

So from the above we get:
infinity + infinity = infinity
infinity + infinity = infinity/2
infinity - 1 = infinity
infinity / 2 = infinity
infinity - infinity = 3

Conclusion: the idea of an actual infinite is logically absurd.

[polymath257]

1. One standard way to show the impossibility of something is a proof by contradiction. If you assume the existence and derive a contradiction, you have established the non-existence. But, in spite of many attempts to show a contradiction in the notion of actual infinities, no such contradiction has ever been found.

1. It is utterly confusing to me why you can't see that your entire #1 is exactly what I have done with my list of 6. 

2. What conflicting answers? Be specific. There are two notions of size relevant to sets: containment and one-to-one correspondence. They are different ways to describe size and yes, they can give different answers. That isn't a contradiction any more than the fact that volume and mass can give different answers to the question of 'how much?'. All that is required to resolve this 'absurdity' is more precise language.

2. Your answer to Hillbert's hotel is that "with infinite sets...". I have shown conclusively that any argument that contains the words "infinite set..." is question begging. You have assumed what you are trying to prove. You need to look that up if you are fuzzy on that. 

The impossibilities of Thomson and Ross-Littlewood are not in the notion of infinity, but the fact that the activities required cannot be done because of relativistic effects.

3. That's nonsense. Relatively has nothing to do with Thomson's two minutes of light switching or Ross-Littlewood's 30 seconds of ball tossing. 

Zeno's paradoxes were *solved* by the introduction of infinities! The infinite divisibility of both space and time nicely solve ALL of the Zeno paradoxes.

4. Except that the Dichotomy paradox example of moving a distance can never start because you always need to traverse the first fraction of the distance--but that fraction is infinitely small. Yet we reach our goals with quite regularity in the real world. So, it would seem you can and cannot traverse an infinite number of points. A contradiction solved by deciding that infinities do not work the same in division as in multiplication, addition and subtraction. It illustrates another aspect of infinity does not translate well into the real world. 

3. Well, one of your objections to the notion of an actual infinity is that it cannot be counted to (which is, truthfully, irrelevant). Neither can Graham's number. So why do you accept one as a possibility and not the other?

What? Do you really think that Graham's number has the same properties of an actual infinity? Of course it is metaphysically possible to count by 1 to Graham's number.

1. And I am pointing out that you failed to demonstrate a contradiction.

2. YOUR claim was that the Hilbert Hotel leads to absurdities. WHAT are the supposed absurdities? Since we are talking about the HH, of course, we are assuming an actual infinity. But that is how proof via contradiction works: you assume the result you want o show is false, then you arrive at a contradiction. You have not done so.

3. Two switch at more than a certain rate would require the switch to move faster than light. That is impossible. Similarly, in the Ross-Littlewood scenario, we cannot move the balls faster than light, so cannot remove and  replace faster than a certain rate.

Hmmm...it seems to me that you have never given your metaphysical axioms....care to show which assumptions you are working with? Are you assuming that everything must be finite (and hence begging the question)?

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Again, what *specifically* is absurd in the HH? Give details.

[SteveII]

2. Your answer to Hillbert's hotel is that "with infinite sets...". I have shown conclusively that any argument that contains the words "infinite set..." is question begging. You have assumed what you are trying to prove. You need to look that up if you are fuzzy on that. 

1. And I am pointing out that you failed to demonstrate a contradiction.

2. YOUR claim was that the Hilbert Hotel leads to absurdities. WHAT are the supposed absurdities? Since we are talking about the HH, of course, we are assuming an actual infinity. But that is how proof via contradiction works: you assume the result you want o show is false, then you arrive at a contradiction. You have not done so.

3. Two switch at more than a certain rate would require the switch to move faster than light. That is impossible. Similarly, in the Ross-Littlewood scenario, we cannot move the balls faster than light, so cannot remove and  replace faster than a certain rate.

Hmmm...it seems to me that you have never given your metaphysical axioms....care to show which assumptions you are working with? Are you assuming that everything must be finite (and hence begging the question)?

---

Again, what *specifically* is absurd in the HH? Give details.

