God Must Exist: Infinite Regression is Impossible

[We_Are_VENOM]

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First off, by "universe", I mean all physical reality govern by natural law. This would include universes that we know/don’t know about.

1. If God does not exist, then the universe is past eternal.

Justification: We know that the universe exist, and if there is no transcendent supernatural cause, then either

A. the universe either popped into being, uncaused, out of nothing.
B. OR, it has existed for eternity.

I think we can safely remove posit A from the equation (unless there is someone who thinks it is a plausible explanation).

Let’s focus on posit B.

Based on posit B, we need not provide any naturalistic explanation as to the cause of our universe, considering the fact that the term “universe” applies (as mentioned earlier) to all physical reality, which means that any naturalistic explanation one provides is already accounted for as “eternal”.

And if God does not exist, then physical reality (the universe) is all there is, and thus must be eternal.

2. If the universe is not past eternal, then God exists.

Justification: If the universe (all physical reality) is NOT eternal, then it had a beginning.

Since natural law (mother nature) cannot logically be used to explain the origin of its own domain, then an external, supernatural cause is necessary.

If “nature” had a beginning, one cannot logically use nature to explain the origin of nature, and to do so is fallacious.

So, where nature stops, supernatural begins.

3. The universe is not past eternal.

Justification: If the universe is past eternal, then the causal chain of events (cause and effect) within the universe is infinite. But this is impossible, because infinity cannot be traversed or “reached”.

If the past is eternal, that would mean that there are an infinite amount of “days” which lead to today. But in order for us to have “arrived” to today, an infinite amount of days would have to be traversed (one by one), which is impossible, because infinite cannot be “reached”.

Consider thought analogy..

Sandman analogy: Imagine there is a man who is standing above a bottomless hole. By “bottomless”, of course if one was to fall into the hole, he would fall forever and ever and ever.

Now, imagine the man is surrounded by an infinite amount of sand, which is at his disposal.

Imagine if the man has been shoveling sand into this hole for an infinite amount of time (he never began shoveling, or he never stopped shoveling, he has been shoveling forever).

Imagine if the man’s plan was to shovel sand into the hole until he successfully filled the sand from the bottom, all the way to the top of the hole.

How long will it take him to accomplish this? Will he ever accomplish this task? No. Why? Because the sand is bottomless, so no matter how fast he shoveled, or how long he shoveled, the sand will never reach the top.

So lets put it all together…

The sand falling: Represents time travel, and the trajectory of the sand falling south of the top represents time traveling into the past, which is synonymous with past eternity.

The man shoveling: Represents the “present”, as the man is presently shoveling without halt. This is synonymous with our present causal reality. We are presently in a state of constant change, without halt.

Conclusion: If the sand cannot reach the bottom of the hole (because of no boundary/foundation) and it can’t be filled from the bottom-up to the present (man), then how, if there is no past boundary to precedent days, how could we have possibly reached the present day…if there is/was no beginning foundation (day).

However, lets say a gazillion miles down the hole, there is a foundation…then the hole will be filled in a finite amount of time, and it will be filled from the bottom-up.

But ONLY if there is a foundation.

Likewise, we can only reach today if and ONLY IF there is a beginning point of reference, a foundation in the distant past.

4. Therefore, an Uncaused Cause (UCC) must exist: As explained, infinite regression is impossible, so an uncaused cause is absolutely necessary.

This UCC cannot logically be a product of any precedent cause or conditions, thus, it exists necessarily (supplementing the Modal Ontological Argument).

This UCC cannot logically depend on any external entity for it’s existence (supplementing the Modal Ontological Argument).

This UCC is the foundation for any/everything which began to exist, which included by not limited to all physical reality…but mainly, the universe an everything in it.

This UCC would also have to have free will, which explains why the universe began at X point instead of Y point...and the reason is; it began at that point because that is when the UCC decided it should begin...and only a being with free will can decide to do anything.

This UCC would have to have the power to create from nothing (as there was no preexisting physical matter to create from, before it was created).

So, based on the truth value of the argument, what can we conclude of the UCC?

1. It is a supernatural, metaphysically necessary being
2. A being of whom has existed for eternity and can never cease existing
3. A being with the greatest power imaginable (being able to create from nothing)
4. A being with free will, thus, a being with a mind

This being in question is what theists have traditionally recognized as God. God exists.

In closing, I predict the whole "well, based on your argument, God cannot be infinite".

