No, I outlined a mapping of a possible finite past, and pointed out there are cosmological models based on a finite past (Hawking, Carroll, and Vilenkin to name 3). I am aware of no such conceptual mapping for an infinite past. — Relativist
OK, then I suggest you quit using "transfinite", because you are only introducing ambiguity. Why then did you say: "The cardinality of the set of natural numbers is the transfinite number aleph-null." If "transfinite" is just an artefact, and transfinites are really infinite, then infinite sets really have no distinct cardinality, they are simply "infinite". — Metaphysician Undercover
Your claim was that an infinite set has a precise and known cardinality. If this is the case then you can show me the relationship between the cardinality of an infinite set, and those other two finite sets, and how the difference between the cardinality of the two finite sets is expressed in the two relationships between each finite set, and the infinite set. — Metaphysician Undercover
You should have specified what you meant by difference. I assumed you were asking how such sets were any different than a purportedly infinite set, so I gave the difference. If you were talking about the difference as in subtraction, then the answer is infinity. If I subtract any finite number from an infinite number, it's not going to change the cardinality. It's only finite numbers whose cardinality decreases when removing finite numbers of elements. If I take the natural numbers and remove the element Zero, it can still be put into a one-to-one correspondence with the even numbers, so this just provably doesn't change the size of the set. — MindForged
And as I said, I don't care if it's a set according to your definition. — MindForged
If you knew the precise cardinality of an infinite set, you'd be able to tell me the relationship between the cardinality of a finite set and that of an infinite set. Obviously you know of no such relationship, as subtracting a finite number from an infinite set does not change its cardinality. There is no such relationship. Therefore my suspicions are confirmed, you really do not know the cardinality of an infinite set. Your claim was a hoax. And so your assertion that "infinite set" is not contradictory is just a big hoax. — Metaphysician Undercover
I know you feel this way, that's why I've proceeded to, and succeeded in demonstrating that "infinite set" is contradictory according to your definition, and the one used by mathematicians. Clearly an "infinite set" is not a well-defined collection in any mathematical sense, because the cardinality of such a set is not at all well-defined. Therefore it cannot be a well-defined collection, mathematically, and cannot be a mathematical "set". — Metaphysician Undercover
If I take the cardinality number aleph-null, the the size of the natural numbers, and remove the element that's the number Zero, the cardinality doesn't change, e.g. — MindForged
I showed the informal proof of it being an infinite set (the one-to-one correspondence argument) and you couldn't even address it. — MindForged
First of all, please refrain from calling me stupid. I could very well be mistaken, and you are welcome to identify flaws in my reasoning or to just disagree since I'm not claiming my position is mathematically provable. But if you'd like to critique me in a reasonable way, please try to understand what I'm saying. — Relativist
There cannot have not been infinitely many paths TAKEN, there are only infinitely many possible paths that could potentially be taken, but it is impossible to actually follow them - no matter how long we have to try. So these paths exist in the abstract, but not in the real world. — Relativist
. Each is a possible path. It's the "taken" you object to? But whenever was a clock attached to a number?there are only infinitely many possible paths — Relativist
The definition of infinity is pretty clear, it's extremely useful in mathematics and science, and it introduces no contradictions into the theorems. — MindForged
The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line.
Do you still believe that there is one clear definition of "infinity" in mathematics? — Metaphysician Undercover
No, it's not just a semi-infinite number line, because that omits the temporal context. Time does not exist all at once, as does an abstract number line.
