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. — Devans99

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... — Devans99

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. — MindForged

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). — MindForged

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?

You might have some kind of superpower. I would check into. You could be investing successfully or winning the lottery before you use your money! — Marchesk

I've got a better idea, spend millions and then win the lottery.

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. — BB100

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. — BB100

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.

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. — Devans99

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 — Devans99

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). — Devans99

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.

Given that existence is more-or-less well defined and well understood, the better question is, what must God be, to be an existing thing? — tim wood

I think that existence is well defined and well understood, is far from the truth.

What I mean is you can divide continuous time to an infinite degree, so it can represent an infinite number of states, which equates to infinite information content. — Devans99

What I mean, is that you're not really dividing the time itself. We say that there are sixty minutes in an hour, sixty seconds in a minute, and so on, but time passes in the exact same way, whether you are counting minutes or seconds, because you are not really dividing the time, you are just counting it, and you may count fast or slow.

If you imagine a system evolving through an infinite number of states over a finite period of time... — Devans99

Actually, I can't imagine such a thing, it appears to be impossible. 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 next. Since there is a finite period of time, there could only be a finite number of changes between states, and therefore a finite number of states.

The Continuum can be modelled by the real numbers between 0 and 1. — Devans99

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? This would make it into a discrete unit.

The Continuum for 1 second of time is identical to the Continuum for 1 year of time in that they are both described by the reals between 0 and 1. So 1 second and 1 year have the same information content. Hence the contradiction. Hence time should be discrete. — Devans99

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.

In contrast, a discrete second of time can be modelled with the natural numbers between 0 and some finite N. Then 1 second contains N possible states, but 1 year contains N*60*60*24*365 possible states; hence no contraction for discrete time. — Devans99

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.

Metaphysician Undercover, that which you say passes means your comparing one event with another which is what a measurement is, to compare like with other like. — BB100

No, I'm not talking about comparing events, I'm talking about simply experiencing time passing. Have you ever sat and listened to music, or watched the sunset? Or maybe you like to meditate? It is not a matter of comparing events, nor describing events, it's just a matter of enjoying the wondrous "passing of time"

This means that any change is just a happening that when observed appears to be from this than there are points in between. But that would mean another event, not one that is part of this. — BB100

Suppose there are "points" in between passing time. How would this means that there are other events corresponding to these points? The points would be between the events, so there wouldn't be any time occurring at the points, nor events occurring at the points.

But the continuous is by definition infinitely divisible: — Devans99

That's contradictory, isn't it? If it were divided, it would not be continuous. The points and divisions exist in principle only. They are in the representation of time, the model of time, not in time itself. If time itself were thus divided it would not be continuous, it would be discrete. To avoid contradiction we could assume that time is discrete, but this is not how time appears to us, it appears to be continuous.

There can be two meanings of time, the measurement of events relative to others, and the description of each event in their order. — BB100

What about the thing which passes, as I sit here. That's how I understand time, not as a measurement of events, nor as a description of events, but what passes, and allows for events to occur.

I was wondering about that: If time is truly continuous then a 1 second interval is graduated as finely as a 1 hour interval (implicit from the definition of continuous). That seems contradictory by itself: suggests the short interval contains as many distinct states (therefore information) as the long interval... — Devans99

That's the consequence of assuming infinite divisibility of the continuous. A 1 second interval is infinitely divisible, as is a 1 hour interval. It's the same issue as the real numbers.

The contradiction is in the assumption that the continuous is divisible. If you can really divide it, then it is not continuous, as per the divisions. If it is really continuous then you can't really divide it as that would make it discontinuous. So there is a separation between the continuous thing, and the divisions which we assign to it.

The continuous thing, being time as it exists passing in the world, is not really separated by those divisions, which we assign to it. The divisions, a second, an hour, etc., are within our descriptions, not within the continuous thing, making a categorical separation between the two, such that we are not really dividing the thing. The divisions are conceptual only, used to facilitate understanding of the thing described, like coordinates of a map.

The present is the last phenomenon that just occurred and if we use time by just relative to what happened that the present is the instaneous point where A just occurred. — BB100

You are creating an artificial discontinuity. There is no real point when A ends. If A, B, and C, are a series of events, there is continuity between them such that any ending of A and beginning of B is a function of the description. We describe things as one event ending, and the next beginning, but in reality there is continuity between them such that the point where one ends and the next begins is arbitrary.