Okay, without appealing to infinite set theory (because that would be question begging) tell me why this all makes sense:

Imagine a hotel with a finite number of rooms. All the rooms are full and a new guest walks in and wants a room. The desk clerk says no rooms are available.

Now imagine a hotel that has an infinite number of rooms. All the rooms are filled up so an infinite number of guests. A new guest walks up and wants a room. All the clerk has to to do is to move the guest in room #1 to room #2 and the guest from #2 to #3 and so on so your new guest can have a room #1. You can do this infinite number of times to a hotel that was already full.

Now imagine instead the clerk moves the guest from #1 to #2 and from #2 to #4 and from #3 to #6 (each being moved to a room number twice the original). All the odd number rooms become vacant. You can add an infinite number of new guests to a hotel that was full and end up with it half empty. 

How many people would be in the hotel if the guest in #1 checked out?

If everyone in odd number rooms checks out, how many checked out? How many are left?

Now what if all the guest above room number 3 check out. How many checked out? How many are left?

So from the above we get:
infinity + infinity = infinity
infinity + infinity = infinity/2
infinity - 1 = infinity
infinity / 2 = infinity
infinity - infinity = 3

Conclusion: the idea of an actual infinite is logically absurd.

Regarding metaphysical axioms for this discussion, just the basics. Existence, consciousness, the Law of Identity, LNC, and LEM. Notice these axioms are self-evident and can not fail to be true and therefore are not in the same class as the math-specific Axiom of Infinity.

[polymath257]

1. And I am pointing out that you failed to demonstrate a contradiction.

2. YOUR claim was that the Hilbert Hotel leads to absurdities. WHAT are the supposed absurdities? Since we are talking about the HH, of course, we are assuming an actual infinity. But that is how proof via contradiction works: you assume the result you want o show is false, then you arrive at a contradiction. You have not done so.

3. Two switch at more than a certain rate would require the switch to move faster than light. That is impossible. Similarly, in the Ross-Littlewood scenario, we cannot move the balls faster than light, so cannot remove and  replace faster than a certain rate.

Hmmm...it seems to me that you have never given your metaphysical axioms....care to show which assumptions you are working with? Are you assuming that everything must be finite (and hence begging the question)?

---

Again, what *specifically* is absurd in the HH? Give details.

Okay, without appealing to infinite set theory (because that would be question begging) tell me why this all makes sense:

Imagine a hotel with a finite number of rooms. All the rooms are full and a new guest walks in and wants a room. The desk clerk says no rooms are available.

Now imagine a hotel that has an infinite number of rooms. All the rooms are filled up so an infinite number of guests. A new guest walks up and wants a room. All the clerk has to to do is to move the guest in room #1 to room #2 and the guest from #2 to #3 and so on so your new guest can have a room #1. You can do this infinite number of times to a hotel that was already full.

Now imagine instead the clerk moves the guest from #1 to #2 and from #2 to #4 and from #3 to #6 (each being moved to a room number twice the original). All the odd number rooms become vacant.

1. You can add an infinite number of new guests to a hotel that was full and end up with it half empty. 

2. How many people would be in the hotel if the guest in #1 checked out?

3. If everyone in odd number rooms checks out, how many checked out? How many are left?

4. Now what if all the guest above room number 3 check out. How many checked out? How many are left?

So from the above we get:
5. infinity + infinity = infinity
6. infinity + infinity = infinity/2
7. infinity - 1 = infinity
8. infinity / 2 = infinity
9. infinity - infinity = 3

10. Conclusion: the idea of an actual infinite is logically absurd.

Regarding metaphysical axioms for this discussion, just the basics. Existence, consciousness, the Law of Identity, LNC, and LEM. Notice these axioms are self-evident and can not fail to be true and therefore are not in the same class as the math-specific Axiom of Infinity.

I want to point out that even asking these questions assumes the set theory you reject. When you talk about the infinite number of rooms of the HH, you are using the methods of set theory.

1. Not quite. The notion of 'half' of the rooms is not well defined in this context. There is a notion of density for subsets of the natural numbers, and that density is 1/2, but that is a different thing *and relies on the set theory you object to*. The notion of 'half' does not apply here unless you define what it means in this context. And *that* requires the set theory you reject.

