Actual Infinity in Reality?

[pocaracas]

1. Infinitesimals are no longer used. They were problematic in several ways and were replaced by the epsilon-delta definition of limit, which does not require them.

Damn kids!
Get off my lawn!!

[RoadRunner79]

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Why does this not work when you do the same process that you used to show an infinity, and is also the one in Zeno’s paradox?  You always go to something different and don’t address this in the dichotomy paradox.

Note for everyone else. You may notice, that the math in Zeno’s paradox will never reach it’s goal. If the math gives you trouble, you can see that this same process is being used to show that there is an infinite amount of numbers, that you cannot get to the destination, then you add another infinite, and can then reach the destination. In more ways than one... what is being claimed simply does not add up.

OK, again, part of the problem is the definition of the term 'infinity'. You claim an infinity cannot have an end and that it cannot be bounded. That is simply false. The sequence of 9's above is a wonderful example.

For each one in the sequence, there are an infinite number that are larger. This is true. So it is an infinite collection (whether potential or actual is irrelevant here).

It is also bounded since everything in that sequence is less than 1. Each and every term of that sequence is smaller than 1.

So, we have an infinity that is bounded.

The resolution is easy: we have two different notions of boundedness here. One has to do with where in the e a term is and the other has to do with where in the interval from 0 to 1 that term is. The sequence is unbounded in the first sense and bounded in the second. But they are two different notions of being bounded, so there is no contradiction there.

When you ask if we can 'get through' a sequence, you have to specify exactly what you mean. Strictly speaking there isn't even a process here. There are two collections: the terms of the sequence and the numbers between 0 and 1. Both are infinite. You don't get to 1 by staying in the sequence. But that isn't relevant if you want to get to one from within the numbers between 0 and 1.

But we *do* get to the destination, which means we *do* go through all those points. If you deny that, exactly which point do we not go through? At which point does the division break down?

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You seem to get distracted easily and go off on tangents, so lets keep this simple.
Let's look to Zeno's paradox and systematically work through this. Do you agree or disagree with the following.

The dichotomy paradox (or runners paradox) demonstrates an infinite quantity of numbers between any two points, if one is able to cut in half the distance from current location, and the end.  One can continue in this process ad infinitum with the half as the current point and a new halfway point and always have a new number.  I believe you will agree here, you in fact used this (or something very similar) to show that there is an infinity of real numbers between any two numbers.  Do you agree?

According to the above, in traveling between two points, you will always have another half way point to overtake, prior to reaching the destination point.  That is you will always have another number between your current location and end.  Do you agree?

You can likewise reverse this process and work towards the beginning of the journey.  You will end up with a similar outcome where you can always have another halfway point, between the start and the current halfway point.

Would you agree with these statements in Zeno's paradox?

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That's ridiculous. 0.99999... will NEVER equal 1. Not even potentially

In no world does that make any sense to assert that.

You're objectively wrong.

It's the same fucking number written differently.

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Poor Steve, and just to spoonfeed/confuse him a little more:

1/3 * 3 = 1
0.333333... * 3 = 0.999999...

1/3 = 0.333333...

1/3 * 3 = 0.333333... * 3 = 1

1/3 * 3 = 0.999999... = 1

---

And to put it logically.... there is no possible space between 0.999999... and 1 to make it a separate number. It's no more or less than 1, ergo it's 1.

On an integer line, there is no possible space between 1 and 2,  or for that matter 2 and 3 to make a separate number.   Are these equal as well?

[polymath257]

One of the issues here, I think, is that there are two definitions of the term 'infinite' at play. I think there are good reasons for rejecting one of them and accepting the other.

Definition 1: Something is infinite if it is unbounded.

Definition 2: Something is infinite if it cannot be put into a correspondence with some counting number.

Confusion between these definitions is at least part of the problem here. So, let's discuss the relative merits of the two: The first assumes a *process* that doesn't stop while the second assumes a completed object that doesn't correspond with some counting number.