If you start counting from the end of event A, your count is completely arbitrary. You are not counting anything real, you are counting properties of your description. If you cannot demonstrate that such "points" are real, there is nothing to indicate that your count is nothing more than fiction. If time is continuous, as it appears to be, then the past is just one big event. If that event continues at the present, then you cannot put an arbitrary end to it, at the present, because this is a false representation.

Simple visual is A is the present and events after it are called A1, A2, A3 ... and so on. There can not be an infinite number of events after A because addition synthesis does not lead to infinity — BB100

Again, your representation of the present, as a point, is not supported by any firm ontology, as the passing of time is considered to be continuous. So any representations, or conclusions, you derive from this are meaningless fiction.

I think you're missing something here. There cannot be an event #1, because as soon as it occurs it's in the past, replaced by another event. This jeopardizes your premise "all events in that past must have been the present at one point."

Any event takes time, so by the end of the event, the beginning is already in the past. This means that any event is divisible. We can divide it into the part which has already occurred, and the part which has not yet occurred. But if we allow that there is a part which is occurring, then this itself is divisible into the part occurred and the part not yet occurred. So if the part which is occurring, is the present, the present cannot be a point because the occurrence itself, what is occurring, is what is at the present, and this is always divisible, unlike the point. And if we divide it as if at a point, then part is in the future, and part is in the past but none of the event is at the present, which is just a point.

the right of the people to keep and bear arms, for the common defense, shall not be infringed". — BB100

How would you interpret "for the common defence"?

There is a set X having the property that ∅ is an element of X, and whenever x is an element of X, then x∪{x} is also an element of X.

This is a very precise formulation which one can show yields a set which is not finite (hence infinite):

As ∅ is in X, then ∅∪{∅}={∅} is an element of X.
As {∅} is in X, then {∅}∪{{∅}}={∅,{∅}} is in X.
As {∅,{∅}} is in X, then {∅,{∅}}∪{{∅,{∅}}}={∅,{∅},{∅,{∅}}} is in X.
...
You see that these elements of X get larger and larger without (finite) bound, and so it stands to reason that such an X must be infinite. — tim wood

I don't see how this proves that n infinite set is possible. It seems to assume that an infinite set is possible, "begs the question". Could you provide a rendition in English?

This is what happens when you don't realize that Even numbers exist, are a proper subset of the naturals, and are provably the same size as the naturals.

0 - 0
1 - 2
2 - 4
3 - 6

If the even numbers (those on the right side) are smaller (as you say proper subsets "clearly are") then point out exactly when the even numbers fail to give a number to match to the naturals. If you can't do that (which you can't) then the only way you can continue is by ignoring the definitions used. So I'm just not bothering anymore. — MindForged

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.

As an ontological principle, it demands a preferred frame. Without that, two events cannot be actually simultaneous. TOR does not assert that preferred frame, so it makes no such ontological assertion. — noAxioms

Yes, I believe that's correct But the truth or falsity of special relativity depends on the truth or falsity of the relativity of simultaneity, which appears to follow necessarily, as a logical conclusion from the stipulation that the speed of light is constant in different frames of reference . We can avoid having to verify the ontological truth or falsity of the relativity of simultaneity by assuming the truth or falsity of the constant speed of light.

Also, while it does describe relative simultaneity, but it doesn't rest on that. It is a conclusion that follows from the constant speed of light measured against any frame. — noAxioms

The problem here is that the available evidence indicates that the speed of light is not constant against any frame of reference. So we have for example, what is known as the expansion of the universe, which is a type of motion of objects, that cannot be called "motion" because it doesn't fit into the confines of special relativity. Therefore we cannot say that the speed of light is constant against any frame, because in some frames such an assumption leaves us with evidence of a further motion which is unaccounted for.

Since this principle, the constancy of the speed of light, is called into question, consequently the relativity of simultaneity is called into question. Our ability to determine the speed of light in many different frames is extremely limited. We need principles to relate one frame to another, and if these principles assume that the speed of light is constant, we're just begging the question. If problems, such as the expansion of the universe, are exposed, then we can conclude that the principles by which we relate one frame to another are inadequate. Therefore we ought to go right back to the ontological principle, simultaneity, and develop an understanding of exactly what it means to be simultaneous. This will give us a better position to determine "the speed of light" with more accuracy, allowing a variable speed of light depending on the frame of reference, determined by a proper understanding of simultaneous.

