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

Your "conceptual mapping" of a finite past was a semi-infinite number line. You say you cannot think of a corresponding "conceptual mapping" for an infinite past? Really?

I am sorry, but this isn't worth my time.

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

Because that's what the numbers are called. http://en.wikipedia.org/wiki/Transfinite_cardinal. Further, your comment that they have no "distinct" cardinality because they're infinite does not follow. The way cardinality is determined is exactly how we know the set of naturals have a cardinality of aleph-null.

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

Man, didn't I just do this? I showed the cardinality of the naturals between 1 and 100 and how we determined that. I also showed previously that the same means of determining cardinality, when applied to the entire naturals, results in a proper subset of the set having the same cardinality as the parent set.

Look, I'll try again for completeness.

"A" = set of naturals between 1 & 100
"B" = set of naturals between 1 & 200

To determine which is larger, we will pair each number from beginning to end with exactly one number from the other set. One element from A mapped to one element from B (A's on the left, B's on the right)c

1 - 1
2 - 2
3 - 3
etc.

99 - 99
100 - 100

Now we've hit a problem. Set A has no more members, we can't pair anything else up with the members of set B. And the reason is perfectly transparent: Set B has a larger cardinality, it has more members. But note what happens if we take the natural numbers (everything 0 and greater) with the even numbers and try to pair them off this way:

0 - 0
1 - 2
2 - 4
3 - 6
Etc.

Neither set ever fails to have members to pair off. That cardinality is infinity. The comparison to the other sets you mentioned, as I've said several times now, is that the naturals cannot be paired off with sets like the one you gave. Because sets like the natural numbers can be put into a one-to-one correspondence with a proper subset of themselves, they cannot be put into such a correspondence with sets, like those you gave, which cannot be put into that correspondence with a proper subset of themselves. Only infinite sets can do this.

You don't seem to understand the issue. You have stated that the cardinality of the set of naturals between 1 and 100 is 100, and that the cardinality of the naturals between 1 and 200 is 200. So I can conclude that the difference between these two cardinalities is 100.

You have also stated that you call the cardinality of the complete set of natural numbers, "aleph-null". If it is true that you know precisely and clearly the cardinality of the complete set of natural numbers, then you ought to be able to tell me the difference between the cardinality of 100 and aleph-null, as well as the difference between the cardinality of 200 and aleph-null, and also show how the difference of 100 which exists between the cardinalities of 100 and 200, is reflected in the difference between the difference between the cardinality of 100 and aleph-null, and the difference between the cardinality of 200 and aleph-null.

Otherwise I will conclude that you were not telling the truth when you asserted that you know precisely the cardinality of the so-called "set" of natural numbers, and also I will conclude that this so-called "set" is not well defined in any mathematical sense, and so is not a "set" at all, under our definition of the term.

'Your "conceptual mapping" of a finite past was a semi-infinite number line. You say you cannot think of a corresponding "conceptual mapping" for an infinite past? Really?'

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. Rather, each future day just has the potential for eventually existing. The mapping of days to a number line is a real time process: the present moves to a new day every 24 hours. Each future day is a future present. At no point will we reach a point in time that is infinitely far into the future from today: each individual future day is a finite distance from the present. What is infinite is that this temporal process is unending. The future procession of time is a journey without end.

Contrast this with the past. 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.

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.

And as I said, I don't care if it's a set according to your definition. Mathematicians don't use your definitions of these terms. They use the ones they stipulate, so that's what I'm obviously going to go with.

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

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.

And as I said, I don't care if it's a set according to your definition. — MindForged

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".

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

You aren't making sense. I just told you the difference. I already walked your through the informal proof, but not once have you actually acknowledged it. 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.

1 - 0
2 - 2
3 - 4
etc

And so subtracting a finite number of elements from an infinite set won't change the cardinality. What relationship are you looking for? Any finite number will be.lesser than the cardinality of any infinite set, so subtraction here won't do anything.

Seriously, you either are terrible at making your point or you are more concerned about this at an ideological level and reflexive reactions than one of philosophy.

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

You did no such thing. You claimed it was ill defined. I showed the informal proof of it being an infinite set (the one-to-one correspondence argument) and you couldn't even address it. There is no known contradiction related to these infinities in modern mathematics. Anyone claiming there "clearly are" contradictions just don't know anything. Goodbye.