So, for example, the sequence

1,2,3,4,5,6

is both a process that stops and it can be put into correspondence with a counting number ( 6 ), so it satisfies both definitions.

But there is a problem with definition 1. It is ambiguous. The question of whether something is 'bounded' depends on the order properties of that object: which things are larger and which are smaller than others. if two *different* notions of order give different answers, the definition needs to take that into account.

So, to do this, we need to say that something is unbounded *in a particular order*. So, for example, the list of objects

.9
.99
.999
.9999
.99999
...

is an infinite collection of objects (whether you want to say potential or actual is irrelevant here). Each one of the sequence has another that is 'more'. So, with the *internal* ordering from the sequence itself, it is 'unbounded'. But, in the order provided by *all* numbers, this sequence *is* bounded: every single thing in that sequence is smaller than 1.

So, if we are using definition 1, is this sequence unbounded (because for each term there is one larger), or is it bounded (because every term is smaller than 1)? The definition is ambiguous.

On the other hand, using the *second* definition, this sequence is infinite: there is no counting number that is in correspondence with it. NOTHING about process or order or anything else is required to get this answer. And the answer is unambiguous.

So, this makes the second definition 'infinitely' preferable. The ambiguity of the term 'bounded' means that many situations will give ambiguous answers. Furthermore, an attention to the discussion of Zeno's paradox shows that exactly this ambiguity is active in the argument. The above sequence is 'both' unbounded and 'bounded'. That is regarded as a contradiction, when it is actually just a shift in definitions of 'bounded'. No contradiction follows.

Next, the emphasis on process in the first definition forces finitistic reasoning: it is 'begging the question'. Instead, considering relevant sets to exist and be complete allows deeper and and more informative analysis. So, for example, we can consider the set

N={x: x is a counting number}

This is a set: we can tell exactly when something is a member and exactly when something is not a member. There is no ambiguity in the membership in this collection. Hence, it is reasonable to consider this a *completed* collection. There is no 'process' involved, just unambiguous questions about membership. And, it is clear, for example, that 82793 is a member and 34.95723 is not. Simple.

Now, a lot of hay has been made about the assumption in mathematics called the 'axiom of infinity', where it is assumed that a completed infinity actually exists. Some claim this to be 'begging the question'. Let's analyze this claim.

Suppose that actual infinities are, in fact, contradictory. Then adding this axiom would inevitably lead to contradictions. In other words, there would be statements *that are well formed* where both p and (not p) can be proven. It is a fact that no such statements have ever been produced.

But we can go farther. If there are no contradictions actually produced, we can safely and without fear of contradiction add this axiom to our system!

Why would we want to do so? Well, first of all, because it leads to simpler proofs of results that are awkward to prove otherwise. Second, because it leads to deeper insights. And third, because it leads to a more precise and symmetric system of thought.

First, even basic facts about the counting numbers, which *can* be proved without the assumption of infinite sets have *much* simpler proofs with that assumption. Many times, even the statement of fundamental ideas (such as mathematical induction) are ponderous and forced in a finitistic system, but natural and easy when infinities are assumed.

Next, the assumption of infinities leads to deeper insights into even finite sets. By having an extra layer of analysis opened up, we find that even statements about *finite* sets can be stated easier proved quicker. By having the infinite to work with, the finite comes into clearer view.

And, finally, having the assumption of infinite sets allows a MUCH more symmetric and complete system of thought. Most of mathematics over the last century and a half has been based on the use of infinite sets and the subject has expanded greatly since that point. As Hilbert said: nobody will remove us from the paradise Cantor opened up to us.

Next, the question of applicability of mathematics to the real world.

Mathematics is a *language*. When and if it is applicable to the real world is a matter of observation and testing. it is something to be decided upon by the scientists, not the philosophers (who have been so wrong so often). If infinite space and time is a *useful* for our investigation of the universe (and they are), then those ideas have to be taken seriously, in spite of the thrashing of obsolete philosophies.

