A simplified argument from motion

[freesoulhope]

The quarks (or anything’s) nature can be explained by what caused it.

The nature of space, cannot be explained by a natural cause. It is quite possible that its actual existence can be explained by a natural cause, indeed; but it doesn’t follow from this observation, that the nature of space or the intrinsic nature of a thing, is the result of previous causes—only its being. An infinite regress, doesn’t explain its nature; it just explains why it is there. The thing that makes a space, space, necesarily transcends the natural world as a foundation, by being the reason for it being space and not a pink elephant. Its like saying, space is caused by space; that may be, but why is space, space, rather then a pink elephant. To say " because it is space" is to deny that the nature of space needs explaning; with no bases for thinking so. If the nature of a thing, cannot be explained by the reality of being caused; then the cause of its nature is not apart of the natural process.

 

[freesoulhope]

Again, define your starting point. There is no “beginning” of the set.

It seems to me, that when you consider an infinite universe, your considering it from the present moment (zero). Your contradicting yourself by using the present moment as your measure for assuming an infinite universe. In fact, you can only reach the finite-infinite, (in either direction) by starting from zero; which contradicts the law of cause an effect. Cuase and effect is a “forward-duration”. The present moment(zero), exists only relative to a veiwer; so to count from that moment back into infinity is unjustified, and is not reflective of the world, but is a subjective perception of the veiwer. For a finite-infinite to be possible; the present moment would have to be static, with two lines of time(the past and present) moving away, in each direction, from its centre (zero); which is what you have to visualise, in order to come to the conclusion that the past is infinite; assuming theres no beginning. The point is, the only way you can visualise a finite-infinite, is by counting back from the present moment; otherwise you would have no good reason to think it possible. Theres no otherway to veiw it. In there, lies the flaw; since there is no reliable way to measure it, with out counting from a static present.

If we begin with no assumtions at all, and ignore the present moment(zero); then we see that things move forward in time and needs other factors in order to be in effect. I see no reason from these facts, to assume that the universe is infinite in duration. But i do see a good reason, from analysing theses factors, to make the assumption that such movement is ultimately the result of an unmoved mover; and this unmoved mover is placed, precisely because without such a factor with are left with an illogical universe.

 

[punkforchrist]

Before I forget, it keeps occurring to me that the impossibility of there being an infinitely old universe is really more appropriate for a discussion on the kalam version of the cosmological argument. This is my own fault for not addressing this and conflating the two versions. I hope everyone will bear with me as I try to bring the discussion back to the traditional argument from motion as detailed by Aristotle and Thomas Aquinas.

1. We observe things that are in motion.
2. Whatever is in motion is moved by another.
3. If there were only potential movers, then there would be no motion.
4. Therefore, there must be a Prime Mover, itself unmoved.

We might replace the problem of an infinite regress with (3) above. What are some reasons for accepting or rejecting this premise?

My own understanding is that if we remove what is actual, then we end up taking away the transition from potentiality to actuality. A chicken egg can only become a chicken by undergoing a number of complex processes. These processes are the actualizations that bring about the state of “chicken-actuality” from a state of potentiality (i.e. the egg). However, these processes too were derived from something; otherwise, we have arrived at the conclusion, (4). But when we remove the purely actual, is it possible to have any motion whatsoever?

I know the examples above have been touched upon earlier in this thread, but I wanted to bring the conversation back to the Aristotelian/Thomistic formulation.

 

[john_doran]

Why can’t it? What is the cardinality of the set of all finite negative integers? Or for that matter (not that it’s relevant here, but anyway), what is the cardinality of the set of all real numbers between 0 and 1?

i don’t understand the point you think you’re making here…

the cardinality of the set of negative integers is http://mathworld.wolfram.com/images/equations/Aleph-0/inline3.gif. the cardinality of the set of reals in the unit interval is c (or http://mathworld.wolfram.com/images/equations/Aleph-1/inline1.gif, if the continuum hypothesis is true).

but for any two members of c, there an infinite number of intervening members. which means to get to one from the other would require traversing an infinite number of members. that’s my point.

SeekingCatholic:

Where’s the contradiction, then?

in the idea of forming an actual infinite set by increments.