My response to that for now is; first admit the validity of the presented argument, and THEN we will discuss why the objection raised doesn't apply to God.

[William]

William: What makes you think I am wanting you to do anything?

I'm probably projecting. When I critique a view, in a forum, it's for the purpose of discussion, especially if I keep responding to a particular person's posts. I'm sorry for that assumption on my part.

That is quite alright. I have provided response and argument in return but as usual, we find no common ground in which to pitch a tent and light a campfire...

If anything, that is what I would like from anyone. Fortunately I have it on a daily basis with the Message Generating Process so it isn't lack of company...I guess I just like to hear from you now and again and remind you of where I am at.

[Bust Nak]

Are you saying that, as a limit, it is a speed? If so, then it seems what we are each saying is clear enough.

Yes, I think we are.

That article does not say that one should bet that way, at least that I saw.

That's fine, the main point is still that an actual infinite universe is coherient.

Instilling a term with a concept that it doesn’t have within the argument being discussed in order to get around the conclusion is the fallacy of equivocation.

Going back to your example:

P1. Man (male) is the only rational animal.
P2. No woman is a man (male).
C. Therefore, no woman is a rational animal.

Me: But it's not the case that male members of humanity is the only rational animal. Man (humanity) is the only rational animal. Am I guilty of instilling the term "man" with a concept (humanity) to get around the conclusion, thus guilty of the fallacy of equivocation?

But, in my argument, it does imply having a boundary. To address that argument, should one wish to logically avoid the conclusion, one needs to address the premise as it is meant.

Discard premise 1. It's not the case that if something is expanding, then it will grow in size (i.e., move beyond its current boundary).

An actual infinite B-theory past is logically possible, as far as I can tell. I still doubt whether it is metaphysically possible, however.

If an actual infinite B-theory past is logically possible, then actual infinity is a coherent concept, which in terms mean infinity as a quantity is coherent. How are you getting out of this conclusion?

But you have admitted that premise 16 isn’t saying anything about an actual infinite. Your premise 17, then, doesn’t logically follow from anything said in 1-15 because none of that is saying anything about an actual infinite.

17 is just rephrasing of 16 though. 16 follows from 1-15, therefore 17 trivially follows.

I guess I’m trying to say that I don’t, as of right now, think it’s logically impossible in as clear a way as a round square or an actual infinite A-theory past is. I still question whether infinity as a quantity is coherent, so I guess I am questioning whether actual infinity is coherent. That a quantity, when subtracted from another quantity can result in different quantities as answers, for instance.

Same as above, how is this consistent with your view that as far as you can tell, an actual infinite B-theory past is logically possible?

I did not state that. I stated that a series being completed is a synonymous phrase to a series having each and every element of that series, as members of that series, being counted through.

Looks like the same thing to me, look at 17) again, each and every single last of {0, ...}'s elements can be counted through. It's not talking about members of a different series, it says each and every element of that series, as members of that series, can be counted through.

Both of those statements are about the entire wall. Your attempt at a proof takes bricks out of the wall, makes statements about the bricks individually, and then jumps to say the same thing is true of the whole wall they were taken out of (or put together to form).

No, my proof takes bricks without property P out of the wall, leaving only bricks with property P in the wall. That's the very opposite of making statements about the bricks taken out, and then jumps to say the same thing is true of the whole - the bricks taken out, does not have the property we are interested in, non is there any putting bricks together to form a wall.

Try this proof.

1) Here is a wall that consist of bricks of all sorts of color.
2) Take all the grey, brown and dull colored bricks out of the wall.
3) All the bricks of the wall are of bright color.
4) The wall is colorful.

Your objections up to this point amounts to the following:
Step 3) is talking about the whole wall - no, it's clearly says bricks of the wall have the property of having bright color.
You are taking bricks out of the wall, make statements about bricks individually, then say the same thing is true of the wall - no, the statement about the bricks taken out is that they are dull color, the very opposite of what I am saying about the wall. Yes I have move from claims about individual bricks to the wall, but they are not the same claims.
The wall is only colorful, if the bricks of the wall, as part of the wall, are of bright color - yes, that's the point, the wall is colorful since the bricks of the wall, as part of the wall, are of bright color.
Steps 3 and 4 might be synonymous. If they are then step 4 is talking about bricks of the wall - okay, then it also follows that in step 3, talking about all the bricks of the wall being colorful is talking about the wall itself being colorful, so what?
If they are not then what does it mean? - well, 3 says all the bricks..., 4 says the wall; one says bright color the other says colorful, so in some sense they are talking about different things? Again so what?