2. An infinite number. More specifically, *in the set theory your reject* the answer is aleph_0.

3. The answer to both is 'an infinite number'. More specifically, *in the set theory you reject*, the answers are both aleph_0.

4. The answer is 3. We are using set theoretic differences here, though, *which you reject*.

5. This is correct, although you haven't defined what addition is for this context and cannot do so without the set theory you reject. But, yes, the union of two disjoint infinite sets is infinite *in the set theory you reject*.

6. As described above, the notion of infinity/2 is not well defined, although *with the set theory you reject* it can be given a meaning and the result is again aleph_0.

7. Again, your demonstration is that when you remove an element from an infinite set (which is a process you reject), you get another infinite set of the same cardinality. That is true.

8. As explained above, infinity/2 is meaningless unless you define your operations. But yes, if you take an infinite set and divide it into two sets of the same cardinalities, then both pieces will have the same cardinality as the original.

9. And once again, you are attempting to generalize from the case of finite sets to that of infinite sets. The notion of subtraction of cardinalities is not well defined. All this means in this context is that if you take an infinite collection and remove an infinite collection, the answer will depend on *how* you remove that infinite collection. That isn't a contradiction.

No, what you have shown is that the operations of division and subtraction of infinites are either not well defined (they depend on specifics of the situation) or they give results you see as paradoxical, even though they are consistent. But guess what? That happens in ordinary arithmetic. What is 3*0? What is 5*0? So is 0/0=3 or is 0/0=5? That we cannot divide by 0 doesn't mean it is contradictory to use the number 0.

So, here is a question: why do you expect the operations of subtraction and division would work the same way for infinite things as for finite things? What is the physical basis for this expectation?

---

Let me ask this: do you think all those things make sense *within* the context of an infinite set theory?

---

Regarding metaphysical axioms for this discussion, just the basics. Existence, consciousness, the Law of Identity, LNC, and LEM. Notice these axioms are self-evident and can not fail to be true and therefore are not in the same class as the math-specific Axiom of Infinity.

 Consciousness is basic for logic? Really?

OK, what are LNC and LEM? Those are abbreviations I am not familiar with.

I would notice that you used the operations of addition, subtraction, and division. Those are not on your list, are they?

[GrandizerII]

You have been shown several times there are no conflicting answers. Different instances of infinite set are going to yield different results. This is logical, not contradictory.

Same infinite collection - same infinite collection is still 0 (empty collection), and always will be.

It's when you subtract one infinite collection from a different infinite collection that you get other [varying] answers, depending on these collections. It's loosely similar to finite (7) - finity (4) = finity (3) => finity - finity = 3???

And what about 0/0? The answer could be any number, and when we don't know exactly which due to lack of contextual contraints, the answer is that it's indeterminate. Same with infinity - infinity.

For every bit of your answers above, you have assumed the Axiom of Infinity. This axiom was not derived from a logical process. It is simply assumed so particular math problems can be conducted on it. It is not proof of anything or gives guidance to anything in the real world (where the thought experiments are conducted).

Irrelevant. You said it leads to conflicting answers. It doesn't. What you need to do is acknowledge what I said about the maths and address that specifically, if you want to eventually show that it's not metaphysically possible that there be an actual infinity.

And it hasn't been shown to be non-logical (in the conventional sense of the word). Your attempt to disprove it has been an utter failure.

So, you have to deal with the items of my list by showing why these six things do not indicate an actual infinity is metaphysically impossible WITHOUT using infinite set theory from mathematics. I have shown that if you use mathematical infinite set theory in your reasoning if an actual infinite can exist, you have begged the question. That is an invalid argument.

If you can't show that an actual infinity is not logically possible, then you haven't shown that it's not metaphysically possible. Using infinite sets helps to show that you have failed to point out where any logical contradiction regarding actual infinities lies.

1. You cannot get to infinity by successive addition.

Irrelevant. If the elements are all there, then they are there. The set is complete.

2. You get absurdities when you propose an infinite number of actual objects (Hilbert's Hotel).

Stop it, Steve. You've been corrected so many times here it's getting tedious at this point. Veridical paradoxes do not imply actual absurdities. The absurdities you're seeing is all in your mind, due to a failure to understand that assuming infinite sets must lead to different expectations from those that we expect to arise from assuming finite sets, and it has to do with the unbounded end(s) of the infinite set, which your intuition currently isn't well-equipped to deal with.