SeekingCatholic:

Reached here from where? From “the beginning”? There is no beginning.

so your position is that the infinite set of temporal moments was never formed - it is simply given.

then that leads to paradoxes like hilbert’s hotel and tristram shandy, which reduce the idea of an actually infinite set of temporal momehts to absurdity.

 

[SeekingCatholic]

I can’t respond to all these posts, so let me restate the problem, and my questions more specifically. As for the KCA, if you succeed in disproving an infinite-past universe then you automatically succeed in proving the argument from motion, so I don’t find it off-topic.

In a universe with an infinite past, I think we are all agreed that we can represent that past by the set of negative integers, that there is an infinite set of such events, they all occurred, and they are all at a finite distance from the present.

The argument goes, if I understand correctly, that there is an infinite set. The infinite set must have been formed by successive addition. But it’s impossible to form an infinite set by successive addition. Therefore, the universe must be finite.

My questions:

1. Why can there not be an infinity of additions?
2. The additions are proceeding forwards in time, so this argument might apply to a future-infinite universe. But in a hypothetical past-infinite universe, the addition is proceeding in the other direction, e.g., not “towards” the infinity but away from it. IOW, if the universe had a beginning it would still have a beginning even after an infinity of additions.
3. Therefore isn’t the real case here that an infinite-past universe can’t be formed starting from an empty set, because what’s the first member? Once you have a first member, you have a beginning. This is restating the argument about an infinite set of moments needing to be traversed before the present, but if that is the case, the present could never arrive.
4. Obviously, if there is an infinite-past set of moments, they have to be given, therefore, they can’t be formed starting from an empty set. But why is it axiomatic to start with an empty set rather than an infinite past? If the argument is that an infinite past is absurd, why is it absurd? Why specifically is Hilbert’s Hotel necessarily an insoluble paradox here? Sure, you get another temporal moment, which I call “0”, and shift every other member down by 1. But why is that “absurd”? Tristram shandy seems to be related to a infinite future, but not necessarily an infinite past. How, specifically, is that relevant here?

 

[john_doran]

In a universe with an infinite past, I think we are all agreed that we can represent that past by the set of negative integers, that there is an infinite set of such events, they all occurred, and they are all at a finite distance from the present.

no. for any member of the set of past moments, there is a moment that is an infinite distance from it, or else how can the set have the cardinality http://mathworld.wolfram.com/images/equations/Aleph-0/inline3.gif?

SeekingCatholic:

The argument goes, if I understand correctly, that there is an infinite set. The infinite set must have been formed by successive addition. But it’s impossible to form an infinite set by successive addition. Therefore, the universe must be finite.

yes.

SeekingCatholic:

1. Why can there not be an infinity of additions?

an infinite set cannot be formed by successive addition in the sense that a set cannot become or be made inifinite by the successive addition of members; no matter how many members one adds to a finite set, it will always be finite.

the idea of completing an infinite set by successive addition is that the set of past moments becomes infinite with the elapsing of this moment now. which is absurd.

the alternative - that the past has just always been actually infinite, has its own absurdities, seen below.

SeekingCatholic:

1. The additions are proceeding forwards in time, so this argument might apply to a future-infinite universe. But in a hypothetical past-infinite universe, the addition is proceeding in the other direction, e.g., not “towards” the infinity but away from it. IOW, if the universe had a beginning it would still have a beginning even after an infinity of additions.

i’m not sure i understand what you’re getting at here.

SeekingCatholic:

1. Therefore isn’t the real case here that an infinite-past universe can’t be formed starting from an empty set, because what’s the first member? Once you have a first member, you have a beginning. This is restating the argument about an infinite set of moments needing to be traversed before the present, but if that is the case, the present could never arrive.
2. Obviously, if there is an infinite-past set of moments, they have to be given, therefore, they can’t be formed starting from an empty set. But why is it axiomatic to start with an empty set rather than an infinite past? If the argument is that an infinite past is absurd, why is it absurd? Why specifically is Hilbert’s Hotel necessarily an insoluble paradox here? Sure, you get another temporal moment, which I call “0”, and shift every other member down by 1. But why is that “absurd”?

it’s absurd to suggest that there are no more people in Hilbert’s hotel after 15 trillion more people enter, than there was before. just as it is absurd to suggest that no more time has elpased now than had elapsed prior to a point in time 15 trillion years ago.

how could such suggestions not be absurd?

