Right. We know that there must be some way to reconcile gravity with quantum mechanics; we just don’t know how. Since gravity involves spacetime itself, one popular guess is that we need to quantize spacetime. This may (or may not) involve some fundamental smallest possible length scale, similarly to how quantized angular momentum involves a smallest possible angular momentum (but then again, quantized energy does not require a fundamental smallest energy; it depends on the system you’re dealing with). If there is a smallest possible length scale, then we don’t know how big it is, but one guess that’s about as good as any is that it’s roughly in the vicinity of the length scale that you get by combining the fundamental constants of gravity and quantum mechanics in the right way.
To sum up, if spacetime is quantized, and if that quantization involves a minimum possible length scale, then our best guess as to that scale is that it might be approximately the Planck scale. There’s a lot of weasel-wording in there.
Sure, you can add to an infinite set. Consider the set of all positive even integers. Do you agree that the set is infinite? Well what about if you add all the negavite numbers to it? Isn’t it still infinite? Haven’t you, indeed, added to it?
“…think of the smallest amount of time that you can. Really small. So tiny that a second would be like a billion years. Got that? Well, the cosmic quantum tick… that’s what the abbot calls it… the cosmic quantum tick is much smaller than that. It’s the time it takes to go from now to then. The time it takes an atom to think of wobbling. It’s-”
“It’s the time it takes for the smallest thing that’s possible to happen to happen?” said Lobsang.
“Exactly. Well done,” said Lu-Tze. He took a deep breath. “It’s also the time it takes for the whole universe to be destroyed in the past and rebuilt in the future.”
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Infinite is not a quantity. There’s no “adding” to it, there’s no “combining” it with something else. It’s like saying “what if I combine the number 10 with coffee?”. Coffee is not a quantity, rendering the sentence meaningless.
There is, but only when the sets you are combining have members in common.
This whole thread is cheerfully skirting about set theory, and Cantor’s work on infinite sets.
The notion of number is usually defined by the cardinality of a set. The cardinality of the union of two sets that have no members in common is the same as the sum of the cardinalities of the two sets. (Indeed the term “sum” may be defined like this.)
Cantor showed how to reason about sets with infinite members, and apropos the current thread, he defined Aleph Null to be the cardinality of the set of integers. If you try to mess with infinitesimal intervals, you will discover that there are lots of such intervals, in fact there are more than Aleph Null of them.
The union of two sets with cardinality of Aleph Null is a set with cardinality Aleph Null. The set of negative integers has cardinality Aleph Null, as does the set of positive integers.
Conventionally we consider the term infinity to be the same as Aleph Null. At least when we talk about integers. That fact that there are more real numbers than this might give one pause for thought. Technically the cardinality of the set of real numbers is larger than Aleph Null, and thus larger than the conventional meaning of infinity.
There have been lots of threads here about the nature of the various forms of infinity. Logically there is no problem reasoning about the idea of a universe with no beginning, Cantor got here long before.
I do not feel set theory, cardinalities, or Cantor’s work thereof is of any particular relevance to the topic of this thread; not all discussion of infinite objects is discussion about cardinalities.
Also, just to latch onto one other minor point, there are many notions of “number”, most having nothing to do with the cardinality of a set (how does one conceptualize -1 or 1/2 or π as cardinalities of sets? Which isn’t to say one cannot give some sort of story relating these to cardinalities, but such a story has no claim to being the raison d’etre of these concepts).
If “logically possible” just means “conceptually imaginable” (and what else would it mean?), then the answer to the OP is clear. There is no doubt that I can tell a story about a world with an infinite past, with no difficulty, years stretching back B.C. without end just as they do A.D. In that sense, there is nothing logically impossible about an infinite past, anymore than there is anything logically impossible about an infinite line.
(There isn’t even anything logically impossible about a world in which an infinite measure of years may pass between different events, though the above model, as many posters have rightly noted, lacks any instance of this)
One may, I suppose, feel the need to impose further constraints on what sorts of conceptual models one is willing to bless as properly interpreting words like “time”, “past”, etc., and in so doing, impose further axioms outlawing such models as above. But I do not feel there is any greatly principled reason to do so in this case; we all are fully capable of reconciling the above model to our ordinary language understanding of these words, any constraints beyond which, whatever they are, aren’t logically mandated.
Depends on which infinite set we are talking about. Not all infinities are equal. It might not be a quantity, but it can be quantified. And, you can combine anything. That’s like saying you can’t combine regular and imaginary numbers. What would we do without complex numbers?
IOW: Yeah, but… even I, with only the most rudimentary of math knowledge am aware of the fact that there is a hierarchy/taxonomy of infinities/infinite sets or number lines, protocols for determining whether two infinite sets are equivalent (hint: mapping), and even methods of combining (arithmetic!) infinite sets and finite sets (hint: surreal numbers).
I feel like I’m in a precarious position. I don’t have anywhere near enough proper math education to say why an infinite past is no problemo at all. But I have just enough to know that intuitively there’s no reason to rule against it.
The 1/n formulation and the ϵ formulation are, dare I say, indistinguishable. I can just define ϵ = 1/n for any positive integer n. I’m not sure what you’re getting at with “What if you pick the distance I moved?” It doesn’t matter who picks the distance, what matters as far as determining infinitesimality is whether it passes the ϵ test.
What I’m saying is that an infinitesimal distance I moved isn’t less than every distance ϵ you could pick; after all, it’s not less than itself. You have to clarify what kinds of distances it’s less than.
Yes, an infinitesimal distance is less than any and every ϵ I can pick. Because I can only “pick”, i.e. write down, actual numbers, like .00005, or .000017. This is completely identical to saying that an infinitesimal distance is less than 1/n for any and every positive integer n.
That’s one account of what you can pick/what “actual numbers” are. But it seems odd to claim the infinitesimal distance itself is not a pickable distance/measured by an actual number; it just happens to be, well, an infinitesimal one. We could devise a notation system for these too, if we liked, depending on what system including infinitesimal quantities we’re interested in.
I mean, fine, whatever, it doesn’t matter. It’s clear enough now what you mean. It’s just not immediately obvious how to interpret phrases like “no matter how small a value of ϵ”; there are non-obvious restrictions you are placing on that selection of ϵ which ought be mentioned.
Um, ok. When I say “there’s an infinite amount of …”, I (often) mean “there’s more … than any finite amount”. What’s wrong with that? There are more integers than any finite number; an infinitely long line is longer than any finite distance, etc.
You conclude that there must be have been a beginning point to time from the fact that “infinite” means “without end”? You oughtn’t be able to conclude anything so nontrivially strong from mere definitional conventions.
If you like, every second that ticks by adds another second onto the heap of seconds that’ve already passed. Fine. But what rule says the heap of seconds that’ve already passed must contain only a finite amount of seconds?
This post nails it, I think, and everyone should read it very carefully. Almost all the “logical” objections we’ve seen rely on assumed axioms like “the interval between any two points in time must be finite.” These axioms are hardly universal and aren’t necessary to most (any?) standard systems of logic, so in any reasonable sense, you can’t use them to form logical objections. If you could, you might as well choose an axiom “Infinite pasts cannot exist.”