3. You get contradictions about how many squares and square roots there must be (Galileo's paradox)
4. Is the vase full or empty in the Ross–Littlewood paradox?
5. Is the lamp on or off in the Thomson's lamp paradox?

Read polymath's repeated responses to these. It's getting tiresome of you to keep repeating stuff which has been debunked in this thread over and over again.

6. It seems we cannot traverse even a finite distance in Zeno's paradoxes

There are several potential solutions to this. In addition to what polymath has already argued, another potential solution is that this local universe is quantized in terms of time or space or both (with the infinity in terms of size and divisibility being a property of the wider cosmos).

---

Yeah, because all these people participating must be so ignorant compared to you that they don't know what it means, and when to spot it.

I looked it up again.... I don't see a problem with my example or the way that it is being used.   I do see a problem in your assertion, in that the reasoning was not that the properties of the parts are to be applied to the whole as I explained.

Stop acting dense. He doesn't know that the universe, like the physical things in it, is limited. He inferred it from what he sees/intuits about the physical things in the universe.

---

BTW, you are missing the whole point of Hilbert's Hotel:

Imagine a hotel with a finite number of rooms. All the rooms are full and a new guest walks in and wants a room. The desk clerk says no rooms are available.

Now imagine a hotel that has an infinite number of rooms. All the rooms are filled up so an infinite number of guests. A new guest walks up and wants a room. All the clerk has to to do is to move the guest in room #1 to room #2 and the guest from #2 to #3 and so on so your new guest can have a room #1. You can do this infinite number of times to a hotel that was already full.

Now imagine instead the clerk moves the guest from #1 to #2 and from #2 to #4 and from #3 to #6 (each being moved to a room number twice the original). All the odd number rooms become vacant. You can add an infinite number of new guests to a hotel that was full and end up with it half empty. 

How many people would be in the hotel if the guest in #1 checked out?

If everyone in odd number rooms checks out, how many checked out? How many are left?

Now what if all the guest above room number 3 check out. How many checked out? How many are left?

So from the above we get:
infinity + infinity = infinity
infinity + infinity = infinity/2
infinity - 1 = infinity
infinity / 2 = infinity
infinity - infinity = 3

Conclusion: the idea of an actual infinite is logically absurd.

Yep, this reponse to Jenny clearly shows that he never did read my responses to the maths. Otherwise, he wouldn't repeat the same old "infinity - infinity = different answer" crap. Steve isn't interested in learning something new if he sees it'll contradict everything he's believed about God.

[SteveII]

Okay, without appealing to infinite set theory (because that would be question begging) tell me why this all makes sense:

Imagine a hotel with a finite number of rooms. All the rooms are full and a new guest walks in and wants a room. The desk clerk says no rooms are available.

Now imagine a hotel that has an infinite number of rooms. All the rooms are filled up so an infinite number of guests. A new guest walks up and wants a room. All the clerk has to to do is to move the guest in room #1 to room #2 and the guest from #2 to #3 and so on so your new guest can have a room #1. You can do this infinite number of times to a hotel that was already full.

Now imagine instead the clerk moves the guest from #1 to #2 and from #2 to #4 and from #3 to #6 (each being moved to a room number twice the original). All the odd number rooms become vacant.

1. You can add an infinite number of new guests to a hotel that was full and end up with it half empty. 

2. How many people would be in the hotel if the guest in #1 checked out?

3. If everyone in odd number rooms checks out, how many checked out? How many are left?

4. Now what if all the guest above room number 3 check out. How many checked out? How many are left?

So from the above we get:
5. infinity + infinity = infinity
6. infinity + infinity = infinity/2
7. infinity - 1 = infinity
8. infinity / 2 = infinity
9. infinity - infinity = 3

10. Conclusion: the idea of an actual infinite is logically absurd.

Regarding metaphysical axioms for this discussion, just the basics. Existence, consciousness, the Law of Identity, LNC, and LEM. Notice these axioms are self-evident and can not fail to be true and therefore are not in the same class as the math-specific Axiom of Infinity.