---

You seem to get distracted easily and go off on tangents, so lets keep this simple.
Let's look to Zeno's paradox and systematically work through this. Do you agree or disagree with the following.

1. The dichotomy paradox (or runners paradox) demonstrates an infinite quantity of numbers between any two points, if one is able to cut in half the distance from current location, and the end.  One can continue in this process ad infinitum with the half as the current point and a new halfway point and always have a new number.  I believe you will agree here, you in fact used this (or something very similar) to show that there is an infinity of real numbers between any two numbers.  Do you agree?

2. According to the above, in traveling between two points, you will always have another half way point to overtake, prior to reaching the destination point.  That is you will always have another number between your current location and end.  Do you agree?

3. You can likewise reverse this process and work towards the beginning of the journey.  You will end up with a similar outcome where you can always have another halfway point, between the start and the current halfway point.

4. Would you agree with these statements in Zeno's paradox?

---

1. yes, between any two real numbers, there are infinitely many real numbers. I agree.

2. Let's be more precise. Yes, whenever you are *between* 0 and 1, say 0<x<1, there is another point y with x<y<1.

3. Let's be more precise. Whenever 0<x<1, there is a point y with 0<y<x.

4. I agree with both of those as I stated them.

OK, so where is the problem?

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You're objectively wrong.

It's the same fucking number written differently.

---

---

And to put it logically.... there is no possible space between 0.999999... and 1 to make it a separate number. It's no more or less than 1, ergo it's 1.

On an integer line, there is no possible space between 1 and 2,  or for that matter 2 and 3 to make a separate number.   Are these equal as well?

No, the difference between them is 1, which is not zero. In the other example, the distance between is zero.

[RoadRunner79]

---

One of the issues here, I think, is that there are two definitions of the term 'infinite' at play. I think there are good reasons for rejecting one of them and accepting the other.

Definition 1: Something is infinite if it is unbounded.

Definition 2: Something is infinite if it cannot be put into a correspondence with some counting number.

Confusion between these definitions is at least part of the problem here. So, let's discuss the relative merits of the two: The first assumes a *process* that doesn't stop while the second assumes a completed object that doesn't correspond with some counting number.

So, for example, the sequence

1,2,3,4,5,6

is both a process that stops and it can be put into correspondence with a counting number ( 6 ), so it satisfies both definitions.

But there is a problem with definition 1. It is ambiguous. The question of whether something is 'bounded' depends on the order properties of that object: which things are larger and which are smaller than others. if two *different* notions of order give different answers, the definition needs to take that into account.

So, to do this, we need to say that something is unbounded *in a particular order*. So, for example, the list of objects

.9
.99
.999
.9999
.99999
...

is an infinite collection of objects (whether you want to say potential or actual is irrelevant here). Each one of the sequence has another that is 'more'. So, with the *internal* ordering from the sequence itself, it is 'unbounded'. But, in the order provided by *all* numbers, this sequence *is* bounded: every single thing in that sequence is smaller than 1.

So, if we are using definition 1, is this sequence unbounded (because for each term there is one larger), or is it bounded (because every term is smaller than 1)? The definition is ambiguous.

On the other hand, using the *second* definition, this sequence is infinite: there is no counting number that is in correspondence with it. NOTHING about process or order or anything else is required to get this answer. And the answer is unambiguous.

So, this makes the second definition 'infinitely' preferable. The ambiguity of the term 'bounded' means that many situations will give ambiguous answers. Furthermore, an attention to the discussion of Zeno's paradox shows that exactly this ambiguity is active in the argument. The above sequence is 'both' unbounded and 'bounded'. That is regarded as a contradiction, when it is actually just a shift in definitions of 'bounded'. No contradiction follows.

Next, the emphasis on process in the first definition forces finitistic reasoning: it is 'begging the question'. Instead, considering relevant sets to exist and be complete allows deeper and and more informative analysis. So, for example, we can consider the set

N={x: x is a counting number}

This is a set: we can tell exactly when something is a member and exactly when something is not a member. There is no ambiguity in the membership in this collection. Hence, it is reasonable to consider this a *completed* collection. There is no 'process' involved, just unambiguous questions about membership. And, it is clear, for example, that 82793 is a member and 34.95723 is not. Simple.