SeekingCatholic:

Tristram shandy seems to be related to a infinite future, but not necessarily an infinite past. How, specifically, is that relevant here?

because if the past is actually infinite, then for any point in the past, tristram shandy will be completing his autobiography. in other words, if the past is actually infinite, then tristram shandy will always be finished writing…which is the very definition of absurdity.

that’s how it’s relevant.

 

[punkforchrist]

it’s absurd to suggest that there are no more people in Hilbert’s hotel after 15 trillion more people enter, than there was before. just as it is absurd to suggest that no more time has elpased now than had elapsed prior to a point in time 15 trillion years ago.

how could such suggestions not be absurd?

I agree, and this is a separate argument in favor of the finitude of the universe’s past. Craig further elaborates, too. Imagine that guests in the infinite hotel begin to check out. In fact, what if all of them check out? Infinity - infinity = 0. Suppose instead that all of the odd-numbered guests check out. Infinity - infinity = infinity. Finally, suppose that all the guests numbered 4 and higher check out. Infinity - infinity = 3. We subtract identical quantities and arrive at contradictory answers! This ought to suggest that infinity exists only in the mind as an idea, and not as something that corresponds to physical reality.

 

[SeekingCatholic]

no. for any member of the set of past moments, there is a moment that is an infinite distance from it, or else how can the set have the cardinality http://mathworld.wolfram.com/images/equations/Aleph-0/inline3.gif?

I define the set “the set of all finite negative integers”. This has the cardinality aleph-null, and every integer is a finite distance from 0, by the definition of “finite integer”. QED.

an infinite set cannot be formed by successive addition in the sense that a set cannot become or be made inifinite by the successive addition of members; no matter how many members one adds to a finite set, it will always be finite.

What about an infinite number of additions? Or, I add the set of all positive integers to the set {0}.

the idea of completing an infinite set by successive addition is that the set of past moments becomes infinite with the elapsing of this moment now. which is absurd.

Yes, I agree. Either is past is infinite or it is not, but that is not changed by anything that happens now. Even an infinite number of additions in the future can’t make the past infinite.

it’s absurd to suggest that there are no more people in Hilbert’s hotel after 15 trillion more people enter, than there was before. just as it is absurd to suggest that no more time has elpased now than had elapsed prior to a point in time 15 trillion years ago.

how could such suggestions not be absurd?

Are there more members in the set of all integers less than 10 compared to the set of all integers less than 0?

Math for finite numbers/sets simply doesn’t apply to infinite ones. Now you can call this “absurd”, but this is just an opinion. It is not a logical proof of impossibility.

because if the past is actually infinite, then for any point in the past, tristram shandy will be completing his autobiography. in other words, if the past is actually infinite, then tristram shandy will always be finished writing…which is the very definition of absurdity.

that’s how it’s relevant.

Only if tristram shandy himself existed from eternity. Easily refuted therefore by stipulating that despite the universe’s eternal existence, no object within the universe exists from eternity but has a beginning and end.

 

[SeekingCatholic]

I agree, and this is a separate argument in favor of the finitude of the universe’s past.

Again, finding something “absurd” is not the same as a logical disproof. What is “absurd” is a matter of opinion. If you have a rigorous, logical disproof I would be delighted to hear it.

Craig further elaborates, too. Imagine that guests in the infinite hotel begin to check out. In fact, what if all of them check out? Infinity - infinity = 0. Suppose instead that all of the odd-numbered guests check out. Infinity - infinity = infinity. Finally, suppose that all the guests numbered 4 and higher check out. Infinity - infinity = 3. We subtract identical quantities and arrive at contradictory answers!

No, all this shows is that infinity - infinity is an undefined quantity. Like zero divided by zero. Just like any number plus infinity = infinity, or any number times zero = zero, so the additive inverse using infinity or the multiplicative inverse using zero are undefined.