Agree, but theory of relativity is not an ontological principle. — noAxioms

One key principle upon which special theory of relativity rests, is the relativity of simultaneity. As I explained above, it is an ontological principle. It is a claim about "being", what is. That this principle cannot be falsified results in your claim that one cannot do a local experiment to determine the preferred frame of reference.

If you insist. Seems to put your presentism on shaky ground then, if relativity contradicts it. It requires you to reject it. Seems harsh. — noAxioms

I never said I support presentism. I'm just following some points discussed in the thread.

Well, I stand on the eternalism side of that fence, so it would not bother me to see presentism be contradictory like that, but since I cannot think of a single falsification test for it, I suspect your analysis is in error. — noAxioms

It's not that presentism itself is contradictory, but to hold special relativity and presentist principles, both together, would be contradictory. If I were presentist I would reject special relativity. But I'm not presentist by any standard definition of presentism, so if I reject special relativity it's for reasons other than those of presentism.

If they're incompatible, there must be some test that falsifies one or the other. — noAxioms

In general, competing ontological principles are incompatible and there is no easy test to falsify one or the other. That's why we tend to hold different metaphysical principles and we are usually incapable of convincing another to change one's metaphysics.

We can make more progress in analysing God if we assume he's constrained by materialistic rules. — Devans99

What would be the point in that? We'd be analyzing a concept of "God" which is inconsistent with the concept of "God" held by theologians. Therefore we wouldn't really be analyzing the concept of God.

The bible says god is eternal but apparently does not clarify which meaning. — Devans99

I think the only way to understand "eternal' in relation to God, is the second way, outside of time. Material things are often said, in theology to be temporal. God is said to be immaterial, and therefore outside of time.

I think what you're described here is an inconsistent set of assertions. You describe an assertion of "the present" like there is one of them, and then go on to describe other different presents, which means there is more than one. That is inconsistent, unless I'm interpreting your words wrong. — noAxioms

Obviously that's not what I meant to imply, as I went on to talk about a multiplicity. If I approach my lawn with the attitude that there is such a thing as "the blade of grass" this ought not imply to you, that I am only talking about one blade of grass.

For what I said to be incorrect, it would similarly need to be self-inconsistent. — noAxioms

No, it is an incorrect interpretation of special relativity. Special relativity cannot be interpreted as allowing for only one present. Either you take the standard interpretation that there is no such thing as the present (eternalism), or if you approach with the assumption that there must be such a thing as "the present", then you find a multiplicity of presents. Likewise if you approach the lawn with the assumption that there must be such a thing as "the blade of grass", you will find that there is a multiplicity of blades of grass.

It uses the word 'ontological'? That would be news to me. That would indeed be a metaphyscial statement. — noAxioms

No, Einstein does not use the word "ontological", but he says that two events which are simultaneous from one frame of reference are not simultaneous from another. Therefore he makes a statement concerning being, what is and is not, existence, and that's an ontological statement.

SR has no concept of 'now at the present'. — noAxioms

Right, that's why the standard interpretation is that of eternalism, no "now".

No. If there is a present, there is probably only one of them, and a frame that does not correspond to it is simply not the preferred frame. Moments that appear simultaneous in the other frames are not really simultaneous. — noAxioms

That is incorrect. If you interpret special relativity with the assumption that there is such a thing as "the present", then "the present" is necessarily specific to the frame of reference. This means that there are multiple "presents" according to multiple frames of reference. Furthermore, events which are simultaneous in other frames, are really simultaneous in those frames. To say this, what you say here, that moments which appear to be simultaneous in other frames "are not really simultaneous" is to violate special relativity which stipulates that they really are simultaneous, according to the frame of reference.

No, it doesn't exclude a preferred frame. It just says you can't do a local experiment to detect it, if there is one. — noAxioms

Again this is the same incorrect assumption. Special relativity, like all relativity theories, dictates that there is no ontologically "correct" frame. If there is a "preferred frame", it is preferred for other reasons. That the correct frame cannot be detected is consequent to the stipulation that there is no correct frame. It is clearly inconsistent with special relativity to believe that there is a correct frame which cannot be detected.

