...that's what a bijection does... — jorndoe

The numbers are infinite. The bijection is necessarily incomplete. Therefore, in this case the bijection "does not", and that contradict "does".

Not sure it's worthwhile mentioning the obvious, but that's what a bijection does — jorndoe

Bijection gives non-sensical answers: it says there are the same amount of even integers as there are integers. But any finite interval; there are twice as many integers as even integers. And the infinite interval is composed of infinitely many finite intervals. Thats a contradiction that proves bijection is plain wrong.

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.

"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.

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. — aletheist

Why is my first statement false?

∞ = ∞ * 100
So an infinite interval is composed of infinity many finite intervals (of 100 length in this case).

I think I will maybe adopt 'things need a non-zero length to exist' as an axiom. — Devans99

Interesting. So consider length, the human concept of length. It has no length, being a non-physical thing. But it exists. And remember that axioms are guesses, that we call axioms partly to make them sound more credible and scientific, but mainly because we cannot prove them, so we assert them instead, without a shred of evidence or justification. If we could prove them, we would. When we can't we pretend: axioms.

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

∞ = ∞ * 100 — Devans99

...always remembering that it is mathematically invalid to divide both sides of the equation by infinity. You did remember that, right? :chin:

Interesting. So consider length, the human concept of length. It has no length, being a non-physical thing. But it exists. And remember that axioms are guesses, that we call axioms partly to make them sound more credible and scientific, but mainly because we cannot prove them, so we assert them instead, without a shred of evidence or justification. If we could prove them, we would. When we can't we pretend: axioms. — Pattern-chaser

I would argue that length is a concept and concepts can exist in our minds only. Maybe I should have said: 'things need a non-zero length to exist in reality'. Anything can exist in our minds. Talking trees and Santa. Reality is material and you cannot have a material thing with length=0.

An axiom should be more than a guess IMO. The original definition of axiom was 'self evident truth'.

..always remembering that it is mathematically invalid to divide both sides of the equation by infinity. You did remember that, right? :chin: — Pattern-chaser

Sorry I have given up on the maths of infinity. 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... sort of my point... every way we turn, infinity leads to contradictions. It's too illogical to be a real world concept.

Sorry I have given up on the maths of infinity. 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... sort of my point... every way we turn, infinity leads to contradictions. It's too illogical to be a real world concept. — Devans99

Quite right. But mathematicians needed infinity, so they shoe-horned it into their arithmetic and algebra, even though it is obvious that it doesn't and can't fit there, as you describe. :up: But them's the rules (of how infinity is handled), so your attempt to ridicule it is pointless. The whole idea is ridiculous but necessary, so that's that. <wry grin + shrug>

An axiom should be more than a guess IMO. The original definition of axiom was 'self evident truth'. — Devans99

And a self-evident-truth is ... a guess. For if we could manage anything more - ideally, proof - then we would. And when we can't, we guess. Self-evidently, of course. :wink:

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).

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). — aletheist

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

That is actual impossibility, not logical impossibility. — aletheist

When definitions deny the possibility of something due to contradiction, this is a logical impossibility, like a square circle is a logical impossibility. That one could make a bijection of infinite numbers is logically impossible because the definition of "infinite" (what it means to be infinite), contradicts the definition of "bijection" (what "bijection" means). Therefore "infinite bijection" is excluded as a possibility because it is contradictory, i.e. logically impossible.

I don't know what you would be referring to with a distinction between "actual impossibility" and "logical impossibility", because all impossibilities are logical impossibilities. The only way that we have of demonstrating, or knowing that, something is impossible is through logic. Therefore actual impossibilities are logical impossibilities, because all impossibilities are logical impossibilities. An actual impossibility might be one based in sound logic, while a not-actual impossibility might be one based in unsound logic.

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.

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.

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. — aletheist

How is this relevant?

A square circle is logically impossible because the definition of a square and the definition of a circle are mutually exclusive — aletheist

Right, let's hold that thought.

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. — aletheist

The incompatibility is not between "integer" and "even number", it is between "pairing" and "infinite". "Pairing" is a task which requires completion. If the task is incomplete, they are not actually paired, and the "pairing" attempt is a failure. "Infinite" denies the possibility of completion, therefore the "pairing", is of logical necessity, in the case of the infinite, a failure. Therefore either the integers are infinite in which case pairing is impossible, or the integers are not infinite, in which case pairing is possible.

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.

You two don't know MU and Devan. They will just insist there is a contradiction. When you ask then to formally show the contradiction, they will just say it's weird, or that it's not possible to actually map two infinite sets or something like that. They won't actually address the point because they've misunderstood some fundamental things, from confusing distinct modalities to the reasons why mathematicians were rationally forced to accept infinity. MU in particular is so off base that he rejects the mathematical definition of a set ("Sets must be constructed by literally putting things together BY DEFINITION").

I think quantum mechanics, taken seriously, is an argument against the continuous nature of reality period, including space-time. There are no dimensionless points and thus no durationless moments. Space and time cannot be infinitely divided and have any meaning to the terms remain.