Now, a lot of hay has been made about the assumption in mathematics called the 'axiom of infinity', where it is assumed that a completed infinity actually exists. Some claim this to be 'begging the question'. Let's analyze this claim.

Suppose that actual infinities are, in fact, contradictory. Then adding this axiom would inevitably lead to contradictions. In other words, there would be statements *that are well formed* where both p and (not p) can be proven. It is a fact that no such statements have ever been produced.

But we can go farther. If there are no contradictions actually produced, we can safely and without fear of contradiction add this axiom to our system!

Why would we want to do so? Well, first of all, because it leads to simpler proofs of results that are awkward to prove otherwise. Second, because it leads to deeper insights. And third, because it leads to a more precise and symmetric system of thought.

First, even basic facts about the counting numbers, which *can* be proved without the assumption of infinite sets have *much* simpler proofs with that assumption. Many times, even the statement of fundamental ideas (such as mathematical induction) are ponderous and forced in a finitistic system, but natural and easy when infinities are assumed.

Next, the assumption of infinities leads to deeper insights into even finite sets. By having an extra layer of analysis opened up, we find that even statements about *finite* sets can be stated easier proved quicker. By having the infinite to work with, the finite comes into clearer view.

And, finally, having the assumption of infinite sets allows a MUCH more symmetric and complete system of thought. Most of mathematics over the last century and a half has been based on the use of infinite sets and the subject has expanded greatly since that point. As Hilbert said: nobody will remove us from the paradise Cantor opened up to us.

Next, the question of applicability of mathematics to the real world.

Mathematics is a *language*. When and if it is applicable to the real world is a matter of observation and testing. it is something to be decided upon by the scientists, not the philosophers (who have been so wrong so often). If infinite space and time is a *useful* for our investigation of the universe (and they are), then those ideas have to be taken seriously, in spite of the thrashing of obsolete philosophies.

---

You seem to get distracted easily and go off on tangents, so lets keep this simple.
Let's look to Zeno's paradox and systematically work through this. Do you agree or disagree with the following.

1. The dichotomy paradox (or runners paradox) demonstrates an infinite quantity of numbers between any two points, if one is able to cut in half the distance from current location, and the end.  One can continue in this process ad infinitum with the half as the current point and a new halfway point and always have a new number.  I believe you will agree here, you in fact used this (or something very similar) to show that there is an infinity of real numbers between any two numbers.  Do you agree?

2. According to the above, in traveling between two points, you will always have another half way point to overtake, prior to reaching the destination point.  That is you will always have another number between your current location and end.  Do you agree?

3. You can likewise reverse this process and work towards the beginning of the journey.  You will end up with a similar outcome where you can always have another halfway point, between the start and the current halfway point.

4. Would you agree with these statements in Zeno's paradox?

---

---

1. yes, between any two real numbers, there are infinitely many real numbers. I agree.

2. Let's be more precise. Yes, whenever you are *between* 0 and 1, say 0<x<1, there is another point y with x<y<1.

3. Let's be more precise. Whenever 0<x<1, there is a point y with 0<y<x.

4. I agree with both of those as I stated them.

OK, so where is the problem?

So as we progress from say 0 to 1, there is always another point Y between X and 1 where X<Y<1.  
If you must pass through each of these points, you will never reach 1, because there is always another point Y which must be met.

Therefore Zeno concluded that if you can never end your journey or on the inverse (which isn't quite as intuitive) never begin your journey,
then motion is impossible.  If you have to complete a endless number of points before you can get to the end (1) then you will never be at the end (1)

Do you agree?   If not, where do you think that the error is?

---

On an integer line, there is no possible space between 1 and 2,  or for that matter 2 and 3 to make a separate number.   Are these equal as well?