Yes, multiple approaches (I count three), or interpretations of time, but they are no presents, or one present. No view has multiple presents. — noAxioms

Suppose "now", the present, is the principle which defines what is happening, the events which are occurring, presently. "Simultaneous" implies that numerous events, are occurring at the same time. The different events are necessarily at different places, if they are at the same time. Under the precepts of special relativity, depending on the frame of reference, the events which are occurring simultaneously, now at the present, vary. Therefore the interpretation of now varies according to the frame of reference, such that we can have multiple presents. There is no interpretation of special relativity which renders "one present", because this implies a preferred frame of reference, which is strictly excluded by special relativity.

They both work, but the present view demands a preferred foliation (which typically corresponds to an inertial frame only locally) and only one frame, coupled with a current event, defines local events that currently exist. — noAxioms

Special relativity specifically disallows a preferred frame of reference, that is the fundamental principle of "relativity". So any interpretation of time which uses such a preferred foliation is inconsistent with special relativity..

Strictly, the experiment shows that we cannot know if event A caused event B, or B caused A. The meaning of "cause" breaks down here. — Banno

Right, the conceptions of time and space utilized by physicists are inadequate, such that they cannot distinguish the temporal order of such events. Physicist have no standard principles whereby they can get beyond the deficiencies of special relativity, which sees simultaneity as reference dependent. It appears like some physicists might take Einstein's relativity theories as the be all and end all to understanding the relationship between space and time.

5. Relativity suggests the existence of multiple presents, whereas Presentism demands one present
Suggests, yes, but not asserts. It works either way. This point is actually about presentism. — noAxioms

As I understand special relativity, it is stipulated that simultaneity is dependent on the frame of reference. There appears to be two approaches to this assertion. One is that there are multiple presents, depending on frame of reference, the other that there is no such thing as the present. So I do not see how you can make special relativity consistent with presentism.

Ask Banno, the master of large post counts.

Incidentally, it's the threads which have a lot of wandering and nonsense, which get the large post counts.

I do not know how to define a good topic. It seems to be the case that [what seem to be] good topics presented properly can be derailed by posters who wander off into the weeds. Whether it is the posters or the topic that is the problem could be tested by posting a splendidly well-structured OP, then allowing posters from Group A, who have a record of responding to posts in an on-point, orderly manner to respond in one thread. Group B, who have a record of responding to posts in a fashion that disrupts orderly discussion would be allowed to respond to the topic in a second thread. — Bitter Crank

How would you distinguish members of group A from members of Group B? I think posters such as myself, stay on topic when the topic is sufficiently interesting to us. But when we are bored, like I am right now, we'll partake in threads which are lacklustre and of little interest. Then we'll wander off into the weeds.

What!?? I'm right! You're wrong! is not the proper approach?

So what are you selling ricky?

MindForged's problem is in the assumption that there is such a thing as an infinite set.

A set is infinite if it's members can put into a one-to-one correspondence with a proper subset of itself. So we know the natural numbers are infinite because, for example, there's a function from a set to a proper subset (read: non-identical) of itself like the even numbers. For every natural number, you're always able to pair it up with an even number and there's no point at which one of the subset cannot be supplied to pair off with the members of the set of naturals. — MindForged

Clearly, for any set of natural numbers, a proper subset is always smaller. That is always the case, and there is never an exception. In order for the proper subset to have an equal cardinality, it must either be the original set, or contain numbers which are outside the original set. Then it is not a proper subset. Therefore an infinite set, under that definition is impossible.

So for example, the set of even numbers must contain numbers outside the set of natural numbers in order for it to have an equal cardinality. Counting by twos requires that you count twice as high as counting by ones, in order to have the same number of members in your set. But this set of even numbers, which has numbers higher than the set of natural numbers, in order to have an equal cardinality, is not a subset of that set of natural numbers.

You need to realise that you were told the wrong things about infinity at school and free your mind of Cantor’s muddled dogma. — Devans99

Exactly, with smoke and mirrors Cantor created the illusion of coherency, but he was really a master of deception. A mathematical magician is nonetheless, a magician. We need to see through the smoke and mirrors to root out the contradictions which lie within his fundamental assertions.

As each is infinite they cannot be consider as "sets", because "set" implies finitude. We went through all this days ago, about the time you left the discussion in disgust. I hope you do so again, because you seem to have no serious input in this matter

To call either set infinite is to presuppose you know what exactly you mean by "infinite" in context. — tim wood

Have you read any of my posts? I insist that it is contradictory to say that a set is infinite. So no, I am not saying any set is infinite.


