Do you believe there can be an Actual Infinite

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If you could have a finite period of time, it would be impossible to have an infinite number of states in that time because it would require time to change from one state to the nex

It is possible because of the nature of continuous: to move from point 0 to point 1, you first have to travel through point 0.1, then 0.01, then 0.001, then 0.0001 and so on to infinity. Each distinct point represents a different state with different distinct time and space co-ordinates.

I don't believe that you can model a continuum in this way, because you are assigning ends to it. What principle allows you to put a beginning and an end to a continuum?

Any given finite distance we can represent by the reals between 0 and 1. For instance the distance of 2 miles maps like this:
0 mile -> 0.0
1 mile -> 0.5
2 miles -> 1.0

This is contradiction. You are saying that the continuum is made up of discrete units, "1 second", "1 year". To say that is to deny that it is a continuum

I mean that we can use arbitrary units to sub-divide the continuum (but it is not actually made of discrete units).

Right, if you could model time as discrete units you would not run into these problems. The problem though, is that we experience time as continuous, and we've found no natural divisions to form the basis for the finite N, the number of discrete units per second. I don't think the Planck unit provides us with this.

A movie seems continuous but most are a discrete 60 frames a second.
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It is possible because of the nature of continuous: to move from point 0 to point 1, you first have to travel through point 0.1, then 0.01, then 0.001, then 0.0001 and so on to infinity. Each distinct point represents a different state with different distinct time and space co-ordinates.

As I explained, those points are only conceptual. They do not actually exist in the thing they are being applied to. So you're just restating Zeno's paradox.

Any given finite distance we can represent by the reals between 0 and 1. For instance the distance of 2 miles maps like this:
0 mile -> 0.0
1 mile -> 0.5
2 miles -> 1.0

Again, you're conflating the map with the substance.

I mean that we can use arbitrary units to sub-divide the continuum (but it is not actually made of discrete units).

Right, so any description of a division in the continuum is not real or else the continuum would not be a continuum. The divisions are conceptual only or else the so-called continuum would be made of discrete units.
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That's how it is when someone has a Belief about something — SteveKlinko
The traditional Christian view of God is that he is eternal and infinite. I wonder if some people are still religiously invested in infinity? I suspect some atheists might likewise be 'religiously' invested in infinity as a mechanism to explain the apparent fine tuning of the universe for life?

Every time you really work out a problem or analyze a little Deeper it is always found that Infinity is a big problem — SteveKlinko
Wikipedia lists a few (but there are more):

In cosmology they have this paradox:

https://en.wikipedia.org/wiki/Measure_problem_(cosmology)

The solution is a finite universe but cosmologists press on regardless...

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Watching a sunset means to observe a sun from one event being not below horizon to is one. Remember that is comparing and observation is a phenomenon therefore you are part of the event as well as thoughts that you think of. It's just comparing a prior to the now.

I disagree that there is any comparison involved in these activities which I call enjoying the passing of time. Maybe you carrying out similar acts of comparison, but not me, so they are different acts.

Also when I mean points I meant if you were to describe a ball at the top of your house and than it was on your porch, it would be referring to the events that happened in between those two like being at certain relative distances at certain events. This also goes to my point of not infinite past for there can not be an infinite events inverween the ball at the top and to the prch for addition synthesis from a point never leads to infinity.

This is why Aristotle concluded that there is a categorical difference between being and becoming, which cannot be reconciled: the two are in compatible. If change is represented as two distinct states of being (the ball on the roof, and the ball on the porch for example), then to account for the change between these two states we need to introduce a third state which is neither the one nor the other. Now we have an intermediate state, and we need to account for the change between the first and the intermediate, as well as the intermediate and the other, so we introduce two more states. This would result in an infinite regress of states. There appears to be an infinite number of states between any two states, if change is represented as different states. So he proposed that "becoming" (active change) is categorically different from "being" (states of existence), and activity cannot be represented by states.