In 18) is “can be completed” a synonymous phrase to “move through all members” in 16)?

I think they are the same thing. I asked you what does completing a series mean, if not moving through all the members? You told me all members have to be treated as a whole. All members of a series sounds like treating them as a whole to me.

If it is a synonymous concept, then it is a concept about the bricks of the wall having property P and, thus, a rephrasing of 16) just like 17) was.

It is also about the whole wall having property Q, it is the synonymous with its bricks having property P. That's why I originally stopped at 16, because it is synonymous with the series being completed.

[The Tanager]

Going back to your example:

P1. Man (male) is the only rational animal.
P2. No woman is a man (male).
C. Therefore, no woman is a rational animal.

Me: But it's not the case that male members of humanity is the only rational animal. Man (humanity) is the only rational animal. Am I guilty of instilling the term "man" with a concept (humanity) to get around the conclusion, thus guilty of the fallacy of equivocation?

No, you are rightly questioning the truth of P1 and providing a replacement premise that is true, which will then lead to a different conclusion. It didn’t seem like you were doing that with my argument. I thought you were saying that there are two sound arguments that use some of the same terms with equally valid (but different) definitions.

Discard premise 1. It's not the case that if something is expanding, then it will grow in size (i.e., move beyond its current boundary).

Why? Because if one set is in one-to-one correspondence with another set, then it has the same size as that set?

If an actual infinite B-theory past is logically possible, then actual infinity is a coherent concept, which in terms mean infinity as a quantity is coherent. How are you getting out of this conclusion?

Yes, you have me questioning if this is even logically possible.

17 is just rephrasing of 16 though. 16 follows from 1-15, therefore 17 trivially follows.

Looks like the same thing to me, look at 17) again, each and every single last of {0, ...}'s elements can be counted through. It's not talking about members of a different series, it says each and every element of that series, as members of that series, can be counted through.

How can it be just a rephrasing when you’ve said 16 is not about the elements as members of the series, but 17 is? You are adding something, not just rephrasing.

Try this proof.

1) Here is a wall that consist of bricks of all sorts of color.
2) Take all the grey, brown and dull colored bricks out of the wall.
3) All the bricks of the wall are of bright color.
4) The wall is colorful.

Your objections up to this point amounts to the following:

Step 3) is talking about the whole wall - no, it's clearly says bricks of the wall have the property of having bright color.
You are taking bricks out of the wall, make statements about bricks individually, then say the same thing is true of the wall - no, the statement about the bricks taken out is that they are dull color, the very opposite of what I am saying about the wall. Yes I have move from claims about individual bricks to the wall, but they are not the same claims.
The wall is only colorful, if the bricks of the wall, as part of the wall, are of bright color - yes, that's the point, the wall is colorful since the bricks of the wall, as part of the wall, are of bright color.
Steps 3 and 4 might be synonymous. If they are then step 4 is talking about bricks of the wall - okay, then it also follows that in step 3, talking about all the bricks of the wall being colorful is talking about the wall itself being colorful, so what?
If they are not then what does it mean? - well, 3 says all the bricks..., 4 says the wall; one says bright color the other says colorful, so in some sense they are talking about different things? Again so what?

No, my objections do not amount to those things. Parts-to-whole does carry over properties in some cases, but not others. For instance:

1’) Here is a wall that consists of bricks of all sorts of lengths.
2’) Take all the bricks that aren’t 4 inches long out of the wall.
3’) All the bricks of the wall are 4 inches long.
4’) The wall is 4 inches long.

I haven’t given this point enough study and thought, but what jumps out from our two examples is the difference between quality and quantity when moving from parts to the whole. The quality transfers, but the quantity doesn’t. Regardless, you’ve still got work to do.

It is also about the whole wall having property Q, it is the synonymous with its bricks having property P. That's why I originally stopped at 16, because it is synonymous with the series being completed.

When I questioned 16 as saying something about the whole wall, you said that it’s only talking about the bricks, though.

[The Tanager]

That is quite alright. I have provided response and argument in return but as usual, we find no common ground in which to pitch a tent and light a campfire...

If anything, that is what I would like from anyone. Fortunately I have it on a daily basis with the Message Generating Process so it isn't lack of company...I guess I just like to hear from you now and again and remind you of where I am at.

I enjoy discussing our views together. Sometimes, we think we are talking about the same thing, but then end up talking past each other. I’m always open to talking about our views.