What point is it you imagine that you're making? And to be sure, if the ball is small enough, then you hold the infinite paths in your hand. Further, stop to think. There are numbers and quantities manifest in the universe so large they can only be "conceptualized," but they are also just large, ordinary integers. Do the things and objects they quantify not exist because you cannot collect them? What nonsense! I personally do not mind ignorance; we're all ignorant; I'm ignorant. Stupid is when you know better and insist on your ignorance. Most people try not to be stupid. Stop being stupid and learn!

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.

I am trying to show that there is a distinction between abstractions and the ontic objects of the real world. 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.

Those large numbers and quantities of things that manifest in the universe are countable: if we can conceive of one number (i) we can conceive of each number that follows (i+1). Infinity is not a number, in that sense. Each natural number can be reached by successive addition; infinity cannot be reached. Transfinites have mathematical properties just as do groups, rings and fields in abstract algebra, so having mathematical relations does not imply they have a referrent in the real world.

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

That is what is nonsense. There is no such thing as "the size of the natural numbers", unless the natural numbers are not infinite.. If the natural numbers are infinite, they are boundless and therefore cannot have a "size". To say that the natural numbers are infinite, and also that they have a size of aleph-null is just contradiction, because an infinite thing is boundless and cannot have a size. If you assign a size to something you do not consider it to be infinite, (hence the term "transfinite", instead of "infinite"), because to say that a boundless (infinite) thing has a size is contradiction.

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

All you have demonstrated is that a so-called infinite set cannot have a definite cardinality. Instead of proving the reality of an infinite set, what this demonstrates is that "infinite set" is self-contradictory.

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

Sorry. I must have you confused with MU. If your point is that you can have one hundred marshmallows, or a million or a billion, but you cannot actually have an infinite number of marshmallows, then I have no problem, but why would you bother? And apparently you'll concede (10^100,000,000) marshmallows because they "are countable." Countable? Perhaps conceivably, theoretically, or in principle, but not otherwise.

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

Why do they have to be taken to be real? If they're not taken are they not real? How many ways to your mom's house? Do the ways you have not yet taken not exist? Of the paths on the sphere-like object, all of them are real. Any could be taken - what matters it if some are not?

If I have a ruler, are you going to tell me that some, or any, of the possible measurements between zero and one aren't available because not real or don't "have a referent in the real world"?

Whatever it is you're getting at, unless it is the presumed impossibility of infinity marshmallows, seems fatally elusive. It seems on the one hand you can't have that many things, but you step from that to not being able to have the number itself. Or, wait, you acknowledge

there are only infinitely many possible paths — Relativist

. Each is a possible path. It's the "taken" you object to? But whenever was a clock attached to a number?

I'll give you credit for understanding that "infinity" usually refers to some transfinite cardinal.

Before we close this discussion MindForged, remember the reason why I first engaged you on this thread. It was this statement:

The definition of infinity is pretty clear, it's extremely useful in mathematics and science, and it introduces no contradictions into the theorems. — MindForged

I didn't agree with your claim that the definition of infinity in mathematics is clear and unambiguous. So consider this quote from Wikipedia:

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?

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". But the transfinite cardinals and transfinites ordinals are indisputably numbers. Infinity is well understood in mathematics. Infinite sets have a cardinal number which is infinite, but that has nothing to do with what's referred to in limits.

So yes, I think this has run its course for me. Infinity in that other sense essentially means "larger than anything else you have", not necessarily a specific number.

Do you still believe that there is one clear definition of "infinity" in mathematics? — Metaphysician Undercover

"Infinity," like "existential," is a word with multiple meanings and applications. But that does not mean that a given meaning is indefinite or unclear. It does mean that the word ought to be defined in its context. 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?

And mathematicians that I've read are uniform in saying that infinity itself is not itself a number.

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.

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

Neither does the past, whether finite or infinite, according to the A theory of time, which you brought up for no apparent reason. The A theory of time is a red herring; this metaphysical position is irrelevant to the argument that you are trying to make, which is:

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

We've been over this already: this is the same question-begging argument that you made at the beginning of the discussion. 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).

Look, you don't have an argument here; you are just stating and restating your conclusion in slightly different ways. You aren't the first to fight this hopeless fight, of course: the a priori denial of actual infinities is as old as Aristotle; Kant tried to make an argument very similar to yours, and others have followed in his step, including most recently theologian W. L. Craig, who employs a raft of such arguments as part of his Kalam cosmological argument for the existence of God. But nowadays these arguments do not enjoy much support among philosophers (see for instance Popper's critique, if you can get it, or any number of more recent articles).