Accordingly, your description of the ball on the roof, and then the ball on the porch, cannot be used to describe the activity which is the ball moving from the roof to the porch. And no matter how you try to describe this activity through intermediate states, you are only asking for an infinite regress of intermediate states. This problem of infinite regress indicates that it is not a correct description, to describe an activity in terms of states.
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Most descriptions are either based on a sequence of events or have similar concepts to one. Like falling is describing a sequence of an object going towards the earth or other such object. It's all comparison. The only ones that are non observation are like one or unlike which are not describing phenonomen as the others do. Also the film analogy I gave makes it so that you can not have an infinite number of events in between and in the past. Past meaning events that are before now. It is also a finite number of events since that which happens in sequence can can lead to infinity.
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So you're just restating Zeno's paradox

Its similar, but I was highlighting not an infinite number of steps; instead infinite information density. There is a limit already established on information density:

https://en.wikipedia.org/wiki/Bekenstein_bound

If you buy 4D space-time, then information is not transitory, it has permanent residence in a region of space-time, so I would expect information density to apply over a time period as well as a volume of space.

Say we have a system composed of 1 particle that travels 1 meter in 1 second. If space is continuous, how many different states does the system go through? IE If the particle is travelling along the X-axis, the states are just the different positions it occupies x=0 x=0.1 etc...

The answer is infinite states and log(number states) = information.

It is the exact same kind of infinity you get for a system of 1000 particles traveling 1 mile in a 1 day. Do you see the paradox?
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Not lead to infinity, miswrote at the last part part.
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Clearly, your sets as written do not indicate that the right is a subset of the left. The left contains 4 and 6, which are not contained in the right. It is not a subset.

You have got to be kidding me. Both the left and right contained 4 and 6, your just had to continue the mapping a few more spots. (4 & 6 appeared on the right side earlier because the right side was only even numbers, so obviously the natural numbers take longer to get to the even numbers since it also has the odd numbers).
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If you buy 4D space-time, then information is not transitory, it has permanent residence in a region of space-time, so I would expect information density to apply over a time period as well as a volume of space.

Actually I don't buy 4D space-time. I think it is a mistake to unify the concepts of space and time in this way, they need to remain separate. And you haven't really explained what you mean by "information density" so I'm sort of lost here.

Say we have a system composed of 1 particle that travels 1 meter in 1 second. If space is continuous, how many different states does the system go through? IE If the particle is travelling along the X-axis, the states are just the different positions it occupies x=0 x=0.1 etc...

Please read what I wrote to BB100 in my last post, about the incompatibility of activity and states. It really doesn't make sense to describe activity as an infinite number of states between state 1 and state 2.

You have got to be kidding me. Both the left and right contained 4 and 6, your just had to continue the mapping a few more spots.

No, I'm not kidding at all, I think you must be totally confused, or ignorant, or something like that. If I continue the mapping a few more spots, the even numbers go to 8, 10. But these numbers will be outside of the set of natural numbers, on the left. Don't you see that with this type of mapping, the set of even numbers will always contain numbers outside the set of natural numbers which it is mapped to, so it is impossible for it to be a subset?

(4 & 6 appeared on the right side earlier because the right side was only even numbers, so obviously the natural numbers take longer to get to the even numbers since it also has the odd numbers).

Right, so no matter how you lay it out, if you maintain equal cardinality the set on the right side will always contain numbers which are not contained in the set on the left side. Surely you can acknowledge this. So do you agree with me that it is impossible that the set on the right side is a subset of the set on the left side? If not, why not?
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What is the meaning of actual infinity?

Do you mean physically manifest? If so, space may be infinite.
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What is the meaning of actual infinity?

Do you mean physically manifest? If so, space may be infinite.

I'm using Aristotle's definition:

https://en.m.wikipedia.org/wiki/Actual_infinity

I think space is finite:

- The universe is expanding; if infinite then there is nowhere to expand to
- Actual infinity does not occur mathematically and maths reflects the real world
- Actual infinity does not occur in nature
- Empty space has vacuum/dark energy associated with it so empty space is not empty. We want to avoid the total energy content of the universe being infinite.
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Actually I don't buy 4D space-time

Do you buy special relativity? He only has two axioms and both sound very reasonable:

https://en.wikipedia.org/wiki/Postulates_of_special_relativity

And there is a huge volume of experimental evidence to back it up?