[Bust Nak]

No, you are rightly questioning the truth of P1 and providing a replacement premise that is true, which will then lead to a different conclusion. It didn’t seem like you were doing that with my argument. I thought you were saying that there are two sound arguments that use some of the same terms with equally valid (but different) definitions.

Perhaps you mean this: I hold that NASA and mathematicians both have sound arguments re: infinity, that uses same words along the lines of "size," or "boundary."

Why? Because if one set is in one-to-one correspondence with another set, then it has the same size as that set?

That's one reason, another reason (or another implication of the same reason) is something that doesn't have a boundary still have a size.

Yes, you have me questioning if this is even logically possible.

It's having the opposite effect to what I intended.

How can it be just a rephrasing when you’ve said 16 is not about the elements as members of the series, but 17 is? You are adding something, not just rephrasing.

Sounds like there is some miscommunication. I am sure I said 16 is about the elements as members of the series. I also said it's synonymous with the series being completed.

No, my objections do not amount to those things. Parts-to-whole does carry over properties in some cases, but not others... The quality transfers, but the quantity doesn’t. Regardless, you’ve still got work to do.

In this case it does, since a series being complete is synonymous with all members of a series, as members of a series, having the property of can be moved through.

When I questioned 16 as saying something about the whole wall, you said that it’s only talking about the bricks, though.

I am pretty sure I did not say only about the bricks, I certainly have no intention of ruling out it talking about the wall. As I kept asking you, what does completing the series (wall) even mean, if not all the members (bricks of the wall) being moved through.

[The Tanager]

No, you are rightly questioning the truth of P1 and providing a replacement premise that is true, which will then lead to a different conclusion. It didn’t seem like you were doing that with my argument. I thought you were saying that there are two sound arguments that use some of the same terms with equally valid (but different) definitions.

Perhaps you mean this: I hold that NASA and mathematicians both have sound arguments re: infinity, that uses same words along the lines of "size," or "boundary."

Other than being more specific, how is this different than what I said? Are you saying the definitions in my argument are valid or not?

Why? Because if one set is in one-to-one correspondence with another set, then it has the same size as that set?

That's one reason, another reason (or another implication of the same reason) is something that doesn't have a boundary still have a size.

On the first, mathematicians still talk about different sizes of infinite sets, though. Even just thinking about {0, …} and {-1, …} we can phrase talk about that in a way to get one-to-one correspondence and in another way to not get one-to-one correspondence. So, do those sets have the same size or not? Or are we talking about two different meanings for ‘size’?

On the second, we also run into two contradictory statements. The thing without a boundary (assuming such a thing could exist) that has a size can intuitively grow in its size (since sizes are things that can grow), but a growth in size also intuitively includes moving beyond its current boundary.

Yes, you have me questioning if this is even logically possible.

It's having the opposite effect to what I intended.

Yes, I realize that.

Sounds like there is some miscommunication. I am sure I said 16 is about the elements as members of the series. I also said it's synonymous with the series being completed.

Then I don’t accept 16 as true. What I see 1-15 doing is making a statement that element A can be counted when a part of a finite series, element B the same, element C, etc. Those say nothing about bringing these elements together into a series and being able to complete (i.e., count through) the new series. That jump is what is lacking in your proof.

In this case it does, since a series being complete is synonymous with all members of a series, as members of a series, having the property of can be moved through.

Then show it. Whether that step is 16 or 17 or this new 4-piece formulation.

I am pretty sure I did not say only about the bricks, I certainly have no intention of ruling out it talking about the wall. As I kept asking you, what does completing the series (wall) even mean, if not all the members (bricks of the wall) being moved through.

When looking at {3, 4}, I agree that ‘3’ and ‘4’ are numbers that are moved through in that finite series. Because every number in that finite series is moved through, the series is completed.

Your proof establishes that ‘3’ and ‘4’ (and all numbers) are the types of things that can be moved through. Completing the series is about whether all of the elements gathered together in that series are moved through, not whether they are moved-throughable types of things. You have shown they have the potential to be a part of a series that can be completed, but not that all series that contain them are completable.

[Bust Nak]

Other than being more specific, how is this different than what I said? Are you saying the definitions in my argument are valid or not?

You said there was some sort of contradiction, I am saying there isn't. I am saying your definitions are different to that of NASA and mathematicians. I am saying the only reason why you think there is a contradiction is because you are mixing and matching definitions.