Here are the two distinct reasons why the natural numbers and the real numbers are considered to be infinite. In the case of the natural numbers, we can start counting, one, two, three, four, etc,, and never reach completion. Therefore we say that the natural numbers are infinite because they can never be completed. In the case of the real numbers we cannot even start to count, because if we start at one we have already missed an infinity of numbers. And no matter what number we put as the first number after zero, we have already missed an infinity of numbers. Therefore we say that the reals are infinite because we cannot even start to count them.

So the two, the naturals, and the reals, are both said to be infinite because they are uncountable. One is uncountable because we can never reach the end to counting them, the other because there can be no beginning to counting them. Yes, they are different infinities, for this very reason, but it is incorrect to say that one is larger than the other because they are both uncountable and therefore both immeasurable.

You did not give any counterargument here that the real between zero and one are either finite or unbounded. I gave an argument for why it was both, and thus why something can be finite and bounded. — MindForged

It is the number of intervals between zero and one which is unbounded and infinite. You gave no argument that this number is bounded. You have stated an arbitrary boundary of zero and one, but this does not bound the infinite. You could have set your boundaries as 10 and 20, or 200 and 600, or zero and the highest natural number. These boundaries do not bound the infinite itself. So you have provided no argument that the thing which is infinite, the numbers between the boundaries, is bounded. There is an unbounded number of possible places between any two designated real numbers

The "thing" which is infinite is the number of reals, the thing which is bounded is the number of reals. — MindForged

No, clearly the number of reals is not bounded, so get your facts straight. By no means is that number bounded. You appear to be confusing the symbols, the numerals 0 and 1, with the numbers which are assumed to lie between them. The "thing" which is infinite is the number of real numbers between 0 and 1,and this is not bounded, just like the number of reals between 2 and 1000, or whateve,isnot bounded. The number of reals is in no way bounded, just like the number of naturals is in no way bounded. The boundaries are in the definitions by which they are produced, but the definitions are made such that the numbers themselves are not bounded. The two systems, the naturals and the reals, are just two distinct ways of expressing the same infinite numbers. Remember, separate the numerals (as part of the description) from the numbers which are signified by the description. The description, "reals between 0 and 1" signifies infinite numbers without boundary.

If you ignore the last 150 years of math you can believe this, but Cantor's diagonal argument is pretty clearly proof of this. — MindForged

There is a real problem with this so-called proof. It's called begging the question. By assuming that the natural numbers are a countable "set", it is implied that the naturals are not infinite. It is impossible to count that which is infinite. By definition, that which is infinite is uncountable, and that's why I've argued that the natural numbers cannot be a set. What Cantor needed to do was prove that the naturals are a set. But this would be impossible, because as I've explained to you (many times, in many ways) "infinite set" is self-contradictory.

So, as I've explained already, to say that the natural numbers are infinite, and to say that they are a countable set, is contradictory. Therefore we must give up one or the other. If we accept Cantor's proof, then we must accept Cantor's premise, which denies the infinity of the natural numbers. We cannot have both, the infinity of the natural numbers, and Cantor's proof, because Cantor assumes the finitude of the natural numbers as a countable set.

Now, as I explained to andrewk, there is good reason to maintain the infinity of the natural numbers, because this allows that every object is countable. This allows us to measure every object. But if we allow that the natural numbers are an object (set), then all we have done is created an object which cannot be counted (measured), because despite the fact that we might claim that the natural numbers are countable, they remain uncountable. So Cantor has created an object, the countable set of natural numbers, which cannot be counted, or measured in any way, because it is a fictitious object, because the natural numbers are really infinite and cannot be counted, nor can they be an object. That is why set theory ought to be dismissed so that we can go back to a true infinity of natural numbers, and allow that every object may be counted and measured, instead of allowing the existence of objects which cannot be counted or measured, this renders the world as unintelligible.

It's not an assumption if you can prove it. Seriously, assume there is some limit to how many reals there are between any two naturals. A simple expansion can be done to yield a new natural. Ergo on pain of contradiction the initial supposition must be false. There is no smallest real. — MindForged

I agree, but as I explained, the thing which is infinite is not the same thing as the thing which is bounded. Therefore the limits expressed are irrelevant to the infinity expressed, and the infinity is unbounded. Therefore your argument that there can be a bounded infinity is not sound.