And it makes sense from first principles:

1. Something can’t come from nothing
2. So base reality must have always existed
3. If base reality is permanent it must be timeless (to avoid actual infinity)
4. Time was created and exists within this permanent, timeless, base reality
5. So time must be real, permanent and finite

And you haven't really explained what you mean by "information density" so I'm sort of lost here.

The amount of information you can get into a volume of space-time by regarding the spacial co-ordinates of particles as information:

- So in discrete space-time, I could represent a particle's position with (0.35, 0.60, 0.90, 0.20); terminating decimals / rational numbers - a finite amount of information.

- But in continuous space-time, the particle's position is represented by (0.353534..., 0.604836..., 0.903742..., 0.736363...); non-terminating real numbers - an infinite amount of information.

An infinite amount of information in a finite volume of space-time is nonsense and leads to paradoxes...

This is why Aristotle concluded that there is a categorical difference between being and becoming, which cannot be reconciled: the two are in compatible. If change is represented as two distinct states of being (the ball on the roof, and the ball on the porch for example), then to account for the change between these two states we need to introduce a third state which is neither the one nor the other. Now we have an intermediate state, and we need to account for the change between the first and the intermediate, as well as the intermediate and the other, so we introduce two more states. This would result in an infinite regress of states. There appears to be an infinite number of states between any two states, if change is represented as different states. So he proposed that "becoming" (active change) is categorically different from "being" (states of existence), and activity cannot be represented by states.

There is only an infinite regress of states if space-time is continuous.
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Aristotle handled the topic of infinity in Physics and in Metaphysics. He distinguished between actual and potential infinity. Actual infinity is completed and definite, and consists of infinitely many elements. Potential infinity is never complete: elements can be always added, but never infinitely many — Wikipedia

Can you explain the last bit? "elements can always be added, but never infinitely many"?

I guess my understanding of infinity is very simple. From what I read from wikipedia the actual and potential distinction is important to Georg Cantor's conception of infinity but I don't get it.

If you do can you explain it to me?
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Can you explain the last bit? "elements can always be added, but never infinitely many"?

Best way to think of Potential Infinity is a temporal, iterative process. Examples are counting, or walking. Both can potentially go on for ever but never actually do go on for ever.

Actual Infinity is then the result of carrying on that iterative process for ever. So for example, writing out ALL of the natural numbers on a piece of paper would be Actual Infinity (impossible).

On the maths side:

- Potential infinity is closest to the limit concept. IE limits increase towards but never actually attain infinity
- Actual infinity is the concept used in set theory/Cantor's work. Its defined to exist by an axiom. The axiom basically says the set of natural numbers exists as a completed set. They don't prove anything. The axiom is provably wrong so set theory is flawed IMO.
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Do you buy special relativity? He only has two axioms and both sound very reasonable:

I don't think that it's reasonable to believe the speed of light to be constant. If something was going very fast, close to the speed of light for example, relative to something else, then I don't think the motion of light would be the same relative to these two things. Actually, I think that relativity theory in general, while it may be adequate for modeling many motions, is ontologically deficient.

1. Something can’t come from nothing
2. So base reality must have always existed
3. If base reality is permanent it must be timeless (to avoid actual infinity)
4. Time was created and exists within this permanent, timeless, base reality
5. So time must be real, permanent and finite

I see a logical flaw here. You have a base reality which is permanent and timeless. You have a time which is created, comes into existence, within this permanent, timeless base reality. But then you conclude that time is permanent. That is contrary to your premise, that time is created.

The problem is to get beyond temporal existence, to understand that which is timeless. If you start to describe the timeless using the same terms we use to describe the temporal, you are bound to be contradictory. So even to say that the timeless is "prior" to the temporal, or that time "emerges", or is "created", from the timeless, is contradictory because these terms imply temporality, and these temporal attributes are assigned to the timeless in order for time to come into existence.