On the first, mathematicians still talk about different sizes of infinite sets, though. Even just thinking about {0, …} and {-1, …} we can phrase talk about that in a way to get one-to-one correspondence and in another way to not get one-to-one correspondence. So, do those sets have the same size or not? Or are we talking about two different meanings for ‘size’?

Not sure what you mean by that, how can they be phrased in such a way where there is not an one-to-one correspondence?

On the second, we also run into two contradictory statements. The thing without a boundary (assuming such a thing could exist) that has a size can intuitively grow in its size (since sizes are things that can grow), but a growth in size also intuitively includes moving beyond its current boundary.

So abandon whatever definition that lead to the idea that growth in size also intuitively includes moving beyond its current boundary.

Then I don’t accept 16 as true. What I see 1-15 doing is making a statement that element A can be counted when a part of a finite series, element B the same, element C, etc. Those say nothing about bringing these elements together into a series and being able to complete (i.e., count through) the new series...

Okay, with that in mind...

1) Let X be the set (finite or infinite) of all integers > 0 that as part of {0, ...} cannot be counted to from 0. (define X)
2) X is not empty (assumption)
3) Let x0 be the lowest member of X (define x0)
4) Let y be the integer x0-1 (define y)
5) y as part of {0, ...} can be counted to from 0 (from 4)
6) With both as parts of {0, ...}, x0 can be counted to from y (from 4)
7) x0 as part of {0, ...} can be counted to from 0 (from 5 and 6)
8) contradiction, therefore the assumption is false (from 1, 3 and 7)
9) X, the set of all integers > 0 that as part of {0, ...} cannot be counted to from 0, is empty (from 2 and 8)
10) all integers >= 0 falls into 1 of 2 categories: as part of {0, ...} can be counted to from 0, or as part of {0, ...} cannot be counted to from 0 (premise)
11) {0, ...} with members of X removed, leaves a set that contains only members that as part of {0, ...} can be counted to from 0 (from 10)
12) {0, ...} with members of X removed is {0, ...} (from 9)
13) {0, ...} contains only members that as part of {0, ...} can be counted to from 0 (from 11 and 12)
14) an integer that as part of {0, ...} can be counted to from 0, is an integer that as part of {0, ...} can be moved through (premise)
15) {0, ...} contains only members that as part of {0, ...} can be moved through (from 13 and 14)
16) you can move through all members of {0, ...} as part of {0, ...} (from 15)

Now can you accept 16) is synonymous with the series being completed?

[The Tanager]

You said there was some sort of contradiction, I am saying there isn't. I am saying your definitions are different to that of NASA and mathematicians. I am saying the only reason why you think there is a contradiction is because you are mixing and matching definitions.

I’m saying that using the terms with the meanings I am giving them, there would be a contradiction in saying the universe is expanding and has the size of actual infinity.

You seem to be saying that using those same terms with different meanings, there is no contradiction between saying the universe is expanding and has the size of actual infinity.

If so, then I agree with you. What I have then said is that this is irrelevant to whether there is a contradiction based on the meanings I give the terms. Do you agree?

If so, then do you think my meanings are wrong, my argument is invalid, one of the premises is untrue, or that my argument is a sound one?

If not, then how are these other meanings relevant to my argument?

Not sure what you mean by that, how can they be phrased in such a way where there is not an one-to-one correspondence?

The zeros of each set correspond, the ones of each set correspond, the twos of each set correspond, ad infinitum, leaving ‘-1’ as an extra member of one set that doesn’t correspond to anything in the other set.

So abandon whatever definition that lead to the idea that growth in size also intuitively includes moving beyond its current boundary.

If you give me a good reason to, other than something like “so we can say there doesn’t appear to be a contradiction.”

1) Let X be the set (finite or infinite) of all integers > 0 that as part of {0, ...} cannot be counted to from 0. (define X)
2) X is not empty (assumption)
3) Let x0 be the lowest member of X (define x0)
4) Let y be the integer x0-1 (define y)
5) y as part of {0, ...} can be counted to from 0 (from 4)
6) With both as parts of {0, ...}, x0 can be counted to from y (from 4)
7) x0 as part of {0, ...} can be counted to from 0 (from 5 and 6)
8) contradiction, therefore the assumption is false (from 1, 3 and 7)
9) X, the set of all integers > 0 that as part of {0, ...} cannot be counted to from 0, is empty (from 2 and 8)
10) all integers >= 0 falls into 1 of 2 categories: as part of {0, ...} can be counted to from 0, or as part of {0, ...} cannot be counted to from 0 (premise)
11) {0, ...} with members of X removed, leaves a set that contains only members that as part of {0, ...} can be counted to from 0 (from 10)
12) {0, ...} with members of X removed is {0, ...} (from 9)
13) {0, ...} contains only members that as part of {0, ...} can be counted to from 0 (from 11 and 12)
14) an integer that as part of {0, ...} can be counted to from 0, is an integer that as part of {0, ...} can be moved through (premise)
15) {0, ...} contains only members that as part of {0, ...} can be moved through (from 13 and 14)
16) you can move through all members of {0, ...} as part of {0, ...} (from 15)