I want to point out that even asking these questions assumes the set theory you reject. When you talk about the infinite number of rooms of the HH, you are using the methods of set theory.

That's ridiculous. HH posits an infinite amount of real objects and then illustrate how it behaves absurdly when you attempt to make changes. Clear proof that an actual infinity contains absurdities that cannot be reconciled without applying a series of assumptions to make sense of it. It is in trying to make sense of it that YOU keep bringing in set theory. 

1. Not quite. The notion of 'half' of the rooms is not well defined in this context. There is a notion of density for subsets of the natural numbers, and that density is 1/2, but that is a different thing *and relies on the set theory you object to*. The notion of 'half' does not apply here unless you define what it means in this context. And *that* requires the set theory you reject.

2. An infinite number. More specifically, *in the set theory your reject* the answer is aleph_0.

3. The answer to both is 'an infinite number'. More specifically, *in the set theory you reject*, the answers are both aleph_0.

4. The answer is 3. We are using set theoretic differences here, though, *which you reject*.

5. This is correct, although you haven't defined what addition is for this context and cannot do so without the set theory you reject. But, yes, the union of two disjoint infinite sets is infinite *in the set theory you reject*.

6. As described above, the notion of infinity/2 is not well defined, although *with the set theory you reject* it can be given a meaning and the result is again aleph_0.

7. Again, your demonstration is that when you remove an element from an infinite set (which is a process you reject), you get another infinite set of the same cardinality. That is true.

8. As explained above, infinity/2 is meaningless unless you define your operations. But yes, if you take an infinite set and divide it into two sets of the same cardinalities, then both pieces will have the same cardinality as the original.

9. And once again, you are attempting to generalize from the case of finite sets to that of infinite sets. The notion of subtraction of cardinalities is not well defined. All this means in this context is that if you take an infinite collection and remove an infinite collection, the answer will depend on *how* you remove that infinite collection. That isn't a contradiction.

I think you literally invoked set theory in every reply. So again you use a mathematical assumption to explain why we get 5 absurd results when moving people around. That is question begging and is insufficient as an explanation let along a defense of an actual infinity. 

No, what you have shown is that the operations of division and subtraction of infinites are either not well defined (they depend on specifics of the situation) or they give results you see as paradoxical, even though they are consistent. But guess what? That happens in ordinary arithmetic. What is 3*0? What is 5*0? So is 0/0=3 or is 0/0=5? That we cannot divide by 0 doesn't mean it is contradictory to use the number 0.

No, what HH shows is that the concept of infinity is not equipped to explain real objects. The conclusions are true, yet they are absurd. Regarding "0", you cannot parody HH with 0 and show anything remotely similar. As such, the comparison you are attempting makes no progress toward "normalizing" the absurd conclusions. 

So, here is a question: why do you expect the operations of subtraction and division would work the same way for infinite things as for finite things? What is the physical basis for this expectation?

For the very obvious reason that you need to be able to subtract and divide real objections in the real world. If you can't do that with infinite quantities, you cannot have an actual infinity of real objects. This is not rocket science. For like the ninth time, don't you find it interesting that you cannot find an article defending an actual infinity of real objects to post for us to prove your point? 

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Let me ask this: do you think all those things make sense *within* the context of an infinite set theory?

Real objects MUST be able to be divided and subtracted because they are tangible things and you HAVE TO be able to do these simple operations with tangible things. 

---

Regarding metaphysical axioms for this discussion, just the basics. Existence, consciousness, the Law of Identity, LNC, and LEM. Notice these axioms are self-evident and can not fail to be true and therefore are not in the same class as the math-specific Axiom of Infinity.

 Consciousness is basic for logic? Really?

OK, what are LNC and LEM? Those are abbreviations I am not familiar with.

I would notice that you used the operations of addition, subtraction, and division. Those are not on your list, are they?

Yes a consciousness is an awareness and ability to perceive reality and therefore necessary in any discussion of reality. 

Law of Non-contradiction and the Law of the Excluded Middle. 

Operations of addition, subtraction and division are results of real objects being moved about. They are not axiomatic. I don't have to assume any additional truths before conducting the operations.