No, the difference between them is 1, which is not zero. In the other example, the distance between is zero.

I would agree, if they are the same, then they are equal.... that there is no space between them could be confusing.

[polymath257]

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1. yes, between any two real numbers, there are infinitely many real numbers. I agree.

2. Let's be more precise. Yes, whenever you are *between* 0 and 1, say 0<x<1, there is another point y with x<y<1.

3. Let's be more precise. Whenever 0<x<1, there is a point y with 0<y<x.

4. I agree with both of those as I stated them.

OK, so where is the problem?

So as we progress from say 0 to 1, there is always another point Y between X and 1 where X<Y<1.  
1. If you must pass through each of these points, you will never reach 1, because there is always another point Y which must be met.

Therefore Zeno concluded that if you can never end your journey or on the inverse (which isn't quite as intuitive) never begin your journey,
then motion is impossible.  If you have to complete a endless number of points before you can get to the end (1) then you will never be at the end (1)

Do you agree?   If not, where do you think that the error is?

Please be more precise: exactly what process do we use to go from 0 to 1? Be specific.

But yes, at any point there is another point to go through. And yes, we manage to go through all of them.

The problem is that you are assuming we cannot compete an infinite process. Look at my comments on the definition of 'infinity' above.

[Brian37]

I am not asking if there is a concept in mathematics that deals with infinity or if there exists sets with an infinite number of members (although you might use the concept in a larger argument). I am not asking if we can theoretically divide something an infinite amount of times (although you might use the concept in a larger argument). I am not asking about a potential infinite. 

I am asking about an actual infinite of something concrete (not abstract). Can it logically exist? Why or why not? 

No mention of God either. Philosophy subforum--let's stick with pure metaphysics.

"metaphysics" is NOT a real science. It is simply mental masturbation. 

Quantum physics IS a real science.

Not that you will consider this, but who gives a flying fig anyway? You have a specific pet god you gap fill with and think no other religion tries to square itself with science. Hate to burst your bubble, but Muslims, and Jews and Hindus and even Buddhists also attempt to get science to point to their clubs.

And do not hand me any garbage about not having an agenda in that you are not talking about your God. Maybe not in this thread, but this is just a distraction for you to attempt to sound neutral and objective. In other threads and on other websites for that matter you will try to point to your pet God.

But we can play your game here.

1. Stick to metaphysics = Allah, what, that doesn't make sense to you now?

2. Stick to metaphysics = Yahweh, how about now?

3. Stick to metaphysics = Buddha, how about now?

4. Stick to metaphysics = Hindu creator God Brahma, how about now?


You only start threads like this to try to look neutral and objective. But you are not the only religion, or the first theist we've seen to attempt this fake neutrality. 

I see no conflict between infinite and finite and regardless there does not need to be a god of the gaps either way as a starting point.

I see "infinite" as simply the non cognitive fluctuation between a finite "off" to a finite "on". As long as you are breaking it up the cycle can continue forever between a finite on and a finite off without suffering the problem of infinite regress. But regardless, humans make up god claims as a reflection of their own attributes, every single one claimed in our species history, both polytheist and monotheists.

Humans are merely a temporary blip in our own point of view arrow in our universe.

[polymath257]

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One of the issues here, I think, is that there are two definitions of the term 'infinite' at play. I think there are good reasons for rejecting one of them and accepting the other.

Definition 1: Something is infinite if it is unbounded.

Definition 2: Something is infinite if it cannot be put into a correspondence with some counting number.

Confusion between these definitions is at least part of the problem here. So, let's discuss the relative merits of the two: The first assumes a *process* that doesn't stop while the second assumes a completed object that doesn't correspond with some counting number.

So, for example, the sequence

1,2,3,4,5,6

is both a process that stops and it can be put into correspondence with a counting number ( 6 ), so it satisfies both definitions.

But there is a problem with definition 1. It is ambiguous. The question of whether something is 'bounded' depends on the order properties of that object: which things are larger and which are smaller than others. if two *different* notions of order give different answers, the definition needs to take that into account.