This is where dualism assists us because it allows us to separate the two aspects of reality, so that we do not end up assigning temporal properties to the non-temporal. To do this, we need a clear definition of time, which separates time (as immaterial) from the material existence of thing which exist, and move in time. Time itself is now the medium between the timeless and the temporal existence of material things. That's why I said we need a proper conceptual separation between time and spatial existence.

The amount of information you can get into a volume of space-time by regarding the spacial co-ordinates of particles as information:

- So in discrete space, I could represent a particle's position with (0.35, 0.60, 0.90); terminating decimals / rational numbers - a finite amount of information.

- But in continuous space, the particle's position is represented by (0.353534..., 0.604836..., 0.903742...); non-terminating real numbers - an infinite amount of information.

An infinite amount of information in a finite volume of space-time is nonsense and leads to paradoxes...

I agree with you on this matter. Now look at what happens if we separate space from time, and postulate a discrete space with a continuous time. The particle has an infinite number of possible locations because the time between t1 and t2 is infinitely divisible. However, the discrete space limits these possibilities to a finite number. So within the immaterial realm, which is represented conceptually as the realm of time passing, there are infinite possibilities, the information is infinite. But in actuality, the possibilities (and therefore information) are limited by the true nature of space.
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I don't think that it's reasonable to believe the speed of light to be constant.

A universal speed limit makes sense for any universe; if you allow objects to be accelerated upto an infinite velocity (as in Newtonian mechanics), then you get bizarre paradoxes like objects suddenly disappearing from the universe.

If you doubt the speed of light is constant you are also dismissing much of modern science:

https://en.wikipedia.org/wiki/Speed_of_light#Measurement

I see a logical flaw here. You have a base reality which is permanent and timeless. You have a time which is created, comes into existence, within this permanent, timeless base reality. But then you conclude that time is permanent. That is contrary to your premise, that time is created.

Think about a timeless base reality; there is no such thing as transitory information here; there is no time; all information is permanent.

How would time be created/made real in such a base reality? Each moment in our time must have been mapped to a co-ordinate in timeless, permanent, base reality. Hence past, present, and future are equally real.

I agree with you on this matter. Now look at what happens if we separate space from time, and postulate a discrete space with a continuous time. The particle has an infinite number of possible locations because the time between t1 and t2 is infinitely divisible. However, the discrete space limits these possibilities to a finite number. So within the immaterial realm, which is represented conceptually as the realm of time passing, there are infinite possibilities, the information is infinite. But in actuality, the possibilities (and therefore information) are limited by the true nature of space.

The problem is if you assume time is immaterial, you get eternal existence then you get all the paradoxes listed in this OP:

https://thephilosophyforum.com/discussion/4158/nine-nails-in-the-coffin-of-presentism/p1
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I don't think an axiom is right or wrong. By definition it is accepted as true. If course we can reject it but that doesn't invalidate the theorems that derive from the axiom.

By the way I still don't understand the difference between actual and potential infinity
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don't think an axiom is right or wrong. By definition it is accepted as true. If course we can reject it but that doesn't invalidate the theorems that derive from the axiom.

We use maths to reason about the material world. Infinity is impossible in the material world. Maths should not reflect whats impossible. We have theories in cosmology based on an infinite universe and those theories are wrong because the axiom of infinity is wrong.

By the way I still don't understand the difference between actual and potential infinity

Potentially infinite is like this list:

1, 2, 3, 4, 5, ...

It potentially goes on for ever... but in reality there is not enough paper to write it down in full.

Actually infinite you have to imagine the list going on for ever written down somewhere; the whole thing; every possible natural number.
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A universal speed limit makes sense for any universe; if you allow objects to be accelerated upto an infinite velocity (as in Newtonian mechanics), then you get bizarre paradoxes like objects suddenly disappearing from the universe.

It's not that there is a universal speed limit which I doubt, it is that there is something (light) which has the same speed in every frame of reference. I think that it is quite plausible that there is a universal speed limit, but we do not know enough about the universe to be able to determine it.

