The only reason why no one can actually pair the integers is because they are stated to be infinite, and by this definition, it is impossible to do such. Therefore it is logically impossible to do such. — Metaphysician Undercover

Sorry, that is not how logical impossibility is defined. It would have to be something that is impossible for anyone even to conceive (like a square circle), not something that is merely impossible for anyone to do. Again, the latter is actual impossibility, which has absolutely no relevance whatsoever to pure mathematics.

What I have shown is that I cannot understand mathematics because the language of mathematics contradicts my native language, English. This renders mathematics as incoherent and unintelligible to me. — Metaphysician Undercover

My native language is also English, and I see no contradiction whatsoever with the language of mathematics that we have been discussing here. The same is true of any and every English-speaking mathematician in the world. In any case, it frankly seems rather foolish to keep making definitive (and incorrect) pronouncements about a subject that, by your own admission, you cannot even understand.

But that's irrelevant because I was only arguing logical impossibility all along, which as I explained is the only real form impossibility. — Metaphysician Undercover

One more time: The fact that no one can actually pair all of the integers with corresponding even numbers has no bearing whatsoever on its logical possibility.

It is the definition of "infinite" which necessitates that pairing infinite sets is impossible. — Metaphysician Undercover

Actually impossible, but not logically impossible. Just ask a mathematician.

How is this relevant? — Metaphysician Undercover

It illustrates that actual impossibility does not entail logical impossibility.

"Pairing" is a task which requires completion. — Metaphysician Undercover

No; the whole point here is that pairing the members of infinite sets cannot actually be completed, yet it is still logically possible.

How about cannot treat infinity as a quantity because it is not a quantity? — Devans99

You might finally be on to something there, depending on exactly what you mean by it.

I'm trying to present an argument based on the rules of infinity and then to arrive at a contradiction. — Devans99

It would help if you actually understood the rules of infinity that mathematicians recognize, which are different from the rules of finite quantities; then perhaps you would finally realize that they are not contradictory at all.

As usual, equating the logical with the actual leads to absurdity. Logical possibility is much broader than actual possibility.

If pigs had large and powerful wings, then pigs could fly. The truth of this hypothetical proposition is not affected by the fact that pigs do not actually have large and powerful wings. If one were to pair all of the integers with the even numbers, then one would never run out of even numbers while still having integers left. Again, the truth of this hypothetical proposition is not affected by the fact that one cannot actually pair all of the integers with even numbers.

A square circle is logically impossible because the definition of a square and the definition of a circle are mutually exclusive. There is no such incompatibility between the definition of an integer and the definition of an even number; in fact, the alleged paradox is rooted in those very definitions, which place no finite limitation on either set.

What is the point of having a quantity (infinity) that you can do nothing with mathematically; you cannot add/subtract/multiply/divide without hitting a contradiction. — Devans99

You can do all kinds of things with infinity mathematically, but what you cannot do is treat it as if it were just another quantity. Infinity is a different kind of thing from any discrete number, no matter how large (or small).

Because an infinite interval is not composed of infinitely many finite intervals. Until you understand that, we will continue going in circles.

Would you tell me that you were convinced of X even though you knew of reasons to doubt X? — Metaphysician Undercover

Are you still convinced of your definition of "convinced," even though you know of reasons to doubt your definition of "convinced"--such as @DingoJones using that word differently, with no intention to deceive?

I suggest that our strongest convictions are precisely the ones we know of reasons to doubt, but we find those reasons unconvincing.

"Infinite" is commonly defined in such a way that it is impossible, by definition, to pair up infinite things, because the task would never be complete. — Metaphysician Undercover

That is actual impossibility, not logical impossibility. It is completely irrelevant to pure mathematics--the science of drawing necessary conclusions about formal hypotheses--whether anyone could ever actually pair up the members of infinite sets.

And the infinite interval is composed of infinitely many finite intervals. Thats a contradiction that proves bijection is plain wrong. — Devans99

If your first statement were true, then your second statement would also be true. But your first statement is false, so your second statement is also false.