Now can you accept 16) is synonymous with the series being completed?

1)-15) still give a fact about the integers (i.e., how long one brick in a brick wall is). They are things that can be counted to/completed. The justification for your argument would mean 16) also gives a fact about the integers. 16) needs to make a statement about the series as a series (i.e., how long the brick wall is). Counting to a number from 0 is different from completing a series that contains those kinds of numbers.

And don’t forget that you’ve still got a step to show it applies to this kind of series {..., 0}, since that is what an A-theory infinite past would be like.

[Bust Nak]

If so, then I agree with you. What I have then said is that this is irrelevant to whether there is a contradiction based on the meanings I give the terms. Do you agree?

Yes.

If so, then do you think my meanings are wrong, my argument is invalid, one of the premises is untrue, or that my argument is a sound one?

Not wrong as such, just incompatible thus argument invalid. The same way "man" as in male adult and "man" as in mankind are both perfectly fine meaning.

The zeros of each set correspond, the ones of each set correspond, the twos of each set correspond, ad infinitum, leaving ‘-1’ as an extra member of one set that doesn’t correspond to anything in the other set.

So shift everything along 1 space, and you have a match.

If you give me a good reason to, other than something like “so we can say there doesn’t appear to be a contradiction.”

Not having a contradiction is the best reason.

1)-15) still give a fact about the integers (i.e., how long one brick in a brick wall is). They are things that can be counted to/completed. The justification for your argument would mean 16) also gives a fact about the integers. 16) needs to make a statement about the series as a series (i.e., how long the brick wall is).

Facts about integers in a series does tell you something about the series as a series though, just as bricks in a wall, tells you something about the wall, as a wall. That's the point when I asked you if you accept 16) as synonymous with the series being completed? You did say "a series being completed is a synonymous phrase to a series having each and every element of that series, as members of that series, being counted through."

Counting to a number from 0 is different from completing a series that contains those kinds of numbers.

Is it? Is a brick being red different from a wall containing those kinds of bricks, being red?

And don’t forget that you’ve still got a step to show it applies to this kind of series {..., 0}, since that is what an A-theory infinite past would be like.

That can wait, it's trivial once you accept {0, ...} can be completed.

[The Tanager]

Not wrong as such, just incompatible thus argument invalid. The same way "man" as in male adult and "man" as in mankind are both perfectly fine meaning.

My argument is that these concepts are incompatible. So, how would these concepts being incompatible make that argument invalid?

So shift everything along 1 space, and you have a match.

As I said, you can phrase things in such a way to both get ‘one-to-one correspondence’ and not get it. Why choose one phrasing over the other? If it’s “to get the conclusion I want,” then that’s not a rational reason for choosing one phrasing over the other.

Not having a contradiction is the best reason.

Phrasing things to avoid a possible contradiction isn’t rational support for there not being a contradiction.

Facts about integers in a series does tell you something about the series as a series though, just as bricks in a wall, tells you something about the wall, as a wall.

Is it? Is a brick being red different from a wall containing those kinds of bricks, being red?

No, they sometimes do and sometimes don’t. The fact that every brick is red does tell us that the brick wall is red. The fact that every brick is 8 inches long certainly does not tell us that the brick wall is 8 inches long.

That's the point when I asked you if you accept 16) as synonymous with the series being completed? You did say "a series being completed is a synonymous phrase to a series having each and every element of that series, as members of that series, being counted through."

Yes, but then you’ve assumed that “each and every element of that series, as members of that series, being counted through” is synonymous with “being able to count from 0 to each and every element of that series” and it’s not.

That can wait, it's trivial once you accept {0, ...} can be completed.

I’m not sure it is, since your whole proof rests on starting at 0 and all starting points will be taken away, but perhaps you are right. You can keep trying to support your claim about {0, …}.