This ought to suggest that infinity exists only in the mind as an idea, and not as something that corresponds to physical reality.

Or, it suggests that a finite mind is incapable of really grasping and comprehending a real infinite.

 

[freesoulhope]

Only if tristram shandy himself existed from eternity. Easily refuted therefore by stipulating that despite the universe’s eternal existence, no object within the universe exists from eternity but has a beginning and end.

The universe is its members. The universe is made up of its members. So how can the universe be infinite, but its members be finite? If one cannot reach an actual infinite counting from the present moment into the past, it means that one cannot form an actual infinite from additive numbers. An actual infinite does not exist. A chain of addition and duration can only ever reach a finite number from the present moment in either direction. It doesn’t matter how long the universe is, the number will be finite. You are simply assuming that the universe has come from infinity; you cannot show how it is possible that an infinity of additive numbers have reached this present moment, because in order to do that you have to start from some where; which one cannot do if the Universe has always existed.

 

[punkforchrist]

Again, finding something “absurd” is not the same as a logical disproof. What is “absurd” is a matter of opinion. If you have a rigorous, logical disproof I would be delighted to hear it.

The contradiction lies in the fact that when we subtract identical quantities, we arrive at contradictory answers.

No, all this shows is that infinity - infinity is an undefined quantity. Like zero divided by zero. Just like any number plus infinity = infinity, or any number times zero = zero, so the additive inverse using infinity or the multiplicative inverse using zero are undefined.

When we subtract things like books and hotel guests, do we really believe the answer can be undefined? Further, zero is the absence of a quantity and does not have any inherent being itself.

Or, it suggests that a finite mind is incapable of really grasping and comprehending a real infinite.

I understand what you’re saying, but this can be used for anything. A finite mind is incapable of comprehending a square-circle, too, but that doesn’t mean square-circles might exist.

 

[SeekingCatholic]

The contradiction lies in the fact that when we subtract identical quantities, we arrive at contradictory answers.

But the “quantities” aren’t identical in every respect. The set of all hotel guests is not identical to the set of all even-numbered hotel guests. The set of all hotel guests is not identical to the set of all hotel guests with room numbers greater than 10. All infinities aren’t identical. Thus, there is no contradiction.

When we subtract things like books and hotel guests, do we really believe the answer can be undefined?

This is a mere argument from incredulity, based on the fact that our experience of books and hotel guests is based on a finite number of books written or rooms in a hotel. If we subtract an infinite number of books from an infinite number of books, the answer is undefined. We need to know more about the specific infinities involved.

Further, zero is the absence of a quantity and does not have any inherent being itself.

In abstract mathematics, certain expressions can be undefined (zero divided by zero or infinity minus infinity), which does not in any way mean that the mathematics is wrong.

But of course your claim is that the mathematical concepts of “zero” and “infinity” don’t represent “real” concepts. With respect to “zero”, it’s debatable. I can define a “rectangle” as a “rectangular prism with zero height”. “Absolute zero” in temperature is a real temperature, it can’t be said there is “no” temperature. An object launched at the escape velocity from earth has zero total energy (kinetic plus gravitational binding). A motionless object has zero velocity. I claim infinity must also be real. There are an infinity of points on a line of finite length.

But, in any event, once you want to get into the “real world” you can’t just use “infinity” in an equivocal manner; you need to define “infinity of what” and also “the type of infinity of what”. Just like you can’t use “zero” in an equivocal manner, you need to define “zero of what”.

I understand what you’re saying, but this can be used for anything. A finite mind is incapable of comprehending a square-circle, too, but that doesn’t mean square-circles might exist.

A square circle is a contradiction. Infinities are merely paradoxical. Unless you can come up with an argument why there is an inherent contradiction.

 

[punkforchrist]

But the “quantities” aren’t identical in every respect. The set of all hotel guests is not identical to the set of all even-numbered hotel guests. The set of all hotel guests is not identical to the set of all hotel guests with room numbers greater than 10. All infinities aren’t identical. Thus, there is no contradiction.

