# Argument from Motion

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Here is another classic example which might help think about it.

We start with the natural numbers A = {1,2,3,...}

Now using those we construct two new sets. To construct the first set we use f(n) = n, for each element in A, which we will call B, B = {1,2,3,...}

For the other new set we construct it with g(n) = 2n for each element in A, which we will call C, C = {2,4,6,...}

The number of elements of B and C (and their cardinality) must be the same since each set has a matching element for each element of A. Or in other words, there are just as many even numbers as their are natural numbers (which certainly seems counter-intuitive to me, but as I've long said, common sense is worthless here).

But:

B-C = { 1,3,5,7,9,... } = { " The set of odd numbers " } != {}

And:

(x is in C) => ( x is in B ) ( i.e., all members of C are in B )

You have got to get this out of your head if you want to make any progress. [handwaving]When we remove a pair of elements from B and C, it does not matter if they are equal in value; we are not interested in the sum of these sets, we are only interested in number of elements each has.[/handwaving]. And yes, an infinite set subtracted from an infinite set can result in another infinite set, with ALL 3 sets having the same cardinality (like the example you just gave with the odd numbers).

The difference of two sets being a non- empty set, yet having the same cardinality?

I apologize, but I can't just get it out of my head and be satisfied with it. I know where you are coming from as well with the bijection, but it seems that B should be, in some sense, more than C. ( It has all odds and evens; C only has all the evens )

Also, the statement:

(x is in C) => ( x is in B ) ( i.e., all members of C are in B )

is to show that C satisfies one of the two axioms to be a proper subset of B. If C had some members that are not in B, then it would be a problem, just like the " if it exists " in the girl example.

skd6348:

If by "cardinality" you are referring to a count, then an infinite set has no cardinality.

If you are talking about Cantor's sizes of infinity, then there is no change of cardinality even as Nyarlathotep said.

Why isn't their a change in cardinality, when something new definitely has been added to P?

This is a good question and it comes back to the bijection.

Let's start with A = {1, 2, 3, ...} and B = {1, 2, 3, ...}. Of course it is trivial that these have the same cardinality, but lets go through it. The bijection is easy, it is just f(n) = n.

Now lets modify B by adding an element a, which does not exist in A. B = {a, 1, 2, 3, ...}. Do these have the same cardinality? Yes. What is the bijection from A to B?

Let's add another, B = {a, b, 1, 2, 3, ...}. Again they have the same cardinality. What is the bijection from A to B?

This pattern can be repeated as many times as we like to "patch over" any finite number of added elements.

I guess what you are saying is like:

" If we have a 1:1 function from a set A to B, and B to A as well, then A and B have the same cardinality. "

I don't share this view. While it works perfectly if both A and B are finite, infinite set comprehensions are a different thing.

As B-A = { a } ( In your first example ), this seems to mean something else.

As in the girl example, is she removes the exact same set of members from both A and B, and if one is empty now, but the other is not,..... then the other must have more members.

I don't strictly believe that B={a,1,2,3,... } and A={1,2,3,... } have different cardinalities, but that I do find problems in that they have the same cardinality.

We have three options:

1) Same cardinality.

2) Different cardinalities.

3) The concept of cardinality is inapplicable to infinite sets.

Well then I don't know what to tell you.

well set theory for starters.

I already hinted to set theory in my 3rd example:

" 3) Various conceptual sets, like the set of natural numbers, set of multiples of 5, .... "

Once again, these are NOT actual infinities. Actual infinities are like: " Infinite number of particles in a Universe "....

What I'm talking about, more specifically, is the idea that there are an infinite number ( or duration ) of past events...

set theory considers the set of all natural numbers to be an actual infinity.

Under what definition is it considered actual infinity? The set of natural numbers isn't really an object, but just an abstract notion...

Consider this. Is is possible for a super computer to store the entire list of natural numbers in digital form.

No, because that would require the computer to have infinite memory...

THIS is the type of infinity which is being suggested by the Universe having no beginning...

Once again, I'd clarify that sets of the number line, and other series, can be infinite. In fact, I use the convergence properties of infinite series myself in programming.

But merely pointing to sets like of natural numbers, integers, etc. will not prove that scenarios like I + 2 = I are not paradoxical.

I'd repeat again, UNDER what rational, or axioms, do you find I+2=I to be consistent, BUT I-I=0 to be a mistake?

It is considered actually infinite because it already been completed.

" Completed " in what sense? Could you elaborate? Maybe using the set of natural numbers:

N = { 1,2,3,4,... }

I'll try to elaborate my point of view...

1) N IS complete in the sense that it can be expressed as a set. This is only conceptual, and the abstract realm is different from objects, like my laptop for instance..

2) N is complete ( actually closed ) in the sense that addition and multiplication of any two of the elements in N would yield a Third member from the N itself... But division or subtraction will not necessary yield from N...

3) N is complete in the sense that any member, say 7, does not arrive through a non-ending process IN time... Once again, only as an abstraction...

4) BUT, N is NOT complete in the sense that a process, if required to generate and store the elements of N in a real data bank, like a computer, would eventually end...

i.e.' the existence, as expressed information, of each and every one of the elements of N, individually, is not possible.