So, to do this, we need to say that something is unbounded *in a particular order*. So, for example, the list of objects

.9
.99
.999
.9999
.99999
...

is an infinite collection of objects (whether you want to say potential or actual is irrelevant here). Each one of the sequence has another that is 'more'. So, with the *internal* ordering from the sequence itself, it is 'unbounded'. But, in the order provided by *all* numbers, this sequence *is* bounded: every single thing in that sequence is smaller than 1.

So, if we are using definition 1, is this sequence unbounded (because for each term there is one larger), or is it bounded (because every term is smaller than 1)? The definition is ambiguous.

On the other hand, using the *second* definition, this sequence is infinite: there is no counting number that is in correspondence with it. NOTHING about process or order or anything else is required to get this answer. And the answer is unambiguous.

So, this makes the second definition 'infinitely' preferable. The ambiguity of the term 'bounded' means that many situations will give ambiguous answers. Furthermore, an attention to the discussion of Zeno's paradox shows that exactly this ambiguity is active in the argument. The above sequence is 'both' unbounded and 'bounded'. That is regarded as a contradiction, when it is actually just a shift in definitions of 'bounded'. No contradiction follows.

Next, the emphasis on process in the first definition forces finitistic reasoning: it is 'begging the question'. Instead, considering relevant sets to exist and be complete allows deeper and and more informative analysis. So, for example, we can consider the set

N={x: x is a counting number}

This is a set: we can tell exactly when something is a member and exactly when something is not a member. There is no ambiguity in the membership in this collection. Hence, it is reasonable to consider this a *completed* collection. There is no 'process' involved, just unambiguous questions about membership. And, it is clear, for example, that 82793 is a member and 34.95723 is not. Simple.

Now, a lot of hay has been made about the assumption in mathematics called the 'axiom of infinity', where it is assumed that a completed infinity actually exists. Some claim this to be 'begging the question'. Let's analyze this claim.

Suppose that actual infinities are, in fact, contradictory. Then adding this axiom would inevitably lead to contradictions. In other words, there would be statements *that are well formed* where both p and (not p) can be proven. It is a fact that no such statements have ever been produced.

But we can go farther. If there are no contradictions actually produced, we can safely and without fear of contradiction add this axiom to our system!

Why would we want to do so? Well, first of all, because it leads to simpler proofs of results that are awkward to prove otherwise. Second, because it leads to deeper insights. And third, because it leads to a more precise and symmetric system of thought.

First, even basic facts about the counting numbers, which *can* be proved without the assumption of infinite sets have *much* simpler proofs with that assumption. Many times, even the statement of fundamental ideas (such as mathematical induction) are ponderous and forced in a finitistic system, but natural and easy when infinities are assumed.

Next, the assumption of infinities leads to deeper insights into even finite sets. By having an extra layer of analysis opened up, we find that even statements about *finite* sets can be stated easier proved quicker. By having the infinite to work with, the finite comes into clearer view.

And, finally, having the assumption of infinite sets allows a MUCH more symmetric and complete system of thought. Most of mathematics over the last century and a half has been based on the use of infinite sets and the subject has expanded greatly since that point. As Hilbert said: nobody will remove us from the paradise Cantor opened up to us.

Next, the question of applicability of mathematics to the real world.

Mathematics is a *language*. When and if it is applicable to the real world is a matter of observation and testing. it is something to be decided upon by the scientists, not the philosophers (who have been so wrong so often). If infinite space and time is a *useful* for our investigation of the universe (and they are), then those ideas have to be taken seriously, in spite of the thrashing of obsolete philosophies.

---

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I would agree, if they are the same, then they are equal.... that there is no space between them could be confusing.

Well, that was clearly the meaning of what was said: zero distance, which implies equality.

[Brian37]

What do you think Stevie?

(this feels like a gotcha attempt, therefore god)

Yep. I cannot begin to count the number of theists who have tried to pull the "I am not mentioning God, since I got on line in 01, only to go "GOTCHA!" .