As for physicists and cosmologists, to whom you have appealed as well, most don't even take such a priori arguments seriously, though a few have condescended to offer a critique (such as the late great John Bell, back in 1979, responding to the same article as Popper above). As far as cosmologists are concerned, the question is undeniably empirical, and at this point entirely open-ended; see, for example, this brief survey and the following comments from Luke Barnes (who is somewhat sympathetic to your conclusion).

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

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.

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". Varying definitions inevitably lead to contradiction, like my example of the difference between the way that classical mathematics treats the zero and the negative integers, and the way that "imaginary numbers" treats zero and the negative integers. 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.

"Infinity," like "existential," is a word with multiple meanings and applications. — tim wood

I first engaged MindForged on this thread because I objected to the claim that "infinite" has "a clear" definition in mathematics. It now seems like we're all in agreement, that it does not. "Infinite" is like "zero", there are various different conventions in mathematics which give these terms different meaning. The result, I argue, is contradiction within mathematics.

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

Seems you haven't read my posts. The context we're referring to has been stipulated, "mathematics". If the word has varying definitions within the same context, mathematics, then there's a problem with that discipline.

And mathematicians that I've read are uniform in saying that infinity itself is not itself a number. — tim wood

Try telling this to MIndForged, who steadfastly insisted that infinite is a quantity. I agree that many mathematicians would say that infinite is not a number, but MindForged argued set theory in which infinite sets are allowed to have a cardinality. As such, an infinity has a number, a "transfinite" number.

As such, an infinity has a number, a "transfinite" number. — Metaphysician Undercover

A failure to define. Lots of infinities are represented by transfinite cardinals. Infinity in itself doesn't. But you know all this perfectly well.

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

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. Wikipedia is a poor source.

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

In geometry, lines are continuums which are captured in set theory. There is a clear definition of infinity(ies), you just don't seem to get that meaning here is context sensitive. "Boundless" as a definition of infinity is patently stupid because plenty of infinities are bounded. The set of real numbers between 0 and 1 are indisputably infinite, and yet it is bounded between 0 and 1. That's just an obvious reason why that definition won't work. Colloquial definitions are inherently vague and only make sense in certain contexts. Real mathematics is not one of those places these sloppy definitions will work in.

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

I'm sorry, this is not only a misrepresentation of what was said before but is thoroughly ridiculous given it requires pretending words don't intentionally change meaning in the appropriate context. Waste of time.

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

You can say that's a somewhat careless use of the equality sign. The clean way to do it would be something like lim(1 + 1/2 + 1/4 + 1/8 + ...) = 2.

On the other hand, if you think that 1 + 1/2 + 1/4 + 1/8 + ... ~ 2 that means you are fine with infinite numbers.

The clean way to do it would be something like lim(1 + 1/2 + 1/4 + 1/8 + ...) = 2. — Magnus Anderson

To be more precise, because the use of ellipsis can sometimes create ambiguity:
$\sum_{j=0}^\infty 2^{-j}=2$
or, for the benefit of those that have an allergic reaction to the mention of infinity:
$\forall \varepsilon\gt0,\ \exists N\in\mathbb N\textrm{ such that }(n\ge N)\Rightarrow \left(\left|\sum_{j=0}^n 2^{-j} - 2\right|\lt\varepsilon\right)$
which doesn't mention infinity at all.

Somebody above (I forget who) said mathematics is not clear about what 'infinite' or 'infinity' means.

In a sense that's true. The symbol:
$\infty$
which is usually read out loud as 'infinity', has different meanings in different contexts. For example it means something different in the expression:
$\sum_{j=1}^\infty 2^{-j}$
from what it means in the statement:
$\lim_{x\to 0}\frac1{x^2} = \infty$

Both meanings, in context, are very precise and formal. But they need the context to know which meaning is intended.

The word 'infinite' means something different from 'infinity', as one'd expect since the former is an adjective and the latter is a noun.

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).

There are two completely different definitions of 'infinite' when used as a property of a set:

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.

As I recall, it is a common exercise in introductory courses in topology or set theory to show that these two definitions are logically equivalent. The Axiom of Choice may or may not be required. I do not recall.

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

So the contradiction remains unresolved.

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

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.

It is only by removing "well-defined" from the mathematical context, and defining the set by a quality (things with the same property for example) rather than by a quantity, that one can say an infinite set is "well-defined". But that is a category error, as mathematical objects are not defined by qualities. For example, the mathematical difference between a circle and a square, is found in the definitions of lengths, angles etc.. That a circle has a curved line rather than the straight lines, of a square, follows as a consequence, a conclusion from the definition. Even the simple "line" is not defined as "straight", or some such quality, it is defined by points and dimension, which are not qualities. A mathematical definition cannot be based in a quality without being unsound.