The set of naturals has a smaller cardinality than the reals; the former is countable and the latter is uncountable ("unlistable" is probably a better word). So the naturals are bounded, we know numbers which are larger than it so there's a very obvious boundary: — MindForged

I don't believe this. Both the naturals and the reals are infinite, so I believe it is false to say that one is larger than the other. This is where I believe that set theory misleads you with a false premise. I would need some evidence, a demonstration of proof, before I would accept this, what I presently believe to be false. Show me for example, that there are more numbers between 1 and 2, and between 2 and 3, than there are natural numbers. The natural numbers are infinite. So no matter how many real numbers you claim that there are, they will always be countable by the natural numbers.

This is what I've been telling you over and over again. To stipulate that the cardinality of the natural numbers is less than something else, and to also say that the natural numbers are infinite, is contradictory. To say that the number of something is infinite and that there is less of these than something else, is blatant contradiction. If you truly believe (as you appear to), that the natural numbers are infinite, yet there are more real numbers than natural numbers, then you ought to be able to show me the limits, restrictions which have been placed on the natural numbers to allow that there are more reals than naturals. After showing me these limits, explain to me how the natural numbers can be limited in this way and still be infinite.

A set is defined by said rule applying to the objects in question. It has nothing to do with the process of collecting things. The properties in question are possessed by those objects whether or not I accept they do or if I call it something else. Being in the set of African Americans doesn't depend on anything to do what I call them. It doesn't necessitate an essential meaning, just a conventional one which people roughly agree picks out a certain class of objects. — MindForged

You're just repeating your affirmation without answering to my objection. Isn't the act of applying a rule to objects, and determining whether or not the rule applies to them, an act of collecting the ones which the rule applies to? How could a rule be applied to an object without someone applying it to that object? As much as you might insist that "a set is defined by said rule applying to the objects in question", a rule does not apply itself to an object, it must be applied. Do you not agree that someone must apply the rule, and this involves the process of interpreting it, and judging the objects, as I described in my last post? You can't just declare that all red things are members of the set of red things, and expect therefore there is a set of red things, because someone must interpret and define "red", and judge which things fulfil the conditions in order to produce that set.

Being in the set of African Americans doesn't depend on anything to do what I call them. It doesn't necessitate an essential meaning, just a conventional one which people roughly agree picks out a certain class of objects. — MindForged

This is nonsense. What about all those people of mixed ancestry whom some would say are African American, and others would say are not. There is no "set of African Americans" unless "African Americans" has an essential meaning by which every individual might be judged. Otherwise there'd be a number of individuals who may or may not be judged to be members of the set, and the so-called set would not be a set at all due to indefiniteness.

I've just answered this. — MindForged

No, you surely did not answer this. You've gone from saying that there could be a collection without an act of collecting, to saying that a set exists as a set because you stipulate that it does, to now saying that a rule applies to an object without actually being applied to that object. Each time, you claim that something comes into existence, (a collection, a set, the applying of rules to objects), without the act which is implied by the descriptive terms.

Good luck doing that without the rigorous mathematical understanding of infinity as opposed to the vague colloquial understanding. — MindForged

"Rigorous mathematical understanding of infinity". Lol. But if your not joking, you have my sympathy.

The set of reals between 0 and 1 is provably infinite, and clearly bounded. After all, every element in that infinite set is larger than 0 and yet smaller than 1; — MindForged

Does anyone even know what it means to be larger then zero? .So let's leave zero out of this. That there is an infinity of real numbers between any two real numbers is the assumption of infinite divisibility. The possibility for division is assumed to extend infinitely, just like the possibility for adding another natural number is assumed to extend infinitely. That the thing being divided is bounded, is irrelevant to the infinity which involves the act of dividing. So the infinite thing itself, divisibility, is not bounded. Likewise, in the case of the natural numbers, that the one unity being added at each increment of increase is bounded and indivisible, is irrelevant to the infinity which involves the act of increase. That the increasable amount is bounded, restricted to exclude fractions, is not a limit to the infinity itself. Nor is the fact that a divisible unit is bounded a limit or restriction to divisibility.