It is a bait and switch. In my early days of debate, something like this might have stumped me, but years of experience it does not anymore. 

Even if we did agree, which we do not, they're still stuck with "which god".

[RoadRunner79]

So as we progress from say 0 to 1, there is always another point Y between X and 1 where X<Y<1.  
1. If you must pass through each of these points, you will never reach 1, because there is always another point Y which must be met.

Therefore Zeno concluded that if you can never end your journey or on the inverse (which isn't quite as intuitive) never begin your journey,
then motion is impossible.  If you have to complete a endless number of points before you can get to the end (1) then you will never be at the end (1)

Do you agree?   If not, where do you think that the error is?

1.)  Please be more precise: exactly what process do we use to go from 0 to 1? Be specific.

2.)  But yes, at any point there is another point to go through. And yes, we manage to go through all of them.

3.)  The problem is that you are assuming we cannot compete an infinite process. Look at my comments on the definition of 'infinity' above.

1. )  In this case, we are cutting in half the distance between the current position and the end position.  Which you agree, will never result in a number greater or equal to the end positions correct?  In actuality, I believe that one would need to proceed through each point, between 0 and 1 sequentially.  The dichotomy method of half marks allows us an abbreviated way to move forward, to systematically show an infinity and show that we that the end (1) will never be reached.

2.)  If they are without end... how do you go through them all?   That is the question.    (And if you are going to say time again, then how does that help you to get to 1 if X<Y<1.... More or less time, does not help you complete this sequence, it's not a matter of completing it in a timely manner).

3. Using the same method, that you use to show an infinity, also shows that you cannot complete that infinity (hence the concept and the term).   If you disagree with Zeno.... then where?   You agreed to the premises including the math.  Do you think that the conclusion doesn't follow that if X<Y<1  ad infinitum then an infinite number of halfway marks cannot be crossed, therefore the path can never be fully completed?  If so why?

[Brian37]

1.)  Please be more precise: exactly what process do we use to go from 0 to 1? Be specific.

2.)  But yes, at any point there is another point to go through. And yes, we manage to go through all of them.

3.)  The problem is that you are assuming we cannot compete an infinite process. Look at my comments on the definition of 'infinity' above.

1. )  In this case, we are cutting in half the distance between the current position and the end position.  Which you agree, will never result in a number greater or equal to the end positions correct?  In actuality, I believe that one would need to proceed through each point, between 0 and 1 sequentially.  The dichotomy method of half marks allows us an abbreviated way to move forward, to systematically show an infinity and show that we that the end (1) will never be reached.

2.)  If they are without end... how do you go through them all?   That is the question.    (And if you are going to say time again, then how does that help you to get to 1 if X<Y<1.... More or less time, does not help you complete this sequence, it's not a matter of completing it in a timely manner).

3. Using the same method, that you use to show an infinity, also shows that you cannot complete that infinity (hence the concept and the term).   If you disagree with Zeno.... then where?   You agreed to the premises including the math.  Do you think that the conclusion doesn't follow that if X<Y<1  ad infinitum then an infinite number of halfway marks cannot be crossed, therefore the path can never be fully completed?  If so why?

We don't care. Even if we agreed with this argument, which we do not, it still would NOT favor your pet god over any  other. Hawking has already said, and I agree with him, "A God is not required".

1+1= Jesus
1+1= Allah
1+1=Yahweh
1+1= Buddha
1+1= Brahma

You could argue 1+1= Unicorns or Yoda and it would mean the same thing to us.

What would be so frightening to you if you figured out you got sold old mythology? What would be so frightening to you if you found out you really are finite? 

That was then, this is now. It was understandable that BOTH polytheists and monotheists made that crap up because they didn't know any better. Our knowledge of scientific fact has exponentially increased over the past 2,000 years. Just like you don't find it credible to gap fill with Poseidon to claim him to be the cause of Hurricanes.