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

In each of these cases, the thing referred to as an infinite set, ought to be dismissed as not a real set, by failing the criteria of being "well-defined". In the first, cardinality is determined by a bijection with the natural numbers. Without bijection cardinality is indeterminate, the so called "infinite set" is not well-defined, so there is no set. In the second case, the bijection is never complete. The assertion that it is any sort of complete "bijection" is only supported by hidden, undisclosed principles, and is therefore not "well-defined".

Infinity is a Mathematical fiction and should be applied carefully to the World of Physical Things. For example we can say that there are an Infinite number of Natural Numbers. Natural Numbers are Mathematical concepts. But there can not be an infinitely large Pencil in the Universe. A good old fashioned Pencil is made out of a core of Lead or graphite (lets just say Lead). surrounded by a tube of Wood and then a coat of Paint. Take a point exactly in the center of the Lead and then let the Pencil grow in size to Infinity. You will have a Universe that is completely filled with Lead. You can never get to the Wood no matter how far you travel away from the center point (assuming you can travel through Lead). There will be no Wood or Paint in this Universe. The Pencil will become something less than it was when it becomes Infinite. You can not really have an Infinite Pencil.

I think the mathematicians have the definition of Point wrong:

“That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume or any other dimensional attribute’

https://en.m.wikipedia.org/wiki/Point_(geometry)

If a point has no length it does not exist so the definition is contradictory.

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.

Relativist: Consider the future: it doesn't exist.
Sophisticat: Neither does the past, whether finite or infinite, according to the A theory of time, which you brought up for no apparent reason. The A theory of time is a red herring; this metaphysical position is irrelevant to the argument that you are trying to make.

You are free to disagree with my conclusion, since it's not a deductive proof. It's just an explanation as to why I personally consider it more likely the past is finite. A-theory is a critical assumption because under B-theory, all points in time have identical ontological properties. In A-theory, past, present, and future are ontologically distinct. Tomorrow and yesterday have in common the fact that neither exists, but yesterday has the distinction that it actually DID exist. In general, causation is not a transitive relation: A causes B does not entail B causes A. Yesterday caused today, not vice versa - so the relation to the past is different from the relation to 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 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.

You can decide these distinctions are irrelevant, but you cannot claim the distinction isn't there if A-therory is true.

I think the mathematicians have the definition of Point wrong: — Devans99

I'm sorry, wasn't it you that asked not to be called stupid? Who knows, you may be right. Mathematicians since, I dunno, Euclid, have made the same mistake. I'm glad you're here to clear this all up. I can't waithuntil you take on triangles. All real men know triangles have seven sides.

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

This is the difference between an intelligible object and a sensible object. The intelligible object is apprehended directly by the intellect, while the sensible object is perceived by the senses. They both exist. So a non-dimensional point, like other mathematical objects, does not need spatial dimension to exist.

I believe the question you were asking in this thread is whether "actually infinite" is a valid intellectual object. If "actually infinite" were proven to be impossible by way of contradiction, or some other logical proof, we'd be obliged to dismiss it as unintelligible, and therefore not a valid object.

If "actually infinite" were proven to be impossible by way of contradiction, or some other logical proof, — Metaphysician Undercover

- there is a quantity X such that X > all other quantities
- But X+1>X
- Reductio ad absurdum, the actually infinite is not a quantity

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.

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.

But even if we were to try to use the definition you suggest, it would be incorrect to say that infinite sets are not well-defined. In mathematics the words 'well-defined' have a very specific meaning, and they only apply to functions, not properties (aka relations). We say that a function is 'well-defined' if, using the definition to apply it to an element of its domain, there is a unique object that is the image of that element under that function. The notion of being a set, or of having finite cardinality, is a property, not a function, so the notion of 'well-defined' is not relevant.

If you really dislike the concept of infinity, all you need do is reject the 'Axiom of Infinity', which asserts the existence of a set that can be thought of as the set of natural numbers. Without such an axiom, we can have natural numbers as large as we wish, but there is no such thing as the set of all natural numbers. Such an approach to mathematics is consistent, and some people try to limit themselves to that. The trouble is that it is that axiom that gives us the tool of Proof by Mathematical Induction. Without it, there is an enormous volume of important results that we would not be able to us.