The fact remains that we cannot actually pair them because there is an infinite number of them. So the proposition states something which is, by definition, impossible (i.e. it is contradictory). — Metaphysician Undercover

A proposition is not contradictory merely by virtue of stating something that is actually impossible, only if it states something that is logically impossible--which is certainly not the case here. Mathematics has to do with the hypothetical, not the actual.

My argument applies cause and effect consistently throughout. — Devans99

Only by adopting the additional axiom--an assumption, not an argument--that everything is an effect.

Mathematical infinity in set theory is actual infinity. — Devans99

No, it is not. Like all mathematical theories, set theory--especially as applied to infinite sets--is based on certain hypothetical formalizations that may or may not correspond to anything actual.

Mathematical infinity is not actual infinity. Which part of this do you still not understand?

Please pay attention. Arguments for a First Mover consistently affirm that every effect has a cause. What they deny is that everything is an effect; specifically, the First Mover is not an effect and requires no cause.

One more time: Mathematical infinity is not an actual infinity, but it is a real infinity. If we paired up every number with its square, when would we run out of one or the other? Never. How is this a contradiction?

Imagine an eternal being; he would have no birth so could never exist. — Devans99

That does not follow at all. Again, your fundamental assumption is that everything is an effect--i.e., everything has a beginning--which is precisely what arguments for a First Mover deny.

The integers (negative and positive) comprise an infinite linear sequence. What is its first member? Which integer does not have a predecessor?

They do nothing to resolve the paradox of the arrow, so far as I can tell. — MindForged

As summarized by Wikipedia, the arrow paradox states, "If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible." If motion is a more fundamental reality than position, and space-time is a true continuum, then both premises here are false--nothing is ever completely motionless, and time is not composed of discrete instants.

You are claiming ... An equal amount of integers and even numbers in an infinite interval — Devans99

I said nothing whatsoever about "equal amount" or "infinite interval," concepts that mistakenly treat infinity as if it were extremely large, but still finite. How many integers are there? Infinitely many. How many even numbers are there? Infinitely many. If we paired up each integer with an even number, when would we run out of even numbers, but still have integers left? Never.

Note that whether an actual infinity is possible or impossible is completely irrelevant here. This is mathematics, which is the science of drawing necessary conclusions from formal hypotheses. The definitions of "integer" and "even number" constrain us to recognize these somewhat counterintuitive relations between them.

Again, that is not how it works. There are twice as many integers as even numbers within any finite (and even) interval, but neither the set of all integers nor the set of all even numbers is finite. There is no greatest integer, and there is no greatest even number; so for every integer, no matter how large, there is a corresponding even number.

The interval 4 is twice the length of 2 so it should contain twice as many real numbers. — Devans99

That is not how real numbers work. By such (il)logic, there should be twice as many integers as even numbers, which is also not the case. A discrete collection of four items obviously does contain twice as many objects as a discrete collection of two items, but a continuum (such as space-time) does not consist of discrete items at all. The unwarranted axiom here is that reality consists entirely of discrete items and collections thereof.

Now, I do not dispute that actuality consists entirely of discrete items and collections thereof; but I deny that reality--that which is as it is, regardless of what anyone thinks about it--is limited to actuality. Specifically, there are real continua of potential items that are not even remotely exhausted by their discrete instantiations. Being and existence are not coextensive.

That is a very illogical belief IMO. — Devans99

What is the warrant for believing that absolutely everything is an effect caused by something else? How is it any more "logical" than believing that there is one unmoved mover, one uncaused cause, whose being is necessary rather than contingent?

There is just no way a light year has identical granularity and structure to a centimetre — Devans99

The mistake is assuming that, in itself, any arbitrary portion of space-time has any granularity--i.e., discreteness--at all.

Thanks for the conversation though. Happy Xmas. — Devans99

Likewise!

My argument starts with the simple axiom ‘all effects have causes’ and proceeds via deduction to circular time. — Devans99

It also smuggles in the premise that everything is an effect, which is precisely what proponents of an unmoved mover deny.