All infinite sets of aleph-null can be demonstrated to have a one-to-one correspondence. For example, the set of all positive integers {1, 2, 3, 4 . . .} has a one-to-one correspondence with the set of all odd-numbered positive integers {1, 3, 5, 7 . . .}. If we subtract all of the odd-numbered integers from both sets, then we are left with {2, 4, 6, 8 . . .} and 0 (0 not being part of the original set of positive integers). In this respect, the part is as great as the whole, which contradicts Euclid’s Maxim. We can demonstrate other contradictions by creating similar sets as I detailed earlier.

This is a mere argument from incredulity, based on the fact that our experience of books and hotel guests is based on a finite number of books written or rooms in a hotel. If we subtract an infinite number of books from an infinite number of books, the answer is undefined. We need to know more about the specific infinities involved.

Then how do we know actual infinites exist? We do not observe them, and there is no reason to assume they are true based on analytic or transcendental reasoning.

In abstract mathematics, certain expressions can be undefined (zero divided by zero or infinity minus infinity), which does not in any way mean that the mathematics is wrong.

I’m not saying these concepts aren’t fruitful in an abstract mathematical realm (i.e. I’m not saying that the mathematics is “wrong”), but rather that they do not correspond to the physical world.

. . . “Absolute zero” in temperature is a real temperature, it can’t be said there is “no” temperature. . .

No, I would say exactly that absolute zero is the lack of any temperature, which is verified by the lack of motion among protons.

There are an infinity of points on a line of finite length.

This is only true in the mathematical realm, unless you’re a Platonist.

But, in any event, once you want to get into the “real world” you can’t just use “infinity” in an equivocal manner; you need to define “infinity of what” and also “the type of infinity of what”. Just like you can’t use “zero” in an equivocal manner, you need to define “zero of what”.

I agree, but the one-to-one correspondences demonstrate that the part is as great as the whole.

A square circle is a contradiction. Infinities are merely paradoxical. Unless you can come up with an argument why there is an inherent contradiction.

Please see above.

 

[SeekingCatholic]

All infinite sets of aleph-null can be demonstrated to have a one-to-one correspondence. For example, the set of all positive integers {1, 2, 3, 4 . . .} has a one-to-one correspondence with the set of all odd-numbered positive integers {1, 3, 5, 7 . . .}.

Having a one-to-one correspondence is not the same as “identical”.

If we subtract all of the odd-numbered integers from both sets, then we are left with {2, 4, 6, 8 . . .} and 0 (0 not being part of the original set of positive integers).

You’re not performing the identical operation. You’re subtracting “infinity”, true, but not in the same manner. In other words, you’ve “cheated”, mathematically speaking. When you go to infinity, you have to specify exactly how you take the limits. You can get different answers dependent on different ways of taking the limit. Once you specify how you take your limit of subtracting infinity, the supposed “contradiction” disappears.

In this respect, the part is as great as the whole, which contradicts Euclid’s Maxim. We can demonstrate other contradictions by creating similar sets as I detailed earlier.

These aren’t contradictions. They’re paradoxes. Axioms which are true in finite mathematics aren’t true in dealing with infinities.

Then how do we know actual infinites exist? We do not observe them, and there is no reason to assume they are true based on analytic or transcendental reasoning.

We don’t know this. But you are claiming they are impossible to exist. We don’t know that 10^1000 of anything exists. We have no reason to assume that it does. But it is certainly possible.

I’m not saying these concepts aren’t fruitful in an abstract mathematical realm (i.e. I’m not saying that the mathematics is “wrong”), but rather that they do not correspond to the physical world.

That’s something you’ve got to prove. One could say the imaginary (complex) numbers don’t correspond to the physical world, but they can very well model sinusoidally modulated signals, which is why they are used in electrical engineering.

No, I would say exactly that absolute zero is the lack of any temperature, which is verified by the lack of motion among protons.

No, absolute zero means everything is in the ground state - there can still be “motion” (e.g. a non-zero energy level, an example of which would be a quantum harmonic oscillator).

This is only true in the mathematical realm, unless you’re a Platonist.

And you have a disproof of Platonism? If not, you will have to admit that it is possible that real infinities actually exist.

I agree, but the one-to-one correspondences demonstrate that the part is as great as the whole.