If you doubt the speed of light is constant you are also dismissing much of modern science:

Yes, and if you've read my earlier posts I've dismissed a lot of modern mathematics as well as contradictory. The two go hand in hand, mathematics and science, and I find them to be extremely misguided ontologically, like the blind leading the blind.

How would time be created/made real in such a base reality? Each moment in our time must have been mapped to a co-ordinate in timeless, permanent, base reality. Hence past, present, and future are equally real.

The problem is that time passes. At each moment there is something new in the past. To ask how time is created is to ask how time starts passing. Can you imagine a point in time when there was a future with no past?

The problem is if you assume time is immaterial, you get eternal existence then you get all the paradoxes listed in this OP:

I don't see the problem here. How is "eternal existence" different from your assumption of a permanent base reality? The difference is that the ontology I describe provides a proper separation between the "eternal existence", as outside of time, and temporal (material) existence, by placing time as the medium which separates the two. In this way, any physical activity, such as the activity of counting, described in the op cannot be attributed to the eternal existence because this would be a misappropriation of terms, to attribute physical, temporal, activity to that which is outside of time. You provide no such solution, attributing time to the permanent base reality just creates an infinite regress with no way to understand anything beyond the permanent time. Your claim that time is created contradicts your fundamental principles which render time as permanent.
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It's not that there is a universal speed limit which I doubt, it is that there is something (light) which has the same speed in every frame of reference. I think that it is quite plausible that there is a universal speed limit, but we do not know enough about the universe to be able to determine it.

One implies the other.

Yes, and if you've read my earlier posts I've dismissed a lot of modern mathematics as well as contradictory.

I agree with you but I've looked at Special Relativity; it seems sound enough and its accepted by nearly all scientists and philosophers. So whilst science and maths need a thorough review, we must not 'throw the baby out with the bath water'?

Can you imagine a point in time when there was a future with no past?

The start of time. Time is finite and permanent. Has a start and end. Its possible the start and end are joined to form a circle.

How is "eternal existence" different from your assumption of a permanent base reality?

The difference is:

- Eternal existence implies everything has existed for ever within time. Implies time has no start. Implies Actual Infinity. Implies all the paradoxes I listed in the other OP.
- Permanent existence outside of time does not require Actual Infinity.
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Maths should not reflect whats impossible.

I feel maths is a world in itself. It doesn't have any constraints except for those related to logic.

Mathematicians create universes with different axioms and then study them logically. If such a creation finds practical application in the world then well and good but this isn't a necessity.

Strangely, it's more a rule than an exception that mathematical theories have actual real world applications. I don't know if infinity has practical applications but surely it is interesting to realize we can analyze it in an understandable way through set theory or whatever else it is.
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don't know if infinity has practical applications but surely it is interesting to realize we can analyze it in an understandable way through set theory or whatever else it is.

Potential infinity has many practical applications, actual infinity has none.

All theories that use actual infinity are wrong IMO.
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I mentioned already that time can mean two things, the measurement events between each other like a second means a certain number of events of a cesium atom happens, so by nature is relative. this is why special relativity was not that really important because the thought that time was separate from phenomenon was not true. Another meaning of time is the description of events and there occurrence to from one another. The present is what is now, like their is a green fire now and before it was red and after it, it is blue. you can not have an infinite number of events occur after the present for an addition synthesis starting from a point does not ever become infinity. Since all events before the current phenomenon had to be a current phenomenon before becoming one that was before one. since all events in the past are real, they may be represent from the present with the vent before now 1 and the one before that 2 .. . If it was infinite than a point that is equivalent to the amount of even or odd numbered events has to exist. It would be infinity for there are infinite numbers of even and odd numbers. With this there would be an event infinite numbers away from the present and means that an infinite numbers of events occurred to get to now. Which is impossible because an addition synthesis is impossible.
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The start of time. Time is finite and permanent. Has a start and end. Its possible the start and end are joined to form a circle.

s contradictory, to say that something has a start and end and also that it is permanent. If it's circular there is no start or end.

The difference is:

- Eternal existence implies everything has existed for ever within time. Implies time has no start. Implies Actual Infinity. Implies all the paradoxes I listed in the other OP.
- Permanent existence outside of time does not require Actual Infinity.