That's not true. For example, in QM the position and momentum of particles are continuous. Spacetime is also taken to be continuous, time is always taken to be a continuous parameter everywhere in physics. For all the effort out into making space or time discrete, such theories always turn out to be inconsistent somewhere. All (or nearly so) quantum mechanical theories treat spacetime as a continuous parameter, you'd have to go to something much more speculstive like loop quantum gravity to get a discrete structure.

It illustrates that actual impossibility does not entail logical impossibility. — aletheist

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

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

It is the definition of "infinite" which necessitates that pairing infinite sets is impossible. How on earth do you assert that it is still logically possible without changing the definition of "infinite"? That's MindForged's tactic, to produce a different definition of "infinite", but that leaves "infinite" as utter nonsense. As Devan's99 has demonstrated over and over again, the definition of "infinite" employed by set theory is illogical.

They will just insist there is a contradiction. When you ask then to formally show the contradiction, they will just say it's weird, or that it's not possible to actually map two infinite sets or something like that. — MindForged

What a short memory you have. I actually demonstrated the contradiction to you in numerous different ways, because each time I demonstrated it you would change the goal posts in an effort to avoid my demonstration. That's why I had to demonstrate it to you in so many different ways, you kept trying to wiggle out from under the crushing force of blatant contradiction.

As explained in my last several posts, pairing infinite numbers is contradictory due to the definitions of "pairing" and "infinite". If you want to get back under the crushing force of contradiction, and try to wiggle out again, then be my guest, and try to demonstrate to me how this is not contradictory. But since your memory seems to be very short, let me remind you that you were not able to wiggle out last time.

pairing infinite numbers is contradictory due to the definitions of "pairing" and "infinite". — Metaphysician Undercover

why?

Actually, I take that back. Mapping an infinity of one sort against anther is a common mathematical practice. So you are wrong, or talking about something else.

The following temporal duration, A, has an exact length of 3. Proof:

A: "one..........two...........three"

Now does it make sense to dispute this, by arguing that I could have counted the same interval twice as fast?

For mustn't any supposedly 'counterfactual' argument refer to a newly constructed interval, B, and not to the past interval A that no longer exists and therefore cannot itself be re-measured?

B: "one,two,three,four,five,six"

If one accepts the counterfactual argument that A might have been counted differently, then one is led to ask how fast the same interval could have been counted, which leads to the further question as to whether there is a limit. In which case the above statement of A isn't a definition of A but merely one of many possible descriptions of A, namely that it just so happened to begin and end when I was counting.

On the other hand, if one rejects the counterfactual argument then A has an exact length of 3 by definition, and there is nothing more to be said about it.

why?

Actually, I take that back. Mapping an infinity of one sort against anther is a common mathematical practice. So you are wrong, or talking about something else. — Banno

Pairing, and mapping, are all activities just like counting is an activity. You cannot count an infinite number because this is contradictory to the definition of "infinite". You cannot pair an infinite number for the very same reason. You cannot measure an infinite number, nor can you map an infinite number, for the very same reason that these are activities which require completion to be successful.

You might assert that you have mapped an infinite number, and even show me your map. But since I cannot show you the infinite number (because this is contradictory), I cannot show you that your map does not correspond with the infinite number.

All I can do is demonstrate logically that it is impossible to map an infinite number because this is contradictory. Do you recognize the truth of "it is contradictory to claim that you could measure an infinity"? Do you recognize the truth of "to map something requires that it be measured in some way"? What makes you think that mathematicians have done what is logically impossible, mapped infinity? How naïve are you? Suppose I told you, that if you keep going in this direction, counting, you will eventually reach infinity. Would you believe that I have mapped infinity?

As explained in my last several posts, pairing infinite numbers is contradictory due to the definitions of "pairing" and "infinite". — Metaphysician Undercover

Tell me the exact formal definition of a mathematical mapping and infinity within the context of form mathematics and prove the contradiction. Don't do this BS where you talk big but repeatedly leave crucial terms undefined by implicitly assuming colloquial vagueness of the terms when you know full well that's not how definitions work in formal disciplines. You don't have an argument, this is pure bluster on your part.its been known for about a century that the Axiom of Infinity does not add any contradictions to ZF set theory, which on all accounts appears to be consistent. Formally derive the contradiction from the actual definitions used in mathematics or just admit you're straw Manning mathematics.

Not only is there no contradiction entailed, if there were your proof that there was would ensure that you received the Fields Medal. But curious that it will forever be beyond your grasp, almost like you're making fundamental missteps.

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.

All I can do is demonstrate logically that it is impossible to map an infinite number because this is contradictory. — Metaphysician Undercover

And yet we can; and yet we do, map series of infinite numbers, one against the other.

So this comes down to Meta vs. mathematics.

@MindForged is right. What you have shown is that you refuse to understand mathematics.

Tell me the exact formal definition of a mathematical mapping and infinity within the context of form mathematics and prove the contradiction. — MindForged

If you have a problem with my terms (they are English), then address my posts and tell me where the problems are. If my terms are not related to mathematics, then don't worry about them, they pose no threat to this field which you hold sacred.

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. — aletheist

And here's my "one more time". It is only a "fact" by definition, therefore the impossibility is logical. 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.

What you have shown is that you refuse to understand mathematics. — Banno

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. I know that you don't care about this. So be it.