But whether or not sets are bounded or not really has nothing to do with infinity. A set whose members are ever increasing due to some iterative calculation is clearly unbounded, but it's not infinite. Just loop a program which adds new members to an array every iteration; at every iteration the number of members of the array are obviously going to be finite. — MindForged

That's incorrect. Whether or not something is infinite has everything to do with whether or not it is unbounded, because "infinite" is defined as unbounded. Where is your rigorous understanding of infinitiy? And no, an iterative calculation is not unbounded. It is limited by the physical conditions, and the capacity of the thing performing the iteration. That it is so bounded is the reason why it is not infinite.

Not sure whether to be amused or horrified that the leader of the most powerful country in the world has the mental age and emotional maturity of a 12-year-old, and is literally being laughed off the world stage. And that a huge swathe of Americans are just fine with that because, apparently, the only job in the country where no standards of competency at all apply is the Presidency. — Baden

It's a very simple symptom, we value entertainment higher than we value good leadership. And an incompetent leader provides a high quality entertainment. But those who laugh at the leader are not really allowing themselves to be led, they're just being entertained. So the entertaining leader is not really the leader. Who, or what, do you think real leads the most powerful country in the world?

It could have to do with immaterial causality which is known mystically through direct experience (eg miracles), but this is in a sense beyond the confines of regular philosophy and should be treated more phenomenologically. — Nasir Shuja

How is the topic of immaterial causality beyond regular philosophy? Isn't this the essence of free will?

gave a rule that populates members of a set, I do not literally gather abstract objects and place them somewhere. — MindForged

OK, but the question was, how does your giving a rule populate a set? Do you apprehend the issue. Suppose I decree, as you suggest, that all red things are members of a set, the set of red things. How does this declaration make certain things members of that set, while excluding other things? What if there are some things which I would say are red but someone else would say are orange? Is it the case that since I am the one who declared the set, these things are members of the set, because I think that they are red? Don't we need an official definition of "red", an essential meaning of the word? Otherwise my declaration that all red things are members of the set of red things is meaningless, and cannot populate a set because there is nothing here to indicate what it means to be red, and no one to judge which objects fulfill that condition called being red.

Name some condition which applies to all of them or just create an extensional list of the objects. It's seriously simple. — MindForged

You think that it is simple to name some condition which applies to all of a number of objects? This could only work if the condition which is named had an essential meaning, an official definition, allowing that all the things could be judged according to that definition. Even then, the judgement could be mistaken. So this requires two things, essential meaning and flawless judgement. A capable human being might be able to produce a flawless judgement, but where do we get the essential meaning from? Who determines exactly what it means to be red, such that one might be able to understand this essential meaning, and judge for that condition without making a mistake.

Now imagine a Square that is the smallest Square that is not equal to Zero. This thought sends your mind into an endless recursive loop of the Square getting smaller and smaller and we soon realize that it is impossible to imagine such a smallest Square. One thing we can say is that this Square is Infinitely small but is still a Square. In general mathematics this would be called a differential Square or an infinitesimal Square. — SteveKlinko

In order to consider the smallest possible square, we need some ontological principles, principles of physical existence which would dictate how small such an object could be. Otherwise it's just conceptual and there would be no limit to how small it could be. The same is the case for the largest possible square.

However this doesn't work if we want to have objects in our domain of discourse other than natural numbers, because then we need to add a condition 'x is a natural number' to the above induction axiom, which requires referring to the set of natural numbers, whose existence cannot be asserted without the axiom of infinity, or some equivalent.. — andrewk

I think that's all well and good, because no two objects are exactly the same, and the difference between this object and that object is never the same as the difference between that object and another object, contrary to what is the case with numbers. This provides us with a separation between the "ideal realm" of numbers, in which perfection is the ruling principle, and the physical realm of objects, in which the uniqueness of the particular is the ruling principle. Therefore we could have a principle whereby what is true of numbers is not necessarily true of objects.

Consider that the axiom which dictates that the natural numbers are infinite, is very good and useful, because it allows us to count anything and everything. It is very well designed (if "designed" is the right word, because I don't know how it came about, it's just kind of intuitive) because no matter how many sets of things, or individual things we encounter, the numbers can always go higher than the number of things, allowing us to count more things as we encounter them, because numbers are infinite and things are finite. However, if we allow that the numbers themselves are things, as set theory implies, then we have encountered a type of thing which cannot be counted. If we try to count the numbers, there's always more, and we're thwarted. The consequence of this axiom of set theory therefore, which implies that the numbers are objects, is to negate the good and usefulness of the intuitive axiom which stipulates that the natural numbers are infinite. When it is allowed, as a principle in your axiomatic system, that there are objects which are infinite, you lose the capacity to measure all objects, which is what the axiom that the natural numbers are infinite gives us.