You're using a different definition of "eternal" again. We were talking about "eternal" in the sense of outside of time. This is clear from you assumption of a "timeless" base reality. So your argument here for a difference is just equivocation.

If we maintain "outside of time" as our definition of "eternal" and equate this with "eternal existence". we have a problem with your stated claim that time co-exists with this eternal existence. Unless you allow for a dualist separation, or some such thing, you have the contradictory properties of "timeless", and also "time", referring to one and the same reality.

Mathematicians create universes with different axioms and then study them logically. If such a creation finds practical application in the world then well and good but this isn't a necessity.

Strangely, it's more a rule than an exception that mathematical theories have actual real world applications. I don't know if infinity has practical applications but surely it is interesting to realize we can analyze it in an understandable way through set theory or whatever else it is.

This is why we need good ontology, metaphysics, to separate the principles, or axioms which are consistent with the true actual reality, from those which simply appear to be such because they are useful.
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s contradictory, to say that something has a start and end and also that it is permanent. If it's circular there is no start or end.

Not if you are thinking 4D space-time; you have to imagine the universe as a static object in 4D space-time. What is the shape/topology of that object? The start was the big bang; the end is the big crunch. The start and end could coincide to form a circle. All the matter brought back together neatly to the start point. Nature abhors macro-discontinuity so it seems quite likely that the space-time dimensions are joined somehow as I've described.

You're using a different definition of "eternal" again. We were talking about "eternal" in the sense of outside of time. This is clear from you assumption of a "timeless" base reality. So your argument here for a difference is just equivocation.

Sorry, to clarify my language:
- Eternal outside of time. I don't object to this. It can be finite (as in a 4D space time object/block).
- Eternal inside of time. I object to; requires Actual Infinity

we have a problem with your stated claim that time co-exists with this eternal existence. Unless you allow for a dualist separation, or some such thing, you have the contradictory properties of "timeless", and also "time", referring to one and the same reality.

What I mean is time exists inside the timeless base reality. So time is a finite 'thing' existing within a timeless, permanent, finite base reality.
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Not if you are thinking 4D space-time; you have to imagine the universe as a static object in 4D space-time.

See why I don't accept 4D space-time? It results in too many contradictions.

What I mean is time exists inside the timeless base reality. So time is a finite 'thing' existing within a timeless, permanent, finite base reality.

Don't you see the contradiction? You posit a timeless thing which has time within it.
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If I continue the mapping a few more spots, the even numbers go to 8, 10. But these numbers will be outside of the set of natural numbers, on the left. Don't you see that with this type of mapping, the set of even numbers will always contain numbers outside the set of natural numbers which it is mapped to, so it is impossible for it to be a subset?

No it doesn't. You're confusing the listing of the mapping with me populating the set. These numbers are already part of the set. When you say "the natural mumbers" you're ready conceding the issue because you're tacitly acknowledging that there's some common property or rule which makes some numbers "natural numbers". Otherwise your use of the term "natural numbers" is just an empty term, in which case you aren't talking about anything. The even numbers are necessarily part of the natural numbers, it's literally just the naturals without the odd numbers, that's a proper subset. The reason you can't acknowledge this is because then you can't defend this ultrafinitist nonsense.

Right, so no matter how you lay it out, if you maintain equal cardinality the set on the right side will always contain numbers which are not contained in the set on the left side. Surely you can acknowledge this. So do you agree with me that it is impossible that the set on the right side is a subset of the set on the left side? If not, why not?

No because no matter what even number shows us we will always get it in the naturals just a few spots down. I've already explained why not. Speaking of the natural numbers and the even numbers is not me creating said sets by extensionally writing out small parts of the set. That's simply to illustrate the pattern. Unless you seriously think understanding a mathematical relationship requires writing out a entire pattern, this response of yours isn't sensible. It's not a real objection.
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When you say "the natural mumbers" you're ready conceding the issue because you're tacitly acknowledging that there's some common property or rule which makes some numbers "natural numbers".