The information can be in analog form; number of bits is merely a way to quantify it ... all volumes of space-time contain the same amount of information no matter what size is clearly contradictory. — Devans99

As soon as you talk about comparing the "amount" of something, you are quantifying it, and thereby treating it as discrete--i.e., begging the question.

the fact that position is given to infinite precision by the continuum means it is contradictory — Devans99

What part of "motion is more fundamental than position" do you still not understand? Giving the position of something to any degree of precision requires measuring its distance from an arbitrary reference point at an arbitrary instant using an arbitrary unit.

Film is a good analogy for time. — Devans99

Only if you presuppose that time is discrete, like the film in a motion picture.

You did not answer my argument that the information content of a larger region of space-time must be larger than a smaller region of space-time? — Devans99

Yes, I did.

It begs the question to presuppose discrete units of "information" (i.e., points or finite segments) that comprise a "real line." — aletheist

... or in this case, a "region of space-time."

So in both cases our progress through time and space subdivides the continuum to an actual infinity. — Devans99

No, that progress itself through the space-time continuum (i.e., motion) is the fundamental reality; any discrete subdivisions of space and time are our arbitrary constructions.

I see no model of continuity that does not need actual infinity. If you point me to such a model, I stand corrected, but they all seem to use actual infinity. — Devans99

In Peirce's model of a true continuum, the infinity is potential rather than actual. The real is not coextensive with the actual (existence); there are also real possibilities and real (conditional) necessities.

Any real continuum can be subdivided infinity so it it exists in the present or the past, it must support an actually infinite number of sub-divisions. — Devans99

No, any real continuum could potentially be subdivided infinitely; it can never actually be subdivided infinitely. See the difference?

So I'd argue the start of the prime mover arguments uses the axiom 'all effects have causes' and the abandons the axiom at the end of the argument. — Devans99

No, that axiom is maintained throughout. Again, the actual argument is that there must be a cause that is not an effect of some other cause.

And the concept of a first cause is illogical as is an infinite regress. — Devans99

Only in accordance with the unstated (and unwarranted) premise that everything has a cause.

Paradoxes indicate an underlying logic error (actual infinity exists). — Devans99

Contradictions indicate an underlying logic error; paradoxes indicate a need to think more carefully.

How many times must I repeat that I am arguing for real continuity, not actual infinity, and that these are two distinct concepts? that motion (space-time) is more fundamental than position (space) or duration (time) treated separately? that a line does not consist of points, and that a temporal interval does not consist of instants?

My version at least applies the axiom 'all effects have causes' consistently. — Devans99

Only by ignoring the fact that the whole point of arguments for an unmoved mover is that there must be a first cause that is not itself an effect of some other cause. Your axiom is actually that everything has a cause, which is a very different proposition--one that proponents of an unmoved mover would categorically deny.

I've looked at it; its rubbish. — Devans99

No doubt they would say the same about your arguments here.

A point has length 0. How many points on a line segment length 1? 1/0=UNDEFINED. — Devans99

You remain wedded to the mathematics of discrete quantity. Again, there are no points on a continuous line, unless and until we mark them as discontinuities.

I think my reply to Rank Amateur above covers your points. — Devans99

On the contrary ...

So my axiom is to the effect 'all effects have causes'. — Devans99

As eternal and necessary being, the unmoved mover is not an effect; so this "axiom" is irrelevant to the argument.

X in my proof could be real or natural; I made no assumptions. That is a fair proof that actual infinity is not a quantity IMO. — Devans99

All I can suggest at this point is looking into the standard mathematics of infinity. I side with Peirce, rather than Cantor, in denying that the real numbers constitute a true continuum.

Think about a real-life line segment, say the distance between your eyes and the screen; that HAS to be an actual infinity of points/line segments by the very definition of continuous. — Devans99

No, it does not. That is only one way to model it, which I consider incorrect. Another is that the line is a true continuum that does not consist of discrete points or line segments at all. You can arbitrarily mark any such point or segment, but by doing so you introduce a discontinuity into that which is really continuous in itself.