I think this issue of the axiom of infinity may be related to that of omega-completeness, which is about whether there may be natural numbers other than those we get by adding 1 to 0 a finite number of times, ie 'non-standard' natural numbers. Omega-completeness is a very interesting subject, but it usually gets my head all muddled when I try to think about it, if I haven't done so in recent times. — andrewk

It appears like the question of Omega-completeness is just an issue of whether the natural numbers ought to be consider infinite or not. As I said above, I think there is a very good reason to allow infinity for the natural numbers. Where I find a problem is in the idea that the natural numbers are objects, which is the idea that set theory builds on. This causes us to get muddled by the idea of infinite objects, and these are inherently unintelligible. One solution may be that of Omega-completeness, denying that the natural numbers are infinite/ But I don't think that's the best solution.

That is not the mathematical definition of a set. The mathematical definition of a set is that it obeys the axioms of the set theory in which we are working. The most commonly-used set of axioms is Zermelo-Frankel - ZF. The concept of 'collection' does not form part of those axioms.

But even if we were to try to use the definition you suggest, it would be incorrect to say that infinite sets are not well-defined. In mathematics the words 'well-defined' have a very specific meaning, and they only apply to functions, not properties (aka relations). We say that a function is 'well-defined' if, using the definition to apply it to an element of its domain, there is a unique object that is the image of that element under that function. The notion of being a set, or of having finite cardinality, is a property, not a function, so the notion of 'well-defined' is not relevant. — andrewk

Thanks for the information. I know that mathematical definitions are called axioms. The problem that I have been trying to shed light on, is that some axioms contradict other axioms, so that within the field of mathematics in general, there are contradictory axioms. To me, this indicates that the principles upon which these axioms are founded, are not well understood. This is most noticeable in concepts like "infinity", and "zero", but in some cases in modern physics I've noticed that it extends into geometry and dimensionality as well. One case discussed here already at tpf is Euclid's parallel postulate.

If you really dislike the concept of infinity, all you need do is reject the 'Axiom of Infinity', which asserts the existence of a set that can be thought of as the set of natural numbers. Without such an axiom, we can have natural numbers as large as we wish, but there is no such thing as the set of all natural numbers. Such an approach to mathematics is consistent, and some people try to limit themselves to that. The trouble is that it is that axiom that gives us the tool of Proof by Mathematical Induction. Without it, there is an enormous volume of important results that we would not be able to us. — andrewk

Yes, the easy way out would be to reject the axiom of infinity. But to reject an accepted convention without good reason, is generally considered as being irrational. That's why we ought not simply reject the axiom unless it has been demonstrated to be unacceptable.

I do not agree with you about the relation between the axiom of infinity and mathematical induction though. All that mathematical induction requires is that what is true of one number is true of the next, and therefore true of all the following numbers. It doesn't require that the numbers are infinite, because it works also in a descending order, in which there is an end, a lowest number. That the natural numbers are infinite is a separate axiom, unrelated to mathematical induction. We can have an axiom which states that the natural numbers are infinite, without allowing that the natural numbers are a set.

As you point out, we lose quite a lot of mathematics by dropping the Axiom of Infinity. And to me the foregoing arguments for doing so are ridiculous since the claim is contradictions occur with the axiom (they don't). Even Ultrafinitists don't claim that provable contradictions appear, so the justification for dropping the Axiom just looks like philosophical bias more than anything. — MindForged

I am not inclined to drop the idea that the natural numbers are infinite, only the idea that the infinite natural numbers are a set.

As andrewk indicates, if your axiom states that a set may be finite or infinite, then that is what is the case in that axiomatic system. The problem that I see, is that the way "set" is used by mathematicians, as a closed, bounded object, the possibility of an infinite set is precluded. Sets are manipulated by mathematicians, as bounded objects, but an infinite set is not bounded like an object, and therefore cannot be manipulated like an object. This calls into question the understanding of "infinite" which is demonstrated by this axiom of infinity, which stipulates that the infinite natural numbers are a "set".

Far too much rubbish is permitted. — Banno

As the purveyor of a vast quantity of rubbish, you're foolish to call for such restrictions. You might as well request yourself to be banned.