Obviously true when dealing with infinite sets. This is a paradox, not a contradiction.

 

[punkforchrist]

Having a one-to-one correspondence is not the same as “identical”.

They are identical in quantity. Under set theory, there are just as many odd-numbered positive integers as there are all positive integers.

You’re not performing the identical operation. You’re subtracting “infinity”, true, but not in the same manner. In other words, you’ve “cheated”, mathematically speaking.

Subtraction and division are not allowed in Cantorian set theory.

These aren’t contradictions. They’re paradoxes. Axioms which are true in finite mathematics aren’t true in dealing with infinities.

Nothing in reality would prevent us from subtracting or dividing elements, which is another reason set theory does not correspond to the physical world. If actual infinites are going to be an exception, then a reason ought to be offered; otherwise, it is special pleading.

We don’t know this. But you are claiming they are impossible to exist. We don’t know that 10^1000 of anything exists. We have no reason to assume that it does. But it is certainly possible.

10^1000, while enormous, doesn’t involve the kind of paradoxes that infinites do.

No, absolute zero means everything is in the ground state - there can still be “motion” (e.g. a non-zero energy level, an example of which would be a quantum harmonic oscillator).

If absolute zero is being used in the “ground state” sense, then it is not really absolutely zero. The ground state is simply the lowest level of expected energy.

And you have a disproof of Platonism? If not, you will have to admit that it is possible that real infinities actually exist.

Incidentally, the impossibility of actual infinites is used as an argument against mathematical Platonism.

Obviously true when dealing with infinite sets. This is a paradox, not a contradiction.

Could you define how you’re using “paradox”?

 

[john_doran]

I define the set “the set of all finite negative integers”. This has the cardinality aleph-null, and every integer is a finite distance from 0, by the definition of “finite integer”. QED.

that only works if there is a last, finite member of your set. there isn’t, so your proof fails. QED.

SeekingCatholic:

What about an infinite number of additions? Or, I add the set of all positive integers to the set {0}.

in each of these examples, you’re adding an already actually infinite set to a finite set; but you can’t make that set actually infinite by successive addition: it’s already given as infinite.

SeekingCatholic:

Are there more members in the set of all integers less than 10 compared to the set of all integers less than 0?

Math for finite numbers/sets simply doesn’t apply to infinite ones. Now you can call this “absurd”, but this is just an opinion. It is not a logical proof of impossibility.

transfinite math is not absurd considered as the logical entailments of certain mathematical axioms; but it most definitely is absurd when those logical entailments are taken further to have ontological entailments.

SeekingCatholic:

Only if tristram shandy himself existed from eternity. Easily refuted therefore by stipulating that despite the universe’s eternal existence, no object within the universe exists from eternity but has a beginning and end.

now you’re kind of seeing the point: the ***mathematical ***facts yield ontological absurdity - if there is a being that has existed through an actually infinite number of temporal moments, then that being can perform logical absurdities. so, you naturally draw the conclusion that no such being can actually exist.

my answer is to reject the possibility of an actually infinite number of temporal moments, which seems much more plausible a response (if there can be an infinite number of temporal moments, then why not, as a matter of modal fact, a being that exists in each of those moments? to reject that possibility is just arbitrary).

 

[SeekingCatholic]

They are identical in quantity. Under set theory, there are just as many odd-numbered positive integers as there are all positive integers.

Right. So?

Nothing in reality would prevent us from subtracting or dividing elements, which is another reason set theory does not correspond to the physical world. If actual infinites are going to be an exception, then a reason ought to be offered; otherwise, it is special pleading.

All right. Well by “subtraction” here we obviously mean removal of elements from the set. This is what you were referring to when you said “infinity - infinity”.

You can remove elements from an infinite set. You can remove an infinity of elements from the set. The number of elements in the remaining set depends on how the infinity of elements was removed. It’s not enough to say you “subtracted infinity”; it’s necessary to stipulate how the removal was performed.

As for “special pleading”, that’s nonsense. It’s like saying a “reason” needs to be given for discarding Euclidean geometry in Riemannian spaces. The reason is that finite set theory was formulated for finite sets, just like Euclidean geometry was formulated for flat spaces.