Think about the second that just past; by the definition of continuous; its has to include an actually infinite number of moments/periods of time. They just all happened. — Devans99

No, they did not. That is only one way to model it, which I consider incorrect. Another is that any finite interval of time is a true continuum that does not consist of discrete instants at all. You can arbitrarily mark any such instant, but by doing so you introduce a discontinuity into that which is really continuous in itself.

All 'cardinality of the set of natural numbers' moments just happened if you can make sense of that. — Devans99

I can make sense of that, but it is not the case. A true continuum has a cardinality exceeding that of any infinite set. In Peirce's terminology, a true continuum has a multitude exceeding that of any infinite collection. Between any two points that we mark on a line, there is an inexhaustible continuum of other potential points; between any two instants that we mark in time, there is an inexhaustible continuum of other potential instants.

I proposing it on the basis of logic rather than faith. — Devans99

I gather than you mean deductive logic in this context, but that can only guarantee the derivation of true conclusions from true premises; it can neither furnish nor confirm the premises themselves. A deductive argument can be perfectly valid (logically correct), yet unsound (actually false). For example:

• All dogs are orange.
• My pet is a dog.
• Therefore, my pet is orange.

- I do not believe the concept of an unmoved mover is logically sound.
- An infinite regress of movers in time is not logically sound.
- Something from nothing is impossible
- That leaves circular time as the only possible explanation for the start of the universe — Devans99

This conclusion only follows from the three listed premises if there are no other possible explanations for the start of the universe, which seems unlikely. In any case, many careful thinkers throughout history agreed that an infinite regress and something from nothing are both impossible, but found no logical flaw with the concept of an unmoved mover. Your premise for rejecting it seems to be ...

Everything has a cause surely? — Devans99

This is true of everything that has a beginning, whose being is contingent; but it is not true if there is something that has no beginning, whose being is necessary. That is precisely what we mean by an unmoved mover--an eternal being, a necessary being, what in vernacular terms we usually call God.

How on earth could you construct a continuum? It requires us to construct an actual infinity of possible positions for particles to occupy. Thats impossible. — Devans99

Yet again: Because space-time is a true continuum, motion/velocity is a more fundamental reality than either position or duration. We can construct a continuum as the path that a particle traces (or would trace) over an interval of time--which is not a collection (infinite or otherwise) of discrete positions. Besides, how could merely possible positions constitute an actual infinity?

Actual infinity, if it existed, would be a quantity greater than all other quantities ... — Devans99

That definition is incorrect, according to the standard mathematics of infinity; mainly because it begs the question by presupposing the discreteness of quantity. In any case, I am advocating the reality of continua, not the actuality of infinity.

Your continuum is just magic - how can 1 light year be structurally the same as 1 millimetre? — Devans99

An argument from incredulity is not persuasive, and alleging "magic" suggests a lack of interest in engaging in serious philosophical discussion. What exactly do you mean by "structurally the same" in this context?

I don't believe in magic so I have to take a different view - no continuums in nature; they are logically unsound and nature does not do unsound. — Devans99

In what specific sense do you hold that the hypothesis of a real continuum is "logically unsound"? Are you claiming that it is somehow logically impossible, or merely not actual? Either way, why do we nevertheless routinely refer to "the space-time continuum"?

Also it can't be a continuum because actual infinity does not exist. — Devans99

A bare assertion is also not persuasive. Even if I grant the premise, the discontinuity of actuality/existence does not, by itself, rule out the reality of true continua.

If I have a real line length 1 mile, it contains more information than a real line length 1 centimetre. But if they are both continuums then they both contain the same amount of information. Which is impossible. — Devans99

Why? It begs the question to presuppose discrete units of "information" (i.e., points or finite segments) that comprise a "real line." Again, the "parts" of a true continuum are also true continua.

If you imagine a particle in an interval, the position of the particle relative to the beginning of the interval can be regarded as information. In a continuum, that piece of information (particle position) has infinite precision so infinity many bits of information. — Devans99

I repeat: Because space-time is a true continuum, motion/velocity is a more fundamental reality than either position or duration. Marking and measuring the position of a particle at any particular instant imposes an arbitrary discontinuity, like a point on a line.