All computers can store is an iterator, or formula, to generalize the set elements, and execute it ONLY up to a certain level to get a desired element...

Looks like you are doing mental gymnastics, again; which I am not interested in. You told us set theory does not use actual infinities.

When you have made your dismount from the parallel bars, perhaps you should send of copy of your post there so they can fix their website. You should probably include additional copies to every mathematics department in your region of the world; since clearly they have been teaching it wrong all this time! What a wonder service you are going to provide the world! I'm just glad I could be part of it...[/sarcasm]

Hey... calm down. I just asked FOR you to define the terms.

Whether, or not, some website, or book, or author or consensus regards sets like N to be actual infinities, does not prove they actually are. I hope you are not suggesting something to that effect... [ / misuse of authority, majority ]

The reason I stated the FOUR terms about " complete " was to avoid any mental gymnastics.

Because it is much of a matter of definition whether sets are completed are not. Not everyone defines " actual infinity " to be the same way as your link does, and Wikipedia also states there are philosophers who disagree that sets are actual infinities even by this definition...

If you could provide the following, it'd be helpful for the both of us: [ I'm really repeating myself ]

1) " Completed " in what sense? Could you elaborate? Maybe using the set of natural numbers:

N = { 1,2,3,4,... }

2) UNDER what rational, or axioms, do you find I+2=I to be consistent, BUT I-I=0 to be a mistake?

3) CAN one write down the entire sequence of N on any form of information storage? BECAUSE, saying that there is are actually an infinite number / duration of past events is similar to that...

N and " infinite number of past events " are very different. Demonstrating the former does not prove the possibility of the latter.

Similarly, N as a conceptual set, or as a rigorous statement of all its elements are different...

1)it is complete in that the entire set exists already. That we are constructing an infinite set out of finite elements, and the construction process was completed.

2) I never said I-I = 0 is a mistake. I made sure to write what I wanted precisely, please re-read it.

3) that is not a requirement for an infinity to be actual.

I agree, but see I never made that claim either! See a pattern forming yet? The topic came up because you claimed that set of natural numbers is not an actual infinity. You ready to abandon that claim yet? Yeah I didn't think so.

1) Construction process? As in temporally? I don't think N is a result of a process run, and completed, IN time. And that is exactly why it is not possible for any computer to hold all elements of N in memory...

But then again, you could be saying that just in convention... [ That sets are " constructed " ]

2) You said:

"The problem with the manipulation I posted above is that it is incorrect. I sneakily inserted a mistake in the simplification of the right hand side between lines 2 and 3. Infinity - infinity is indeterminate (you can't safely say it equals 0)."

More importantly, under what axioms, or rational, do you consider I=I+2 to be consistent, and " I-I " to be indeterminate?

3) It is not meant to be a requirement. It was being discussed for " infinite past events " scenario.

You should note that whether sets are " actual infinities " or not was only a side matter. Our actual topic is the Prime Mover Argument, and I started my discussion to discuss (7) in the OP. This involves " infinite number of past events", A more thorough discussion on whether this scenario is possible can have useful insight on this, and many other, issues regarding the Universe.

As far as completeness goes, you didn't seem to like what I had to say on the matter, and you didn't seem to like what the Stanford Encyclopedia of Philosophy had to say on the matter. I could link you to some university mathematics websites, but I have a sneaking suspicion how that will go, so I don't know what else to tell you.

I never said I-I was a mistake, and I never said I-I was indeterminate. I said ∞ - ∞ =0 is a mistake because is ∞ - ∞ indeterminate. I was very careful about what I wrote so I wouldn't have to go though a paroxysm of mental gymnastics later.

I agree 100%, it is a side matter. Are you now abandoning your claim that the set of natural numbers is not an actual infinity in set theory (I can't tell so I feel I must ask)?

Let me approach it this way, do you believe that N is " generated " via some source of process completed in time, which has been completed, or was it just a figure of speech?

If we run computer program for an infinite series ( like for the digits in pi ), it will never end. The process has the " potential " to keep going on and on... But can't "actually" generate all the digits of pi, thereby terminating.

I-I refers to " infinity-infinity". I had defined my notations in one of my earliest posts. [ Post 39 ]

As you've said:

" Infinity - infinity is indeterminate (you can't safely say it equals 0). "

About N being actual infinity, I do not believe it is " actual infinity " in the manner used in " Cosmological Arguments ". That manner is no. 4 in my post 74.

This isn't mental gymnastics, you can find that in cosmological arguments. When they say actual infinity cannot exist, they do not mean " conceptual iterators like N ". I thought you'd be using it in the same manner, but later I got unsure so I had to ask.

The definition on Wikipedia is a bit vague. I DON'T believe N is generated through a process that has been completed temporally, BUT that the elements of N are conceptual and exist timelessly...

The only generation that can happen is in minds like ours, or computers, or alike. These cannot end for infinite sets.

V.S.

It seems you have a lot more backflips to do in your gymnastics. Let me know when you are ready to abandon your notion that set theory does not contain actually infinite sets.