If absolute zero is being used in the “ground state” sense, then it is not really absolutely zero. The ground state is simply the lowest level of expected energy.

The temperature is really absolutely zero. Absolute zero degrees Kelvin. Can’t get any lower. And the ground state is not the lowest level of expected energy, but the lowest level of possible energy (lowest eigenstate in quantum harmonic oscillator).

Incidentally, the impossibility of actual infinites is used as an argument against mathematical Platonism.

That’s a circular argument.

Could you define how you’re using “paradox”?

A paradox is something which seems to violate our common-sense intuition of how things should be.

 

[SeekingCatholic]

that only works if there is a last, finite member of your set. there isn’t, so your proof fails. QED.

No, it works even when there isn’t a last finite member of the set. My example shows this. The cardinality of the “set of all finite negative integers” is aleph-null. It’s an infinite set. All members of the set are finite, and therefore at a finite distance from zero. Unless you want to say that the “set of all finite negative integers” must contain an infinite member as an element, which is a contradiction in terms.

in each of these examples, you’re adding an already actually infinite set to a finite set; but you can’t make that set actually infinite by successive addition: it’s already given as infinite.

How is this the case in an infinity of successive additions to {0}? Anyway, I think I’ll drop this point because a past-infinite universe can’t be achieved by additions in the future direction of time.

transfinite math is not absurd considered as the logical entailments of certain mathematical axioms; but it most definitely is absurd when those logical entailments are taken further to have ontological entailments.

OK, well that’s what you need to show then.

now you’re kind of seeing the point: the ***mathematical ***facts yield ontological absurdity - if there is a being that has existed through an actually infinite number of temporal moments, then that being can perform logical absurdities. so, you naturally draw the conclusion that no such being can actually exist.

Right. But the mathematical facts don’t necessarily imply a being that has existed through an infinite number of temporal moments.

my answer is to reject the possibility of an actually infinite number of temporal moments, which seems much more plausible a response (if there can be an infinite number of temporal moments, then why not, as a matter of modal fact, a being that exists in each of those moments? to reject that possibility is just arbitrary).

It’s not arbitrary. Every object we can observe comes into existence, and then out of existence at some point in time. Or at least we can infer that from good data (e.g. stellar formation, etc.) Saying there could be an infinity of temporal moments is consistent with saying that all beings that exist in the universe are nevertheless contingent. The two statements are not logically contradictory, and solve the tristam shandy paradox.

But if you are now resting on the plausibility of the response, we have moved away from direct proof and into inference to the best explanation. I don’t think an infinity of temporal moments the best explanation. But I don’t see a rigorous disproof of its possibility.

 

[john_doran]

No, it works even when there isn’t a last finite member of the set. My example shows this. The cardinality of the “set of all finite negative integers” is aleph-null. It’s an infinite set. All members of the set are finite, and therefore at a finite distance from zero. Unless you want to say that the “set of all finite negative integers” must contain an infinite member as an element, which is a contradiction in terms.

this makes absolutely no sense at all.

look, if the last member of the set of negative integers is finite, then the set has a finite number of members (and the set will not have the cardinality aleph-null). that’s what it means to say that the set of negative integers has no last member.

the set of negative integers is not like the set of reals, with can have a finite “first” and “last” member (0 and 1), but an infinite number of intervening members.

SeekingCatholic:

OK, well that’s what you need to show then.

???

how have i not? how is “there are the same number of people inside that building now that one trillion people have left, as there was before those people left” not an absurd entailment of the reification of actually infinite quantities?

i mean, you yourself recognize the absurdity of the existence of a tristram shandy, and avoid the problem by fiat, and say simply that no such being exists (here and now)…

but maybe we’ve gone as far as we can along this tack, too - i can’t get you to see the absurdity of this any more than i can get someone to see the absurdity of something coming into existence without a cause.