Yes, I agree that there is something common to "the natural numbers", they are the numbers that we use to count things with. They have a common use. However, they are not a whole, in the sense of a unified entity, by the fact that they are infinite. They are "numbers", plural, such that they are by their nature, a multitude, not an individual. An individual is represented by the number "1", "the natural numbers" represents every counting number. "The natural numbers" cannot refer to an individual, to avoid contradiction, because "1" is what refers to an individual.

The even numbers are necessarily part of the natural numbers, it's literally just the naturals without the odd numbers, that's a proper subset.

I do not agree that the even numbers are a "part" of the natural numbers because I don't agree that the natural numbers are a unity, a whole. That status is reserved for "1". We must avoid contradiction.. As I said, the natural numbers are a multitude. They must be in order that we can use them to count a multitude of things. Each one, by its very nature, is necessarily different from each other one, and the difference between each of one them is necessarily the same difference, according to the simple formulae of arithmetic. One such formula is that an even number, is divisible by two, such that every second natural number is also an even number. This doesn't make the even numbers a "part" of the natural numbers, It just tells us which of the natural numbers can also be called even numbers. Some of the natural numbers are also even numbers. This is very basic arithmetic from grade school, remember?

No because no matter what even number shows us we will always get it in the naturals just a few spots down. I've already explained why not. Speaking of the natural numbers and the even numbers is not me creating said sets by extensionally writing out small parts of the set. That's simply to illustrate the pattern. Unless you seriously think understanding a mathematical relationship requires writing out a entire pattern, this response of yours isn't sensible. It's not a real objection.

What you're not doing is showing how the natural numbers can be a "set". That's what I dispute, that the natural numbers may be infinite and also a set, by way of contradiction. You assume that the natural numbers are a set, but that's begging the question. I want to see your reasoning, the logic behind this principle which appears to me to be contradictory.
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Do you believe there can be an actual infinite?

The way the question is phrased invites and argument about the reality of numbers. Essentially it would be the old nominalism debate about particulars and universals, or the physical and the abstract. Or the Platonic debate about the reality of “forms”

It also invites a debate about the relationship between mathematical models and reality. Certainly many of our mathematical models when pushed to the limits give infinity as a result. In general this occurs at the boundary conditions and we engage in manipulations (normalization, etc.) to give useable results. In my view mathematical models are not reality but approximations to reality and the infinities implied in the equations are proof of not infinities in reality but of the limits of mathematical models. Mistaking models for the real is the “fallacy of misplaced concreteness” which seems rather common.

Are “mathematical infinities”, “real” or “actual” is a different question than is space-time infinite and eternal.

Is space-time infinite and eternal?

I mean our best information is there is little meaning to space-time before the “big bang”. That is a horizon beyond which we cannot see and about which discussion is meaningless. Time cannot be meaningfully abstracted from the change by which we measure it. The space-time field is full of quantum fluctuations “foam”, virtual particles appearing and disappearing for change “time” is fundamental to space and they cannot be separated. Space-time is neither eternal nor infinite on the large scale.

Is space-time continuous and therefore infinitely divisible?

On the small scale reality is not continuously or infinitely divisible in any meaningful manner. As we examine the small at the ultimate scale we encounter the quantum world were continuity breaks down and not all positions orbits or values are allowed, theoretical determinism becomes statistical indeterminism, probability clouds give values when measured or interacted with. Our best efforts at unified field theory involve symmetry breaking, limits and constraints not an infinitely divisible continuum. The mathematical models of the very large and the very small are not compatible and thus once again mathematical models are not “the real” only approximations to the “real” which only remain valid within certain scales, limits or constraints. Currently quantum gravity theories would appear to our best chance at unifying gravity with the other three fundamental forces. The implication of quantum gravity would be to reject the continuous divisibility of the space-time continuum.
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This is why we need good ontology, metaphysics, to separate the principles, or axioms which are consistent with the true actual reality, from those which simply appear to be such because they are useful.

I don't know if we have a fix on what reality is. Usefulness and truth may be related. Things are useful because they're real or very close to it in some way.
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