But wouldn't that be merely a dogma? I am not interested at taking FOR GRANTED what certain people say. I already told you:

" Whether, or not, some website, or book, or author or consensus regards sets like N to be actual infinities, does not prove they actually are. I hope you are not suggesting something to that effect... [ / misuse of authority, majority ] "

Furthermore, Wikipedia also states that not all philosophers believe in this:

https://en.wikipedia.org/wiki/Actual_infinity#Classical_set_theory

What I am interested is in discussing THE RATIONAL behind these ideas. Merely quoting others who believe in it is not enough... Also, there is a definition gap, which I explained.

I'm not asking you to take it for granted. It seems you are incapable of listening to me, perhaps you can go to those sites and read the explanation there (but I have my doubts).

Oh so apparenlty you are interested in what websites books and authors have to say, but only when you think it supports your assertions. You win! I'm done.

You've Said:

"I'm not asking you to take it for granted. It seems you are incapable of listening to me, perhaps you can go to those sites and read the explanation there (but I have my doubts)."

I'm already acquainted with set theory, particularly ZFC. But:

1) I don't fully agree with all its foundations.

2) You're links don't provide clear, well founded answers to what I'm asking here. There's a lot of vagueness in there. I simply ask you to give me a to-the-point answer yourself for some of what you've said.

"Oh so apparenlty you are interested in what websites books and authors have to say, but only when you think it supports your assertions. You win! I'm done."

Ummm... OK...?

I already said sources don't matter if they don't have proofs or justifications. That's applied TO my sources too...

The reason I quoted was only to mention that not everyone believes N is an actual infinity. That's not meant to be a proof for N not being actual infinity.

And I quoted Wikipedia's page on Actual Infinity, not some far fetched site.

Here is a part from a discussion of cosmological argument:

"Premise (KS1) claims that an actual infinite is impossible. Presumably, it makes no difference whether what is proposed as actually infinite is a magnitude, a collection, a subdivision[3], or a series of events; all of these we are to consider impossible. Although actual infinites are impossible, there may still be collections, magnitudes, subdivisions, or series which increase (or diminish) without limit. These are potential infinites. Craig explains:

When Aristotle speaks of the potential infinite, what he refers to is a magnitude that has the potency of being indefinitely divided or extended. Technically speaking, then, the potential infinite at any particular point is always finite.[4]

A potentially infinite collection (or magnitude or series) is a sort of work in progress. Although at any particular time the collection is finite, it is continually being augmented, and at no time does this augmentation terminate. The idea behind a potentially infinite collection is not merely that of a finite collection to which some new addition could be made, but rather that of a collection that is growing without limit. In contrast, an actual infinite is described as "a determinate whole actually possessing an infinite number of members."[5] "In an actual infinite, all the members exist in a determinate, completed whole."[6]

According to Craig, while there is no barrier to a potential infinite's existing, the same cannot be said of an actual infinite. Since an actual infinite is an impossibility, the series of past events in the universe cannot be actually infinite. Neither does it make sense to suggest that the series of past events is potentially infinite (in the past) since we cannot say that new past events are being continually added to the beginning of the series (though, of course, it is perfectly possible to add new events to the end of the series). The universe must, therefore, have come to exist at some definite point in the past. "

https://infidels.org/library/modern/eric_sotnak/kalam.html

This is a very different sense than an iterator for N.

An iterator ( in the form of a formula, or a definition ) for N can be provided, but all individual elements cannot be rigorously stated in actuality.

It seems I have to keep repeating the same questions:

1) Under what axioms, or rational, do you consider I=I+2 to be consistent, and " I-I " to be indeterminate?

2) Do you believe that N is " generated " via some source of process completed in time, which has been completed, or was it just a figure of speech?

[ As you've said: it is complete in that the entire set exists already. That we are constructing an infinite set out of finite elements, and the construction process was completed. ]

3) CAN one write down the entire sequence of N on any form of information storage? BECAUSE, saying that there is are actually an infinite number / duration of past events is similar to that...

Nyarlathotep:

As far as I know, the + and - operations are not defined for infinity. So, these expressions are incoherent rather than "wrong."

Right, what I was trying to say is that I-I = 0 is OK, but I-I =0| I = ∞ is dangerous because it is indeterminate. I didn't want to get into the situation where I'd say "I-I is indeterminate"; only to have someone accuse me of contradicting high school algebra.

I had already defined "I" to refer to infinity in our discussion. So I really don't know what to say here...

I mean your statements:

1) "The problem with the manipulation I posted above is that it is incorrect. I sneakily inserted a mistake in the simplification of the right hand side between lines 2 and 3. Infinity - infinity is indeterminate (you can't safely say it equals 0)."

2)" I never said I-I = 0 is a mistake. I made sure to write what I wanted precisely, please re-read it."

3) "I never said I-I was a mistake, and I never said I-I was indeterminate. I said ∞ - ∞ =0 is a mistake because is ∞ - ∞ indeterminate. I was very careful about what I wrote so I wouldn't have to go though a paroxysm of mental gymnastics later."

Seem to conflict if you were taking I to mean infinity ( which I clearly stated in the beginning ).

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