Consider the future: it doesn't exist. — Relativist
The present is the END of a journey of all prior days. That would be the mirror image of reaching a day infinitely far into the future, which cannot happen. A temporal process cannot reach TO infinity, and neither can a temporal process reach FROM an infinity. — Relativist
This is why you don't quote Wikipedia, especially when it's not a topic you're familiar with. The infinity referred to there is not a number. Limits do not diverge to a number per se (or if it does, it's to some transfinite number), they just increase without bound which meets a colloquial meaning of "infinity". — MindForged
"Infinity," like "existential," is a word with multiple meanings and applications. — tim wood
Near as I can tell, and charitably at that, you've taken the word out of its usual context, tried to fit it where it doesn't fit, and reported it back as a problem with the underlying concept. What possible use is that? Why would any intelligent person do that? — tim wood
And mathematicians that I've read are uniform in saying that infinity itself is not itself a number. — tim wood
A failure to define. Lots of infinities are represented by transfinite cardinals. Infinity in itself doesn't. But you know all this perfectly well.As such, an infinity has a number, a "transfinite" number. — Metaphysician Undercover
Exactly, this is what the quotation is saying, "infinite" in calculus and algebra is different from "infinite" in set theory. Set theory has transfinite numbers, alephs, but the definition of "infinite" in calculus and algebra is defined in relation to limits. — Metaphysician Undercover
The point being that there is no clear definition of "infinite" in mathematics as you claim, the definition varies. In geometry for example, a line is endless, infinite. Contrary to your claim, "boundless" is a valid definition of "infinite". — Metaphysician Undercover
So your defence, which was nothing more than an appeal to authority is lame and vacuous. Because the various mathematical authorities have various ways of defining the term, we cannot trust that any of them really knows what "infinite" means. — Metaphysician Undercover
Calculus has problems too. For example the infinite series 1/2^n
1 + 1/2 + 1/4 + 1/8 ... = 2
Logically it’s incorrect to write =2 should be ~2. It’s only a small error but the sum of that series is always less than 2. — Devans99
To be more precise, because the use of ellipsis can sometimes create ambiguity:The clean way to do it would be something like lim(1 + 1/2 + 1/4 + 1/8 + ...) = 2. — Magnus Anderson
This is what I'm talking about. "Infinity" in the context of limits might mean something else (emphasis on "might"), but calculus still uses multiple levels of infinity as understood in set theory, because we understand calculus through set theory. Hell, even in limits I could just assume the infinit there refers to Aleph-null and the calculation is still going to work. All it needs to mean is that it's larger than whatever I'm working with. And Aleph-null is necessarily larger than any finite number. — MindForged
The word 'infinite' is usually only applied to a set, to refer to its cardinality (although it can also be applied to ordinals, but let's not complicate things by worrying about them). — andrewk
1. A set is finite if there exists a bijection between it and a natural number. A set is infinite if it is not finite.
2. A set is infinite if there exists a bijection between it and a proper subset of itself. — andrewk
I agree with what you said, but it's beside the point. We agree that infinity is not reached to or from, but that just implies we need look elsewhere for our conception of an infinite future. The future is NOT the destination, it is the unending causal process following the arrow of time. The concept of "completeness" is key: the process for the future is never complete. On the other hand, the past is certainly complete - there is no continuing process - the process has completed (except for the finite process of appending an additional day every 24 hours). That is another way that the past has ontologically distinct properties from the future.The reason a temporal process will never reach infinitely far into the future is that there is nothing for it to reach: a process can start at point A and reach point B, but if there is no point B, then talk about reaching something doesn't make sense. Turn this around, and you get the same thing: you can talk about reaching the present from some point in the past, but if there is no starting point (ex hypothesi), the talk about reaching from somewhere doesn't make sense, unless you implicitly assume your conclusion (that time has a starting point in the past).
I think the mathematicians have the definition of Point wrong: — Devans99
A point must has length > 0 else it does not exist. With this revised definition of a point we can see that the number of points on any line segment is always a finite number rather than Actual Infinity. — Devans99
If "actually infinite" were proven to be impossible by way of contradiction, or some other logical proof, — Metaphysician Undercover
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.That a set could have an infinite cardinality is what I dispute, as contradictory. "Infinite cardinality" contradicts the definition of "set" as a "well-defined" collection. To be "well-defined" in this mathematical context, of a "set", is to have a definite cardinality, and "infinite" means indefinite.
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