SeekingCatholic:

Right. But the mathematical facts don’t necessarily imply a being that has existed through an infinite number of temporal moments.

exactly. but the mathematical facts don’t imply the existence of an actual infinite number of temporal moments, either…

but if you posit one, then there’s no reason not to posit the other, unless you can demonstrate that it is logically impossible for there to exist a being that exists through an infinite number of temporal moments (without at the same time demonstrating the impossibility of such a series itself)…

SeekingCatholic:

It’s not arbitrary. Every object we can observe comes into existence, and then out of existence at some point in time. Or at least we can infer that from good data (e.g. stellar formation, etc.) Saying there could be an infinity of temporal moments is consistent with saying that all beings that exist in the universe are nevertheless contingent. The two statements are not logically contradictory, and solve the tristam shandy paradox.

that doesn’t solve the paradox, because the paradox is logical; and since there are possible worlds in which a being with an actually infinite temporal extension exists, the putative fact that no such being exists in this world is irrelevant.

basically, the argument i’m making here is modal: it is an argument that an actually infinite series of temporal moments is ***logically impossible ***- that there are no possible worlds where tristram shandy exists (because if he did exist, then he would be able to do perform the logically absurd).

SeekingCatholic:

But if you are now resting on the plausibility of the response, we have moved away from direct proof and into inference to the best explanation. I don’t think an infinity of temporal moments the best explanation. But I don’t see a rigorous disproof of its possibility.

there aren’t “rigorous” disproofs for anything by the standards you’re employing here, i’m afraid.

 

[SeekingCatholic]

this makes absolutely no sense at all.

look, if the last member of the set of negative integers is finite, then the set has a finite number of members (and the set will not have the cardinality aleph-null). that’s what it means to say that the set of negative integers has no last member.

No, it is not what it means.
The set of negative integers has no last (actually, first) member != there is an infinite member of the set.
Your logic goes like this:
The finite set has a finite first member.
The infinite (aleph-0) set cannot have a finite first member.
Therefore, the infinite set must contain an infinite member.
It’s a fallacy: the infinite set contains no first member, period.

how have i not? how is “there are the same number of people inside that building now that one trillion people have left, as there was before those people left” not an absurd entailment of the reification of actually infinite quantities?

I’ll agree it goes against “common sense”. There are lots of things which go against common sense and are nevertheless true. It’s “common sense” there should be no quantum tunnelling. Yet it exists.

i mean, you yourself recognize the absurdity of the existence of a tristram shandy, and avoid the problem by fiat, and say simply that no such being exists (here and now)…

I did more than recognize the “absurdity” of tristram shandy. I recognized logical impossibility. That’s the difference between the two things (subtraction from infinity and tristram shandy).

but if you posit one, then there’s no reason not to posit the other, unless you can demonstrate that it is logically impossible for there to exist a being that exists through an infinite number of temporal moments (without at the same time demonstrating the impossibility of such a series itself)…

Which we just did with tristram shandy. Whether or not there are an infinite number of temporal moments, the existence of a being that exists continuously through an infinite number of temporal moments is logically impossible, as demonstrated via tristram shandy, because that gives him the ability to do the logically impossible. However, the tristram shandy only demonstrates that such a being cannot logically exist, not that a infinite temporal series cannot exist, if all beings exist for only finite periods of time.

that doesn’t solve the paradox, because the paradox is logical; and since there are possible worlds in which a being with an actually infinite temporal extension exists, the putative fact that no such being exists in this world is irrelevant.

This is correct. I just wanted to show the assumption that all beings exist for finite periods of time to not be “arbitrary”.

basically, the argument i’m making here is modal: it is an argument that an actually infinite series of temporal moments is ***logically impossible ***- that there are no possible worlds where tristram shandy exists (because if he did exist, then he would be able to do perform the logically absurd).

I understand, but the modal argument fails. There are no possible worlds in which tristram shandy exists. However part of the subset of possible worlds in which tristram shandy doesn’t exist can be infinite in duration. IOW, there are possible worlds, infinite in temporal duration, in which no tristram shandy exists. Why does he not exist in those worlds? Because his existence is logically impossible in any world, not because the worlds themselves are impossible, infinite in temporal duration but in which every object exists only for finite duration.

Your fallacy is in assuming that allowing one condition necessary for the possibility of tristram shandy’s existence (a universe with infinite temporal duration) is enough to make his existence logically possible. It isn’t.